REPRESENTATIONS OF INFINITE-DIMENSIONAL...

REPRESENTATIONS OF

INFINITE-DIMENSIONAL

LIE ALGEBRAS

Benjamin J. Wilson

Thesis submitted for the degree of

Doctor of Philosophy

Under the cotutelle scheme of the

School of Mathematics and Statistics

University of Sydney, Australia

and the

Instituto de Matematica e Estatıstica

Universidade de Sao Paulo, Brasil

October 2007

Preface

This thesis chronicles three studies in the representation theory of infinite-dimensional

Lie algebras. The first work concerns the imaginary highest-weight theory of a!ne sl(2).

Futorny [14] describes the structure of all but one of the universal objects of the theory,

the imaginary Verma modules. If reducible, an imaginary Verma module V(!) possesses

an infinite descending series of submodules such that the quotient of any submodule by

its successor is isomorphic to a certain module M(!) that depends only upon the highest

weight !. The infinitely recurrent factor M(!) is reducible precisely when ! = 0. The

structure of M(0) is apparently complicated, and not at all understood. The principal

result of the first work is a classification of the irreducible quotients of the submodules

of M(0). This classification complements the result of Futorny to provide a structural

description of the imaginary Verma modules. One may define, for any function on the

integers with values in the field, an irreducible module of level zero for a!ne sl(2):

" : Z ! C defines N(") irreducible module for a!ne sl(2).

The irreducible quotients of the submodules of M(0) are precisely the modules N(")

where " is a linear combination of exponential functions with coe!cients that are inte-

gral, even, and negative. The classification follows from the construction of a family of

singular vectors, and from a description of M(0) in terms of the symmetric functions.

The class of so-called exponential-polynomial modules, which is the class of those mod-

ules N(") defined by an exponential-polynomial function ", therefore contains all the

irreducible quotients of the submodules of M(0). An exponential-polynomial function

" is a function that may be written as a sum in which each summand is a product of

an exponential and a polynomial function. Equivalently, a function " is exponential-

polynomial precisely when the sequence of its values

· · · "("2), "("1), "(0), "(1), "(2), · · ·

solves some linear homogeneous recurrence relation with constant coe!cients. The val-

ues of the function " provide the structure constants of the module N("), and so the

i

ii PREFACE

exponential-polynomial modules may be thought of as modules with structure constants

of a limited complexity. The isomorphism classes of the exponential-polynomial mod-

ules parameterise the isomorphism classes of the modules N(") whose representatives

have only finite-dimensional weight spaces [3] [12]. The open problem of describing the

multiplicities of the weight spaces of an exponential-polynomial module may be resolved

through a study of the highest-weight representations of truncations of the loop algebra.

Any proper quotient of a loop algebra is isomorphic to a truncation of the form

g = g # C[t]/tN+1C[t],

where N is some non-negative integer. In the second work a highest-weight theory for

the truncation g is developed when the underlying Lie algebra g possesses a triangular

decomposition. The principal result is a reducibility criterion for the Verma modules

of g when g is a symmetrisable Kac-Moody Lie algebra, the Heisenberg algebra, or the

Virasoro algebra. This is achieved through a study of the Shapovalov form.

The third work employs the highest-weight theory of truncations of the loop algebra to

describe the multiplicities of the weight spaces of an exponential-polynomial module in

the case where g = sl(2). An exponential-polynomial module N(") may be realized as an

irreducible constituent of a loop module built from an irreducible highest-weight module

L(") for a truncation of the loop algebra. This realization, which is due to Chari and

Pressley [9], expresses the multiplicities of N(") in terms of the multiplicities of L(") via

a certain action of a finite cyclic group. In the particular case of g = sl(2), an expression

for the formal character1 of a module L(") may be deduced from the aforementioned

reducibility criterion for Verma modules of g. The third work develops a theory of semi-

invariants for finite cyclic groups and employs this expression to solve the multiplicity

problem for exponential-polynomial modules.

I am indebted to my supervisors, Alexander Molev at the University of Sydney, and

Vyacheslav Futorny at the Universidade de Sao Paulo, for their academic guidance. I

would like to extend further thanks to Vyacheslav for facilitating my transition to Brazil,

and for his friendship and support during a challenging period. I am very grateful to

Yuly Billig, under whose direction the second work was completed, for his counsel and

encouragement.

1That is, the generating function defined by the multiplicities of the weight spaces.

PREFACE iii

I am fortunate to be part of a wonderful family that has provided unwaivering support

throughout this most extended of adventures, and I am profoundly thankful for their

understanding. A whimsical undergraduate foray into mathematics has a"orded my

passion and profession! Finally, I would like to thank my friends, in whose company the

triumphs and tribulations of this period were marked.

The content of this thesis is original work of which I am the sole author. The use of

existing works is explicitly and duly acknowledged in the text.

Benjamin J. Wilson

Sao Paulo, October 11, 2007.

Contents

Preface i

Chapter 1. Recapitulation 1

1. Exponential-Polynomial Modules and Truncated Current Lie Algebras 1

2. Imaginary Highest-Weight Representation Theory 5

3. Highest-Weight Theory for Truncated Current Lie Algebras 9

4. Characters of Exponential-Polynomial Modules 12

Chapter 2. Imaginary Highest-Weight Representation Theory 17

1. The Canonical Quotient M(0) 17

2. Symmetric Function Realisation 19

3. Singular Vectors 22

4. Irreducible Quotients of M(n) 26

5. Irreducible Subquotients of M(0) 29

Chapter 3. Lie Algebras with Triangular Decomposition 31

1. Lie Algebras with Triangular Decomposition 31

2. Highest-Weight Representation Theory 35

Chapter 4. Highest-Weight Theory for Truncated Current Lie Algebras 43

1. Truncated Current Lie Algebras 43

2. Decomposition of the Shapovalov Form 49

3. Values of the Shapovalov Form 54

4. Reducibility of Verma Modules 63

4.A. Characters of Irreducible Highest-Weight Modules 68

4.B. Imaginary Highest-Weight Theory for Truncated Current Lie Algebras 69

Chapter 5. Characters of Exponential-Polynomial Modules 73

1. Preliminaries 73

2. Loop-Module Realisation of N(") 77

3. Semi-invariants of Actions of Finite Cyclic Groups 80

4. Exponential-Polynomial Modules 84

v

vi CONTENTS

Index of Symbols 91

Bibliography 95

CHAPTER 1

Recapitulation

Section 1 presents the preliminary results on exponential-polynomial modules and on

truncated current Lie algebras that are necessary to describe the results of the thesis in

Sections 2 – 4. The material of Section 1 is principally derived from Billig and Zhao [3],

Chari [6], and Chari and Pressley [9].

1. Exponential-Polynomial Modules and Truncated Current Lie Algebras

The loop-module realisation motivates a study of an exponential-polynomial module via

an irreducible highest-weight representation of a truncated current Lie algebra. In this

section, the exponential-polynomial modules and their corresponding truncated current

Lie algebras are constructed, and the realisation is described. Let k denote a field of

characteristic zero. Denote by g = spank { e,h, f } the three-dimensional Lie algebra sl(2)

with the commutation relations

[ e, f ] = h, [ h, e ] = 2e, [ h, f ] = "2f,

and triangular decomposition

(1.1) g = g+ $ h $ g!, e % g+, h % h, f % g!.

For any Lie algebra a over k, denote by

a = a # k[t, t!1]

the Z-graded loop algebra associated to a, with the Lie bracket

[x # ti, y # tj ] = [x, y ] # ti+j , x, y % a, i, j % Z.

An a-module M is Z-graded if M =!

n"ZMn and

a # tm · Mn & Mm+n, m, n % Z.

The decomposition (1.1) defines a decomposition of g

g = g+ $ h $ g!

1

2 1. RECAPITULATION

as a direct sum of Z-graded subalgebras. We consider g-modules M that are h-diagonal-

isable, i.e.

M ="

!"h!

M! where h|M! = #(h), h % h.

A Z-graded g-module M may be decomposed

M ="

(!,n)"s(M)

M!n , M!

n = M! ' Mn,

as a direct sum of homogeneous components M!n , where

s(M) = { (#, n) % h# ( Z | M!n )= 0 } .

The functional ! % h# given by !(h) = 2 is the positive root of g. The category O,

introduced by Chari [6], consists of those Z-graded g-modules M such that

s(M) &#

""A(!" Z+!) ( Z

for some finite subset A = AM & h#. The morphisms of the category are the homo-

morphisms of Z-graded g-modules. For any Lie algebra a, a map $ : M ! N is a

homomorphism of Z-graded a-modules M and N if $ is a a-module homomorphism and

if there exists k % Z such that $(Mn) & Nn+k, for all n % Z.

1.1. Exponential-polynomial modules. Denote by F the vector space of func-

tions " : Z ! k. For any " % F , the rule

(1.2) " : h # tm *! "(m)tm, m % Z,

defines a homomorphism " : U(h) ! k[t, t!1] of Z-graded associative algebras, where the

grading of U(h) is defined by that of h. Write H(") for im " considered as a Z-graded

h-module via ". Let

F $ = {" % F | im " = k[tr, t!r] for some r > 0 } .

For " % F $, write deg" = r for the degree of ", where im " = k[tr, t!r]. The set F $

consists of those functions whose support is not wholly contained in N or "N. The

module H(") is irreducible and of dimension greater than one precisely when " % F $.

For " % F , consider H(") as a (h $ g+)-module via g+ · H(") = 0. Whenever H(") is

irreducible, the induced g-module

(1.3) Indg

h%g+H(")

1. EXPONENTIAL-POLYNOMIAL MODULES AND TRUNCATED CURRENT LIE ALGEBRAS 3

has a unique irreducible Z-graded quotient, which we denote by N("). It is shown in

[6] that, over an algebraically closed field, any irreducible object of the category O is of

the form N("). For any ! % k&, define the exponential function

exp(!) : Z ! k, exp(!)(m) = !m, m % Z.

A function " % F is exponential polynomial if it can be written as a finite sum of products

of polynomial and exponential functions, i.e.

(1.4)$

""k"

""exp(!),

for some polynomial functions "" % F and distinct scalars ! % k&. Write

E = {" % F | " is exponential polynomial} .

Then E \ { 0 } & F $ & F . The exponential-polynomial functions are those whose succes-

sive values solve a homogeneous linear recurrence relation with constant coe!cients.

A module N(") is exponential polynomial if " % E . It is shown in [3] that the ho-

mogeneous components of an exponential-polynomial module are finite dimensional.

Conversely, for " % F $, the module N(") has finite-dimensional homogeneous com-

ponents only if " % E (cf. [12]). In particular, if k is algebraically closed, then the

exponential-polynomial modules {N(") | " % E } are precisely those irreducible objects

of the category O for which all homogeneous components are finite dimensional.

1.2. Loop modules and truncated current Lie algebras. For " % F , let kv#

be the one-dimensional h-module defined by

h # tm · v# = "(m)v#, m % Z.

Let g+ · v# = 0, and denote by

(1.5) V(") = Indg

h%g+kv#

the induced g-module. This module has a unique irreducible quotient L("). The modules

V(") and L(") are not Z-graded. Suppose that " % E is an exponential-polynomial

function, and write c# % k[t] for the characteristic polynomial1 of the minimal-order

linear homogeneous recurrence relation that is solved by the values of ". It can be

shown that the ideal

g # c#k[t, t!1] & g

1For example, if ! = exp("), then c" = (t ! ").

4 1. RECAPITULATION

acts trivially on the g-module L("). Hence L(") is a module for the “truncation”

(1.6) g(") := g/g # c#k[t, t!1] += g # k[t, t!1]/c#k[t, t!1]

of the loop algebra g. A Lie algebra (1.6) is called a truncated current Lie algebra.

Let M be a g-module. The vector space %M = M&

kk[t, t!1] is a Z-graded g-module

(called a loop module) via

x # a · u # b = (x # a · u) # ab, x % g, u % M, a, b % k[t, t!1].

The Z-grading is defined by degree in the indeterminate t % k[t, t!1]. The loop-module

realisation, due to Chari and Pressley [9], relates the exponential-polynomial modules

to the irreducible highest-weight representations of a truncated current Lie algebra.

Chari and Pressley show that if " % E is non-zero and r = deg", then the Z-graded

g-module 'L(") has precisely r irreducible constituents, all of which are isomorphic to

the exponential-polynomial module N("). Moreover, the weight spaces of N(") are

described in terms of the semi-invariants of an action of the cyclic group Zr on L("). Thus

the exponential-polynomial modules, and in particular their weight-space multiplicities,

may be studied via highest-weight representations of truncated current Lie algebras.

The Chinese Remainder Theorem implies that if " is written in the form (1.4), then

g(") +=!

""k" g(""exp(!))

is an isomorphism of Lie algebras. Remarkably, if k is algebraically closed, then it follows

also that

L(") +=&

""k" L(""exp(!))

is an isomorphism of g-modules (cf. Chapter 5). Therefore, in order to study highest-

weight representations of truncated current Lie algebras, it is su!cient to consider only

truncated current Lie algebras of the form g("), where " = aexp(!) and a % F is a

polynomial function. In any such case,

g(") += g # k[t]/tN+1k[t].

where N = deg a.

2. IMAGINARY HIGHEST-WEIGHT REPRESENTATION THEORY 5

2. Imaginary Highest-Weight Representation Theory

A!ne Lie algebras admit non-classical highest-weight theories through alternative parti-

tions of the root system. Although significant inroads have been made, much of the clas-

sical machinery is inapplicable in this broader context, and some fundamental questions

remain unanswered. In particular, the structure of the reducible objects in non-classical

theories has not yet been fully understood. This question is addressed in this thesis for

a!ne sl(2), which has a unique non-classical highest-weight theory, termed imaginary.

The reducible Verma modules in the imaginary theory possess an infinite descending

series, with all factors isomorphic to a certain canonically associated module, the struc-

ture of which depends upon the highest weight. If the highest weight is non-zero, then

this factor module is irreducible, and conversely. Chapter 2 examines the degeneracy of

the factor module of highest-weight zero. The intricate structure of this module is un-

derstood via a realisation in terms of the symmetric functions. The realisation permits

the description of a family of singular vectors, and the classification of the irreducible

subquotients as a certain subclass of the exponential-polynomial modules.

Let k denote a field of characteristic zero, and let g denote an a!ne Kac-Moody Lie

algebra over k. Write h & g for the Cartan subalgebra and # & h# for the root system.

Denote by g$ the root space associated to any root % % #. So

g = (!

$"! g$) $ h, ad h|g# = %(h), h % h, % % #.

A g-module V is called weight if the action of h upon V is diagonalisable. That is,

V =!

""h! V", h|V$= !(h), h % h, ! % h#.

2.1. Partitions and highest-weight theories. The notion of a highest-weight

module for g depends upon the partition of the root system. A subset P & # is called

a partition of the root system if both

i. P is closed under root space addition, i.e. if %,& % P and % + & % #, then

%+ & % P;

ii. P ' "P = , and P - "P = #.

If #+(') denotes the set of positive roots with respect to some basis ' & # of the

root system, then P = #+(') is an example of a partition. A partition P defines a

6 1. RECAPITULATION

decomposition of the Lie algebra g as a direct sum of subalgebras

g = N! $ h $ N+, where N+ = $$"Pg$, N! = $$"Pg!$.

A weight g-module V is of highest-weight ! % h# with respect to the partition P if there

exists v % V" such that

U(g) · v = V, and N+ · v = 0.

Thus the choice of partition P defines a theory of highest-weight modules. A highest-

weight theory defined by the set of all positive roots P = #+(') with respect to some

basis ' of the root system is called classical. Two partitions are equivalent if they are

conjugate under the action of W ( {±1 }, where W denotes the Weyl group associated

to the root system #. Equivalent partitions define similar highest-weight theories. All

partitions of the root system of a finite-dimensional semisimple complex Lie algebra are

classical, and hence equivalent. In contrast, it has been shown by Jakobsen and Kac [19],

and by Futorny [13], that there are finitely many, but never one, inequivalent partitions

of the root system of an a!ne Lie algebra. Thus any a!ne Lie algebra has multiple

distinct highest-weight theories.

2.2. Imaginary highest-weight theory for a!ne sl(2). Up to equivalence, there

is precisely one non-classical partition, the imaginary partition, of the root system of

a!ne sl(2). The associated imaginary highest-weight theory has been pioneered by Fu-

torny in [14], [15]. These works provide an almost complete understanding of the uni-

versal objects of the theory, the imaginary Verma modules. Let g denote the a!nisation

of g:

g = g $ kc $ kd,

with Lie bracket relations:

[x # tk, y # tl ] = [x, y ] # tk+l + k(k,!l( x | y )c, [ c, g ] = 0,

[ d, x # tk ] = kx # tk, x, y % g, k, l % Z,

where ( · | · ) denotes the Killing form of g. The Cartan subalgebra h & g is given by

h = span {h # t0, c,d } .

Let !, " % h# be such that

!(h # t0) = 2, !(c) = 0, !(d) = 0,

"(h # t0) = 0, "(c) = 0, "(d) = 1.

2. IMAGINARY HIGHEST-WEIGHT REPRESENTATION THEORY 7

Then # = {±! + i" | i % Z } - { i" | i % Z, i )= 0 }. The imaginary partition P & # is

given by

P = {! + i" | i % Z } - { i" | i % Z, i > 0 } .

Thus the associated subalgebras N+,N! of g are given by

N+ = g+ $ [!

j>0 h # tj ], N! = g! $ [!

j<0 h # tj ].

Let ! % h#, and consider the one-dimensional vector space kv" as an (h $ N+)-module

via

N+ · v" = 0, h · v" = !(h)v", h % h.

Let V(!) denote the induced g-module:

V(!) = Indg

h%N+kv".

The g-module V(!) is the universal highest-weight g-module of highest-weight !, and

so is called an imaginary Verma module.

Theorem. [14] Let ! % h#. Then:

i. If !(c) )= 0, then V(!) is irreducible.

ii. Suppose that !(c) = 0. Then V(!) has an infinite descending series of submod-

ules

V(!) = V 0 . V 1 . V 2 . · · ·

such that any factor V i/V i+1, i ! 0, is isomorphic to the quotient of g-modules

M(!) = V(!) / /h # tj · v" | j < 0 0,

up to a shift in the " weight-decomposition. Moreover, if !(h # t0) )= 0, then

M(!) is irreducible.

The value !(d) of the action of d on the generating vector is immaterial to the structure

of the imaginary Verma module V(!). Hence the theorem above provides an almost

complete description of the structure of the imaginary Verma modules for g, lacking

only a statement about the imaginary Verma module of highest-weight zero. Part (ii) of

the Theorem motivates a study of V(0) through its canonically associated and infinitely

occurrent quotient M(0). Chapter 2 is an extensive study of the degeneracy of M(0).

The central element c acts trivially on the module M(0). Hence M(0) may be studied

as a Z-graded module for the loop algebra g. The g-module M(0) may be defined by

(2.1) M(0) = Indg

h%g+ku0,

8 1. RECAPITULATION

where ku0 is the trivial one-dimensional (h$ g+)-module. The module M(0) is Z-graded

by construction, and

s(M(0)) & ("Z+!) ( Z.

Thus M(0) % O. In fact, if ! % h# is such that !(c) = 0, and " % F is given by

"(m) = !(h # t0)(m,0, m % Z,

then the induced module defined by (1.3) and the canonical quotient M(!) are isomor-

phic as Z-graded g-modules.

Throughout the remainder of this section, all modules are Z-graded.

2.3. Symmetric functions and singular vectors. It is apparent from the con-

struction (2.1) that

M(0) ="

n"Z+

M(n) where M(n) = M(0)!n!, n ! 0,

is a decomposition of M(0) as a direct sum of Z-graded h-modules. The h-modules M(n)

have remarkable realisations in terms of the symmetric functions, and play a large part

in the structural description of the g-module M(0). For any positive integer n, let

An = k[z1, z!11 , . . . , zn, z!1

n ]Sym(n)

denote the k-algebra of symmetric Laurent polynomials in the n indeterminates z1, . . . , zn.

The module M(n) may be considered as a graded An-module in such a way that the ac-

tion of h upon M(n) factors through an epimorphism of algebras U(h) " An. Therefore,

it is su!cient to consider the module M(n) as a graded An-module. In fact, it may be

shown that as a graded An-module, M(n) is isomorphic to the graded regular module

for An. Thus, in particular, the h-module structure of M(n) may be understood via the

graded algebra An. The classification of the irreducible quotients of the algebra An,

and the construction of singular vectors of M(0), together permit the classification of

the irreducible subquotients of the g-module M(0).

An element v % M(0) is singular if g+ · v = 0. Non-trivial singular vectors generate

proper submodules. For any positive integer n, let

$n =(

1!i<j!n

(zi " zj)2 % An.

3. HIGHEST-WEIGHT THEORY FOR TRUNCATED CURRENT LIE ALGEBRAS 9

In the theory of symmetric functions, the function $n appears both as the square of the

Vandermonde determinant, and as the discriminant function for degree-n polynomials.

Let sgn()) = ±1 denote the sign of a permutation ).

Theorem. Let n be a positive integer. Then

i. All elements of the h-submodule $n · M(n) are singular.

ii. The space $n ·M(n) is spanned by singular vectors of the form

w(#) =$

%"Sym(n)

sgn())(

1!i!n

f # t!i+%(i) u0,

where # % Zn and u0 denotes the generator of M(0).

Conjecture. Let n be a positive integer, and suppose that v % M(n) is singular. Then

v % $n ·M(n).

2.4. Structure of the canonical quotient M(0). A g-module Q is a subquotient

of a g-module M if there exists a chain of g-modules

M . N . P,

such that N/P += Q. The preceding results may be employed to classify the irreducible

subquotients of the g-module M(0). Let

E(!) = {" % E | "" % "2Z+ for all ! % k& } ,

in the notation of (1.4).

Theorem. For any " % E(!), the g-module N(") is an irreducible subquotient of M(0).

Moreover, if k is algebraically closed, then any irreducible subquotient of M(0) is of the

form N(") for some " % E(!).

3. Highest-Weight Theory for Truncated Current Lie Algebras

Let g be a Lie algebra over a field k of characteristic zero, and a fix positive integer N.

The Lie algebra

(3.1) g = g #k k[t]/tN+1k[t]

over k, with the Lie bracket

(3.2) [x # ti, y # tj ] = [x, y ] # ti+j, x, y % g, i, j ! 0,

10 1. RECAPITULATION

is a truncated current Lie algebra. In Chapter 4, a highest-weight theory for g is de-

veloped when the underlying Lie algebra g possesses a triangular decomposition. The

principal result is the reducibility criterion for the Verma modules of g for a wide class

of Lie algebras g, including the symmetrisable Kac-Moody Lie algebras, the Heisenberg

algebra, and the Virasoro algebra. This is achieved through a study of the Shapovalov

form. In the particular case of g = sl(2), an expression for the formal character of the

irreducible module L(") may be deduced from the reducibility criterion.

3.1. Truncated current Lie algebras. There have been various studies of trun-

cated current Lie algebras and their representation theory in the particular case where

g is a semisimple finite-dimensional Lie algebra. There are applications in the theory of

soliton equations [5] [23] [26], and in this context the Lie algebra g is called a polyno-

mial Lie algebra. The paper [4] describes a construction of g via the Wigner contraction.

Taki" considered this case with N = 1 in [31], and that work was extended in [28], [16],

[17] without the restriction on N. As such, when g is a semisimple finite-dimensional

Lie algebra, the Lie algebra g is often called a generalised Taki! algebra. The category

of modules for a truncated current Lie algebra is examined in [22].

3.2. Highest-weight modules. We assume that the Lie algebra g is equipped

with a triangular decomposition

(3.3) g = g! $ h $ g+, g± =!

&"!+g±&, #+ & h#.

The fundamental definitions and results concerning Lie algebras with triangular decom-

positions and their highest-weight representation theory are the subject of Chapter 3

(here, h0 = h). The exposition follows that of Moody and Pianzola [24], modified in ac-

cordance with our definitions. The triangular decomposition (3.3) of g naturally defines

a triangular decomposition of g,

g = g! $ h $ g+, g± =!

&"!+g±&,

where the subalgebra h and the subspaces g& are defined in the manner of (3.1), and

h & h is the diagonal subalgebra. Hence a g-module M is weight if the action of h on

M is diagonalisable. A weight g-module is of highest weight if there exists a non-zero

vector v % M , and a functional % % h# such that

g+ · v = 0; U(g) · v = M ; h · v = %(h)v, for all h % h.

The unique functional % % h# is the highest weight of the highest-weight module M .

Notice that the support of a weight module is a subset of h#, while a highest-weight is

3. HIGHEST-WEIGHT THEORY FOR TRUNCATED CURRENT LIE ALGEBRAS 11

an element of h#. A highest-weight % % h# may be thought of as a tuple of functionals

on h,

(3.4) % = (%0,%1, . . . ,%N), where /%i, h0 = /%, h # ti0, h % h, i ! 0.

All g-modules of highest weight % % h# are homomorphic images of a certain universal

g-module of highest weight %, denoted by V(%). These universal modules V(%) are the

Verma modules of the highest-weight theory.

3.3. Reducibility of Verma modules. A single hypothesis su!ces for the deriva-

tion of a criterion for the reducibility of a Verma module V(%) for g in terms of the func-

tional % % h#. We assume that the triangular decomposition of g is non-degenerately

paired, i.e. that for each * % #+, a non-degenerate bilinear form

( · | · )& : g& ( g!& ! k,

and a non-zero element h(*) % h are given, such that

[x, y ] = ( x | y )& h(*),

for all x % g& and y % g!&. The symmetrisable Kac-Moody Lie algebras, the Virasoro

algebra, and the Heisenberg algebra all possess triangular decompositions that are non-

degenerately paired. The reducibility criterion is the following.

Theorem. The Verma module V(%) for g is reducible if and only if

/%,h(*) # tN0 = 0,

for some positive root * % #+ of g.

Notice that the reducibility of V(%) depends only upon %N % h#, the last component of

the tuple (3.4).

3.4. Applications of the Theorem. The criterion described by the Theorem has

many disguises, depending upon the underlying Lie algebra g.

Example 3.5. Let g = sl(3) be the Lie algebra of type A2. The diagonal subalgebra

h is two-dimensional, and the root system # & h# carries the geometry defined by the

Killing form. A Verma module V(%) for g is reducible if and only if %N is orthogonal to a

root. This is precisely when %N belongs to one of the three hyperplanes in h# illustrated

in Figure 1(a). The arrows describe the root system.


(a) g of type A2 (b) g of type G(1)2

Figure 1: Reducibility criterion for the Verma modules of g, where g is of type A2 or

G(1)2

Example 3.6. Let g denote the fourteen-dimensional simple Lie algebra of type G2,

and let

g = g # k[s, s!1] $ kc $ kd,

denote the a!nisation of g with central extension c and degree derivation d. Then g is

the Kac-Moody Lie algebra of type G(1)2 . The diagonal subalgebra

(3.7) h = h $ kc $ kd,

is obtained from the diagonal subalgebra h of g. The Lie algebra g has a triangular

decomposition defined by a choice of simple roots. For any & % h#, denote by )& the

restriction of & to h defined by (3.7). A Verma module V(%) for g is reducible precisely

when *%N belongs to the infinite union of hyperplanes described in Figure 1(b), where

the dashed line segment has length |/%N, c0|.

4. Characters of Exponential-Polynomial Modules

Let g denote the Lie algebra sl(2) over an algebraically closed field k of characteristic

zero, and adopt the notations of Section 1. An expression for the formal character of an

exponential-polynomial module L(") is derived in Chapter 5. As described in Subsection

1.2, the loop-module realisation reduces this task to the study of the semi-invariants of a

4. CHARACTERS OF EXPONENTIAL-POLYNOMIAL MODULES 13

finite cyclic group acting on the irreducible highest-weight module L(") for the truncated

current Lie algebra g(") defined by (1.6). A certain generalisation of Molien’s Theorem

in the case of a cyclic group describes the multiplicities of these semi-invariants. The

character of the exponential-polynomial module N(") is then expressed in terms of the

character of L("), obtained in Chapter 4.

For any " % E , write

charN(") =$

k"0

$

n"Z

dimN(")k,n XkZn % Z+[[X,Z,Z!1]],

for the formal character of N("), where

N(")k,n = N(")(#(0)!k)!n k ! 0, n % Z.

For any positive integer r, define the function

+r =$

'r=1

exp(,) % E ,

where the sum is over all roots of unity , such that ,r = 1. The function +r takes the

constant value r on its support rZ. If " % E is non-zero and deg" = r, then it can be

shown that r > 0 and

(4.1) " = +r ·$

i

aiexp(!i)

for some finite collection of polynomial functions ai % F and scalars !i % k&, such that

if (!i/!j)r = 1, then i = j. The formal character of the exponential-polynomial module

N(") is described by the following theorem.

Theorem. Let " % E be non-zero, and write r = deg". In the notation of (4.1),

(4.2) charN(") =1

r

$

n"Z

$

d|r

cd(n)+P#(Xd)

, rd

Zn,

where the inner sum is over the positive divisors d of r, the quantities cd(n) are Ra-

manujan sums, and

P#(X) =

-ai"Z+

(1 " Xai+1)

(1 " X)M,

where M =.

i(deg ai + 1) and the product is over those indices i for which ai % Z+.

The Ramanujan sum cd(n) is given by

(4.3) cd(n) =#(d)µ(d$)

#(d$), d$ =

d

gcd(d, n),


where # denotes Euler’s totient function and µ denotes the Mobius function. The

expression (4.2) is the Ramanujan-Fourier transform of charN(").

It is apparent that cd(·) is a function of period d, and thus it may be deduced from

formula (4.2) that, for any k ! 0, the multiplicity function

n *! dimN(")k,n, n % Z,

has period r = deg". Therefore the character of N(") is completely described by the

array of weight-space multiplicities [dimN(")k,n] where k ! 0 and 0 # n < r. Examples

of these arrays, such as those illustrated by Figures 2(a) – 2(d), may be computed in a

straightforward manner using the formula (4.2). Columns are indexed left to right by n,

where 0 # n < r, while rows are indexed from top to bottom by k ! 0.

Greenstein [18] (see also [8, Section 4.1]) has derived an explicit formula for the for-

mal character of an integrable irreducible object of the category O. These objects are

precisely the exponential-polynomial modules N(") where " is a linear combination of

exponential functions with non-negative integral coe!cients. Indeed, our result may

alternatively be deduced by considering separately the case where N(") is integrable,

employing the result of Greenstein, and the case where N(") is not integrable, using

Molien’s Theorem. Our approach, via a general study of finite cyclic-group actions, has

the advantage of permitting a unified proof. Both approaches employ the explicit ex-

pression of the formal character of an irreducible highest-weight module for a truncated

current Lie algebra described in Chapter 4.

4. CHARACTERS OF EXPONENTIAL-POLYNOMIAL MODULES 15

1 0 0 0 0 0

1 1 1 1 1 1

3 2 3 2 3 2

4 3 3 4 3 3

3 2 3 2 3 2

1 1 1 1 1 1

1 0 0 0 0 0

0 0 0 0 0 0...

...(a) ! = #6

1 0 0 0

2 2 2 2

10 8 10 8

30 30 30 30

86 80 84 80

198 198 198 198

434 424 434 424

858 858 858 858...

...(b) ! = !#4(exp(") + exp(µ))

1 0 0 0 0 0

1 1 1 1 1 1

4 3 4 3 4 3

10 9 9 10 9 9

22 20 22 20 22 20

42 42 42 42 42 42

80 75 78 76 78 75

132 132 132 132 132 132

217 212 217 212 217 212

335 333 333 335 333 333...

...(c) ! = !#6

1 0

2 2

5 3

6 6

9 7

10 10

13 11

14 14

17 15

18 18...

...(d) ! = #2(exp(") ! exp(µ))

Figure 2: Array of weight-space multiplicites of N(")

CHAPTER 2

Imaginary Highest-Weight Representation Theory

Adopt the notation of Section 2 of Chapter 1. In particular, k denotes any field of char-

acteristic zero, g denotes the Lie algebra sl(2), and g denotes the Z-graded loop algebra

associated to g. In this chapter, all modules are Z-graded, unless stated otherwise. For

any x % g and k % Z, write x(k) = x # tk.

1. The Canonical Quotient M(0)

The following preliminary result provides a description of the action of g upon M(0).

Here, and throughout, the use of a hat above a term in a sum or product indicates the

omission of that term. The subalgebra g! is abelian, and so U(g!) may be identified

with the infinite-rank polynomial ring

k[ f(j) |j % Z ].

Proposition 1.1. The following hold:

i. The g-module M(0) is generated by an element u0 such that the action of U(g!)

on u0 is free, whilst the actions of U(g+) and U(h) are trivial.

ii. The g-module M(0) has a basis

(1.2)/

n"0

{(

1!i!n

f(-i)u0 | -1 # -2 # · · · # -n, - % Zn }

iii. The action of g on M(0) is given by

f(k) ·(

1!i!n

f(-i)u0 = f(k)(

1!i!n

f(-i)u0,

h(k) ·(

1!i!n

f(-i)u0 = "2$

1!i!n

f(-1) · · · 'f(-i) · · · f(-n)f(-i + k)u0,

e(k) ·(

1!i!n

f(-i)u0 = "2$

1!i<j!n

f(-1) · · · 'f(-i) · · · !f(-j) · · · f(-n)f(-i + -j + k)u0,

for all - % Zn, n ! 0 and k % Z.

17

18 2. IMAGINARY HIGHEST-WEIGHT REPRESENTATION THEORY

Proof. Part (i) is clear, and part (ii) follows from part (i). To prove part (iii), we firstly

derive some commutation relations in U(g) before considering them in light of parts (i)

and (ii). If L is any Lie algebra and x % L, then the adjoint map

ad x : U(L) ! U(L), ad x : y *! [x, y ] = xy " yx, y % U(L),

is a derivation of the associative product of U(L). That is,

(1.3) [x,(

1!i!n

yi ] =$

1!i!n

y1 · · · yi!1[x, yi ]yi+1 · · · yn.

This formula yields immediately the commutation equation

[ h(k),(

1!i!n

f(-i) ] = "2$

1!i!n

f(-1) · · · 'f(-i) · · · f(-n)f(-i + k)

for all - % Zn, n ! 0 and k % Z. Using formula (1.3) and substituting the above

commutation equation for h(k),

[ e(k),(

1!i!n

f(-i) ] = "2$

1!i<j!n

f(-1) · · · 'f(-i) · · · !f(-j) · · · f(-r)f(-i + -j + k)

+$

1!i!n

f(-1) · · · 'f(-i) · · · f(-r)h(-i + k)

for all - % Zn, n ! 0 and k % Z. These formulae, in consideration of parts (i) and (ii),

immediately imply the formulae of part (iii), and completely describe the action of g on

M(0). $

Corollary 1.4. The g-module M(0) has a decomposition

M(0) ="

n"Z+

M(n),

as a direct sum of modules for h, where

M(n) = span {(

1!i!n

f(#i)u0 | # % Zn } ,

and M(n) = M(0)!n! for any n ! 0.

Remark 1.5. It follows from Corollary 1.4 above, and Proposition 3.6 of [20], that the

only integrable subquotient of M(0) is the trivial one-dimensional g-module.

2. SYMMETRIC FUNCTION REALISATION 19

2. Symmetric Function Realisation

This section presents a realisation of the h-module M(n) as the graded regular mod-

ule of the symmetric Laurent polynomials in n variables. The realisation allows the

classification of the irreducible quotients of the h-module M(n) outlined in Section 4.

Fix a positive integer n. The k-algebra An is Z-graded by total degree. The elementary

symmetric functions $i % An, 1 # i # n, are defined by the polynomial equation

(2.1)n(

i=1

(1 + zit) = 1 +n$

i=1

$i(z1, . . . , zn)ti.

Notice that $n = z1 · · · zn is invertible in An. For any k % Z, let

p(k) = zk1 + · · · + zk

n % An,

denote the sum of k-powers of the indeterminates, and for any - % Zn, write

m(-) =1

n!

$

%"Sym(n)

(

1!i!n

z(i

%(i) % An.

The symmetric polynomial m(-) may alternatively be defined by

m(-) =1

n!

$

%"Sym(n)

(

1!i!n

z(%(i)

i .

The set {m(-) | - % Zn } spans the k-algebra An of symmetric Laurent polynomials.

Lemma 2.2. Let n > 0 and -,# % Zn. Then

m(-) · m(#) =1

n!

$

)"Sym(n)

m(- + #) ),

where #) % Zn is given by (#) )i = #)(i), for all 1 # i # n and . % Sym(n).


Proof. Let -,# % Zn. Then

m(-) ·m(#) =0 1

n!

12$

%"Sym(n)

$

)"Sym(n)

(

1!i!n

z(%(i)+!&(i)

i

=0 1

n!

12 $

%"Sym(n)

$

)"Sym(n)

(

1!i!n

z(%(i)+!(&#%)(i)

i

(substituting . 1 ) for .)

=0 1

n!

12 $

%"Sym(n)

$

)"Sym(n)

(

1!i!n

z((+!& )%(i)

i

(since #!!" = (#! )")

=1

n!

$

)"Sym(n)

m(- + #) ) $

Proposition 2.3. For any positive integer n:

i. The k-algebra An is generated by the set { $i | 1 # i # n } - { $n!1 }.

ii. The k-algebra An is generated by the set of power sums {p(k) | k % Z }.

Proof. Let

A+n = k[z1, . . . , zn]Sym(n), A!

n = k[z!11 , . . . , z!1

n ]Sym(n).

Then

(2.4) An =$

k"0

$!kn ·A+

n ,

and, in particular,

(2.5) An = A!n · A+

n .

Part (i) follows from the Fundamental Theorem of Symmetric Functions and equation

(2.4), while part (ii) follows from the Newton-Girard formulae and equation (2.5). $

Proposition 2.6. For any n > 0, M(n) is a Z-graded An-module via linear extension

of

(2.7) m(-) ·(

1!i!n

f(#i)u0 =1

n!

$

%"Sym(n)

(

1!i!n

f(#i + -%(i))u0, -,# % Zn,

with the Z-grading defined by

deg(

1!i!n

f(#i)u0 =$

1!i!n

#i, # % Zn.

2. SYMMETRIC FUNCTION REALISATION 21

Proof. For any /, -,# % Zn,

m(/) ·0m(-) ·

(

1!i!n

f(#i)u01

=0 1

n!

12 $

%"Sym(n)

$

)"Sym(n)

(

1!i!n

f(#i + /%(i) + -)(i))u0

=1

n!

$

)"Sym(n)

1

n!

$

%"Sym(n)

(

1!i!n

f(#i + (/+ -) )%(i))u0

(substituting . 1 ) for .)

=+ 1

n!

$

)"Sym(n)

m(/+ -) ),·

(

1!i!n

f(#i)u0

=0m(/)m(-)

1·

(

1!i!n

f(#i)u0,

by Lemma 2.2. As the polynomials m(-) span An, linear extension of (2.7) endows

M(n) with the structure of a Z-graded An-module. $

Theorem 2.8. Let n > 0. The action of U(h) on M(n) factors through an epimorphism

' : U(h) " An

of graded algebras defined by

' : h(k) *! "2p(k), k % Z.

That is, if 0 and 1 denote the representations of U(h) and An on M(n), respectively,

then the following diagram commutes:

U(h)"

!! !!

*""

!

!

!

!

!

!

!

!

!

An

+

##

EndM(n)

Proof. The map ' is an algebra epimorphism by Proposition 2.3 part (ii). Let k % Z,

and let 2k = (k, 0, . . . , 0) % Zn. Then

p(k) = nm(2k).

It follows that, for any # % Zn,

p(k) ·(

1!i!n

f(#i)u0 =$

1!i!n

f(#1) · · · !f(#i) · · · f(#n)f(#i + k)u0

and so h(k)|M(n) = '(h(k))|M(n) by Proposition 1.1 part (iii). $


Therefore, for n > 0, it is su!cient to consider M(n) as a Z-graded An-module. Write

Aregn for the regular Z-graded An-module, i.e. for An considered as a Z-graded An-

module under multiplication.

Theorem 2.9. For any n > 0, the map

( : M(n) ! Aregn

defined by linear extension of

( :(

1!i!n

f(#i)u0 *! m(#), # % Zn,

is an isomorphism of Z-graded An-modules.

Proof. The map ( is a bijection, by Proposition 1.1 part (ii). Let -,# % Zn. Then

(+m(-) ·

(

1!i!n

f(#i)u0

,= (

+ 1

n!

$

%"Sym(n)

(

1!i!n

f(#i + -%(i))u0

,

=1

n!

$

%"Sym(n)

m(#+ -%)

= m(-) · m(#) (by Lemma 2.2)

= m(-) · (+ (

1!i!n

f(#i)u0

,.

Hence ( is an isomorphism of Z-graded An-modules. $

Corollary 2.10. Let n ! 0 and let v % M(n) be non-zero and homogeneous. Then

U(h)v and M(n) are isomorphic as h-modules.

Proof. For n = 0, the statement is trivial. For n > 0, it is su!cient to employ Theorem

2.9, and observe that the ring An is an integral domain. $

3. Singular Vectors

The existence of non-zero singular vectors is related to the degeneracy of the module

M(0):

Proposition 3.1. Let n ! 0. Then

i. The set of all singular vectors in M(n) forms an h-submodule.

3. SINGULAR VECTORS 23

ii. If v % M(n) is non-zero and singular, and V = U(g)v, then V has decomposition

V =!

m"n V !m!, V !m! & M(m),

into non-trivial eigenspaces for h(0).

Proof. The set of all singular vectors in M(n) clearly forms a vector space. Now suppose

that v % M(n) is singular. Then

e(k)(h(l)v) = "2e(k + l)v + h(l)e(k)v = 0,

for any k, l % Z. Thus if v is singular, then so is h(l)v, for any l % Z, and so the set

of singular vectors in M(n) forms an h-module, proving part (i). For part (ii), suppose

again that v % M(n) is a non-zero singular vector, and let V = U(g)v. Then

V = U(g) · v = U(g!) # U(h) · v &!

m"n M(m),

since U(g) = U(g!) # U(h) # U(g+). $

Theorem 3.2. For any n > 0 and # % Zn,

w(#) =$

%"Sym(n)

sgn())(

1!i!n

f(#i + )(i))u0 % M(n)

is a singular vector.

Proof. Singularity may be demonstrated directly by applying the formula for the action

of e(k), k % Z, of Proposition 1.1 part (iii). $

Lemma 3.3. For any n > 0, the symmetric function $n is equal to$

%,)"Sym(n)

sgn() 1 .)(

1!i!n

z%(i)+)(i)!2i

up to a change in sign.

Proof. Let )n =.

%"Sym(n) sgn())-

1!i!n z%(i)i . It is not di!cult to show that

)n|zi=zj = 0, 1 # i < j # n,

and that )n|zi=0 = 0 for all 1 # i # n. Hence )n is equal up to sign to(

1!i!n

zi ·(

1!i<j!n

(zi " zj),

by degree considerations. Therefore $n and(

1!i!n

z!2i · )2

n


are equal up to sign, from which the result follows. $

Lemma 3.4. Suppose that g % An and that g|zi=zj = 0. Then (zi " zj)2 divides g in

k[z±11 , . . . , z±1

n ].

Proof. Let )ji denote the ring automorphism of k[z±1

1 , . . . , z±1n ] that interchanges the

variables zi and zj and leaves all other variables invariant. It is easy to convince oneself

that

()ji h)|zi=zj = h|zi=zj , h % k[z±1

1 , . . . , z±1n ].

Now g|zi=zj = 0, and so g = (zi"zj)h, for some h % k[z±11 , . . . , z±1

n ]. Moreover, )ji h = "h,

since )ji g = g. Therefore

h|zi=zj = (")ji h)|zi=zj = "(()j

i h)|zi=zj ) = "(h|zi=zj ),

and so h|zi=zj = 0. Hence (zi " zj) divides h also. $

Theorem 3.5. For any n > 0,

$n · M(n) = span {w(#) | # % Zn } ,

and hence all elements of the non-zero h-submodule $n · M(n) are singular.

Proof. Fix n > 0, and let

Wn = span {w(#) | # % Zn } .

The symmetric function realisation of Theorem 2.9 may be used to demonstrate the

inclusion Wn & $n · M(n). Let # % Zn. Then

((w(#)) =1

n!

$

)"Sym(n)

$

%"Sym(n)

sgn())(

1!i!n

z!&(i)+%()(i))i

=1

n!

$

)"Sym(n)

sgn(.)$

%"Sym(n)

sgn())(

1!i!n

z!&(i)+%(i)i

(substituting ) 1 ."1 for ))

=1

n!

$

)"Sym(n)

sgn(.)F) ,

where, for any . % Sym(n),

F) =$

%"Sym(n)

sgn())(

1!i!n

z!&(i)+%(i)i .

It is not di!cult to verify that

F) |zi=zj = 0, 1 # i < j # n, . % Sym(n).

3. SINGULAR VECTORS 25

Therefore, for all 1 # i < j # n, ((w(#)) is divisible by (zi " zj)2 in k[z±11 , . . . , z±1

n ], by

Lemma 3.4. Thus

((w(#)) = $nh for some h % k[z±11 , . . . , z±1

n ]

since the factors (zi " zj)2 are pairwise co-prime. In fact, h % An, since both ((w(#))

and $n are symmetric. Therefore, ((w(#)) % $n ·An, and hence Wn & $n ·M(n). Now

let %n = (-

1!i!n z2i ) · $n. The factor

-1!i!n z2

i is invertible in An, and so

$n ·M(n) = %n ·M(n),

by Theorem 2.9. In particular, by Corollary 1.4,

$n · M(n) = span {%n ·(

1!i!n

f(#i)u0 | # % Zn } .

By Lemma 3.3, the polynomial %n is equal up to sign to

$

%,)"Sym(n)

sgn() 1 .)(

1!i!n

z%(i)+)(i)i .

Therefore, for any # % Zn,

%n ·(

1!i!n

f(#i)u0

=1

n!

$

+"Sym(n)

$

%,)"Sym(n)

sgn() 1 .)(

1!i!n

f(#i + () 1 1)(i) + (. 1 1)(i))u0

=1

n!

$

+"Sym(n)

$

%,)"Sym(n)

sgn() 1 .)(

1!i!n

f(#i + )(i) + .(i))u0

(substituting ) 1 1"1 for ) and . 1 1"1 for .)

=$

%"Sym(n)

sgn())$

)"Sym(n)

sgn(.)(

1!i!n

f(#i + )(i) + .(i))u0

=$

%"Sym(n)

sgn())w(#())),

where #()) % Zn is given by

#())i = #i + )(i), 1 # i # n,

for all ) % Sym(n). Hence $n · M(n) & Wn. $

Conjecture 3.6. Let n > 0, and suppose that v % M(n) is singular. Then v % $n ·M(n).


4. Irreducible Quotients of M(n)

For any Z-graded k-algebra B, write B(k), k % Z, for the graded components of B.

Proposition 4.1. Let n be a positive integer, and let B be a graded simple quotient of

the graded algebra An. Then B = F[tm, t!m] for some positive divisor m of n and finite

algebraic field extension F of k.

Proof. As $n = z1 · · · zn is invertible in An, it must be that B(n) )= 0. Let m be the

minimal positive integer such that B(m) )= 0, and let u % B(m) be non-zero. Then u is

invertible, since B is simple, and so multiplication by uk is a vector-space automorphism

of B such that

B(l) ! B(km+l), l % Z.

In particular,

(4.2) B(km) = uk · B(0), k % Z.

Suppose that B(l) )= 0, for some l % Z, and let q, r be the unique integers such that

l = qm + r, 0 # r < m.

Then

0 )= u!qm · B(l) = B(r),

and so r = 0 by the minimality of m. Hence

B =!

k"ZB(km),

and in particular m must be a divisor of n. Moreover, by (4.2),

B += B(0) #k k[tk, t!k]

via uk *! tkm, k % Z. As An is finitely generated, by Proposition 2.3 part (i), so is the

k-algebra A(0)n . Hence B(0) is a finite algebraic field extension of k (see, for example, [1],

Proposition 7.9). $

Proposition 4.3. Let n be a positive integer, and suppose that , : An ! k is a non-zero

algebra homomorphism. Then there exist scalars *1, . . . ,*n, all non-zero and algebraic

over k, such that

,(p(k)) =n$

i=1

*ki , k % Z.

4. IRREDUCIBLE QUOTIENTS OF M(n) 27

Proof. Suppose that , : An ! k is a non-zero homomorphism, and let

g(t) = 1 +n$

i=1

,($i)ti % k[t].

As $n = z1 · · · zn is invertible in An, it must be that ,($n) )= 0. Let *1, . . .*n be some

iteration of the scalars defined by

g(t) =n(

i=1

(1 + *it).

Then by equation (2.1),

(4.4) ,($i) = $i(*1, . . . ,*n), 1 # i # n.

The *i are necessarily non-zero since

*1 · · ·*n = $n(*1, . . . ,*n) = ,($n) )= 0.

By Proposition 2.3 part (i), there is a unique algebra homomorphism with the property

(4.4), namely the restriction of the evaluation map

zi *! *i, 1 # i # n.

In particular, ,(p(k)) =.n

i=1 *ki , for all k % Z. $

Recall that E(!) & E is given by

E(!) = {" % E | "" % "2Z+ for all ! % k& } .

For any n ! 0, let

E(!n) = {" % E(!) |$

""k"

"" = "2n } ,

so that E(!) =2

n"0 E(!n).

Theorem 4.5. For any n ! 0 and " % E(!n), the h-module H(") is an irreducible

quotient of the h-module M(n). Moreover, if k is algebraically closed, then any irreducible

quotient of the h-module M(n) is of the form H(") for some " % E(!n).

Proof. The statement is trivial for n = 0, so suppose that n is a positive integer. Let

" % E(!n), and let *1, . . .*n % k& be non-zero scalars such that

"(k) = "2n$

i=1

*ki , k % Z.

Define

3 = 3&1,...,&n : k[z1, z!11 , . . . , zn, z!1

n ] ! k[t, t!1]


by extension of zi *! *it, 1 # i # n. Then

3|An : An ! im3|An & k[t, t!1]

is a homomorphism of graded algebras, and so im3|An may be considered as an An-

module, and as a quotient of the regular An-module Aregn . Therefore im3|An is a quotient

of the An-module M(n), by Theorem 2.9. Now im3|An is an h-module via the map

' : U(h) ! An of Theorem 2.8. For any k % Z,

(3|An 1')(h(k)) = "23|An(p(k))

= "2n$

i=1

*ki tk

= "(h(k)),

where " : U(h) ! k[t, t!1] is defined by (1.2), page 2. Hence im3|An+= H(") as h-

modules, and so H(") is a quotient of the h-module M(n). It is not too di!cult to verify

that H(") is an irreducible h-module for any " % E(!).

Now suppose that k is algebraically closed, and that & is an irreducible quotient of the

h-module M(n). By Theorems 2.8 and 2.9, & is an An-module, and a quotient of Aregn .

Hence there exists a simple quotient B of the graded algebra An

3 : An " B,

such that B += & as An-modules, when B is considered as an An-module via the algebra

epimorphism 3. Moreover, by Proposition 4.1, B = k[tm, t!m] for some positive divisor

m of n. Let , : An ! k be given by

3(x) = ,(x)tdeg x,

for all homogeneous x % An. Then , is a non-zero algebra homomorphism, and so by

Proposition 4.3 there exist non-zero scalars *1, . . . ,*n % k& such that

3(p(k)) = ,(p(k))tk =n$

i=1

*ki tk, k % Z.

Therefore & += H("), where

" : Z ! k, "(k) = "2n$

i=1

*ki , k % Z,

by Theorem 2.8. $

5. IRREDUCIBLE SUBQUOTIENTS OF M(0) 29

5. Irreducible Subquotients of M(0)

An h-module & is weight if h(0) acts by a scalar h(0)|# on &. For any weight h-module

&, write V (&) for the induced g-module

V (&) = Indg

h%g+&, where g+ · & = 0.

Proposition 5.1. Let & be a weight h-module. Then the g-module V (&) has a unique

maximal submodule that has trivial intersection with &.

Proof. Let ! = h(0)|#. Then

V (&) =!

n"0 V (&)("!n)!,

and V (&)" = &. If N & V (&) is a g-submodule that has trivial intersection with &, then

N &!

n>0 V (&)("!n)!.

Hence the same is true of the sum of all such g-submodules. This sum is itself a g-

submodule, and its maximality and uniqueness follow from construction. $

For any h-module &, denote by L (&) the quotient of V (&) by its unique maximal

submodule that has trivial intersection with &. Hence L (&) is an irreducible g-module

if & is an irreducible h-module. In particular, if " % F $, then the h-module H(") is

irreducible, and so L (H(")) = N(").

Theorem 5.2. For any " % E(!), the g-module N(") is an irreducible subquotient of

M(0). Moreover, if k is algebraically closed, then any irreducible subquotient of M(0)

is of the form N(") for some " % E(!).

Proof. Let n be a positive integer, and let " % E(!n). Theorem 3.5 guarantees the

existence of a non-zero singular vector v % M(n). The h-module U(h)v contains only

singular vectors, and is isomorphic to M(n), by Corollary 2.10 and Proposition 3.1. Let

P = U(g)v. By the Poincare-Birkho"-Witt Theorem,

U(g) · v = U(g!) # U(h) # U(g+) · v

= U(g!) # U(h) · v.

Therefore P = U(g!)P!n! where P!n! = U(h)v += M(n). Hence, by Theorem 4.5, there

is an epimorphism of h-modules

P!n! " H("),


which extends to an epimorphism of g-modules

P " N(").

Thus N(") is an irreducible subquotient of M(0).

Now suppose that k is algebraically closed, and that N is an irreducible subquotient of

M(0). The support of N is a subset of the support "Z+! of M(0). Let n denote the

minimal non-negative integer such that N!n! )= 0. Then g+ · N!n! = 0. Thus there is

an epimorphism of g-modules N " L (N!n!), and since N is irreducible, this map is

an isomorphism. Therefore, the weight space N!n! is an irreducible h-module. Indeed,

a proper h-submodule of N!n! generates a proper g-submodule of L (N!n!) += N . Now

let P $ & P be g-submodules of M(0) such that N = P/P $. Then N!l! = P!l!/P $!l!,

for all l ! 0, and in particular N!n! is a subquotient of the h-module M(n). Therefore,

by Corollary 2.10 and Theorem 4.5 there exists " % E(!n) such that N!n! += H(") as

h-modules. Thus there is an isomorphism of g-modules

N += L (N!n!) += L (H(")) = N("),

which completes the proof of the Theorem. $

The following Corollary is immediate from [3] and the inclusion E(!) & E .

Corollary 5.3. Suppose that k is algebraically closed. Then the homogeneous compo-

nents of any irreducible subquotient of M(0) have finite dimension.

CHAPTER 3

Lie Algebras with Triangular Decomposition

This chapter develops the technology necessary for the study of the highest-weight theory

of truncated current Lie algebras undertaken in Chapter 4. The notion of a Lie alge-

bra with triangular decomposition is introduced, and several examples are considered.

Fundamental results in the highest-weight representation theory are then described,

concluding with a proof of Shapovalov’s Lemma. The content of this chapter is entirely

derivative of the book of Moody and Pianzola [24]. Let k denote a field of characteristic

zero.

1. Lie Algebras with Triangular Decomposition

Let g be a Lie algebra over the field k. A triangular decomposition of g is specified by a

pair of non-zero abelian subalgebras h0 & h, a pair of distinguished non-zero subalgebras

g+, g!, and an anti-involution (i.e. an anti-automorphism of order 2)

4 : g ! g,

such that:

i. g = g! $ h $ g+;

ii. the subalgebra g+ is a non-zero weight module for h0 under the adjoint action,

with weights #+ all non-zero;

iii. 4|h = idh and 4(g+) = g!;

iv. the semigroup with identity Q+, generated by #+ under addition, is freely

generated by a finite subset {*j }j"J & Q+ consisting of linearly independent

elements of h#0.

This definition is a modification of the definition of Moody and Pianzola [24]. There,

the set J is not required to be finite, root spaces may be infinite-dimensional, and h0 = h.

We distinguish between h0 and h in order to include Example 1.6.

31

32 3. LIE ALGEBRAS WITH TRIANGULAR DECOMPOSITION

Write Q =.

j"J Z*j. Call the weights #+ of the h0-module g+ the positive roots, and the

weight space g& corresponding to * % #+ the *-root space, so that g+ = $&"!+g&. The

anti-involution ensures an analogous decomposition of g! = $&"!$g&, where #! = "#+

(the negative roots) and g!& = 4(g&) for all * % #+. Write # = #+ -#! for the roots

of g. Consider Q+ to be partially ordered in the usual manner, i.e. for -, -$ % Q+,

- #Q+ -$ 23 (-$ " -) % Q+.

We assume that all root spaces are finite-dimensional, and that #+ is a countable set.

For clarity, a Lie algebra with triangular decomposition may be referred to as a five-tuple

(g, h0, h, g+,4).

Example 1.1. Let g be a finite-dimensional semisimple Lie algebra over C, with Cartan

subalgebra h and root system #. Then

g = h $ (!

&"! g&).

Let ' be a basis for #, and let Q+ be the additive semigroup generated by '. Write

#+ = # 'Q+, and let g+, g! be given by

g+ =!

&"!+g&, g! =

!&"!$

g&,

where #! = "#+. Then g+ is a weight-module for h0 = h with weights #+, and

g = g! $ h $ g+.

All root spaces are one-dimensional. For any * % #+, choose non-zero elements

x(*) % g&, y(*) % g!&.

An anti-involution 4 on g is defined by extension of

4|h = idh, 4(x(*)) = y(*), 4(y(*)) = x(*), * % '.

Thus (g, g+, h, h,4) is a Lie algebra with triangular decomposition. The semisimple

finite-dimensional Lie algebras over C are parameterised by Euclidean root systems, or

equivalently by the Cartan matrices. The Serre relations permit the construction of any

such Lie algebra from its Cartan matrix, and this construction works over an arbitrary

field k of characteristic zero. The preceding assertions hold also for the Lie algebras over

k constructed in this manner. Here and throughout, semisimple finite-dimensional Lie

algebra means a Lie algebra over k defined by a Cartan matrix and the Serre relations.

Example 1.2. It shall be convenient to consider the following particular case of Example

1.1 in greater detail. Let g denote sl(3), the finite-dimensional semisimple Lie algebra

over k with root system A2. Denote by !1, !2 the simple roots, by

x(!1), x(!2), y(!1), y(!2), h(!1), h(!2)

1. LIE ALGEBRAS WITH TRIANGULAR DECOMPOSITION 33

the Chevalley generators, and by h = kh(!1) $ kh(!2) the Cartan subalgebra, so that

!1(h(!1)) = !2(h(!2)) = 2, !1(h(!2)) = !2(h(!1)) = "1.

Then the root system is defined by # = #+ - #!, where #+ = {!1,!2,!1 + !2 } and

#! = "#+. Write

x(!1 + !2) = [ x(!1), x(!2) ], y(!1 + !2) = [ y(!2), y(!1) ],

h(!1 + !2) = h(!1) + h(!2).

Then for each * % #+, the elements x(*), y(*),h(*) span a subalgebra of g isomorphic

to sl(2). The anti-involution 4 fixes h point-wise, and interchanges x(*) with y(*) for

every * % #+. Write

g& = kx(*), g!& = ky(*), * % #+,

and g± = $&"!+g±&. Then g+ is a weight-module for h0 = h with weights #+, and

g = g! $ h $ g+.

The semigroup Q+ is generated by ' = {!1,!2 }. Note that the h(*) defined here are

only proportional to the elements h(*) defined later on.

Example 1.3. Let g be the Kac-Moody Lie algebra over k associated to an n ( n

generalised Cartan matrix (we paraphrase [20]). Let h denote the Cartan subalgebra,

and # the root system. Then

g = h $ (!

&"! g&),

and all root spaces are finite-dimensional. The collection * of simple roots is a linearly-

independent subset of the finite-dimensional space h#. Let Q+ denote the additive

semigroup generated by *, let #+ = # 'Q+, and write

g+ =!

&"!+g&, g! =

!&"!$

g&,

where #! = "#+. Then g+ is a weight-module for h0 = h with weights #+, and

g = g! $ h $ g+.

If ei, fi, 1 # i # n, denote the Chevalley generators of g, then g+ and g! are the

subalgebras generated by the ei and by the fi, respectively. An anti-involution 4 of g is

defined by extension of

4|h = idh, 4(ei) = fi, 4(fi) = ei, 1 # i # n,

(this 4 di"ers from the 4 of [20]). Thus (g, g+, h, h,4) is a Lie algebra with triangular

decomposition.


Example 1.4. Let g denote the k-vector space with basis the symbols

{Lm | m % Z } - { c } ,

endowed with the Lie bracket given by

[ c, g ] = 0, [ Lm,Ln ] = (m " n)Lm+n + (m,!n&(m)c, m, n % Z,

where & : Z ! k is any function satisfying &("m) = "&(m) for m % Z, and

&(m + n) = 2m+nn!m &(n) + m+2n

m!n &(m), m, n % Z, m )= n.

If & = 0, then the symbols Lm span a copy of the Witt algebra. The Virasoro alge-

bra is the only non-split one-dimensional central extension of the Witt algebra, up to

isomorphism [21], and is typically defined with &(m) = m3!m12 . Let

g± =!

m>0 kL±m, h0 = h = kL0 $ kc,

and let " % h# be given by

"(L0) = "1, "(c) = 0.

Then g = g! $ h $ g+, and g+ is a weight module for h0 = h, with weights

#+ = {m" | m > 0 } .

The semigroup Q+ is generated by ". An anti-involution 4 is given by

4(c) = c, 4(Lm) = L!m, m % Z,

and in this notation g is a Lie algebra with triangular decomposition.

Example 1.5. Let a denote the k-vector space with basis the symbols

{ am | m % Z } - {", d } ,

endowed with the Lie bracket given by

[ am, an ] = m(m,!n", [", a ] = 0, [ d, am ] = mam, m, n % Z.

The Lie algebra a is called the extended Heisenberg or oscillator algebra. Let

a± =!

m>0 ka±m, h = ka0 $ k" $ kd,

and let " % h# be given by

"(a0) = "(") = 0, "(d) = 1.

Then a = a! $ h $ a+, and a+ is a weight module for h0 = h, with weights

#+ = {m" | m > 0 } .

2. HIGHEST-WEIGHT REPRESENTATION THEORY 35

The semigroup Q+ is generated by ". An anti-involution 4 is given by

4(") = ", 4(d) = d, 4(am) = a!m, m % Z,

and in this notation a is a Lie algebra with triangular decomposition.

Example 1.6. Let g be a k-Lie algebra with triangular decomposition, denoted as

above, and let R be a commutative, associative k-algebra with 1 (e.g. R = k[t]/tN+1k[t],

N > 0). Write g = g #k R, and similarly for the subalgebras of g. Then g is a k-Lie

algebra with Lie bracket

[x # r, y # s ] = [x, y ] # rs, x, y % g, r, s % R,

and contains g as a subalgebra via x *! x # 1. Moreover, g = g! $ h $ g+, and h0 & h

are non-zero abelian subalgebras of g. The subalgebra g+ is a weight module for h0 with

weights coincident with the weights #+ of the h0-module g+, and (g+)& = (g&+). So g

and g share the same roots # and root lattices Q, Q+. The anti-involution 4 of g is

given by R-linear extension

4 : x # r *! 4(x) # r, x % g, r % R,

and fixes h point-wise. Thus (g, h0, h, g+,4) is a k-Lie algebra with triangular decompo-

sition.

2. Highest-Weight Representation Theory

Throughout this section, let (g, h0, h, g+,4) denote a Lie algebra with triangular decom-

position. The universal highest-weight modules of g, called Verma modules, exist and

possess the usual properties. An extensive treatment of Verma modules and the Shapo-

valov form can be found in [24]; we present only the definitions and the most important

properties.

2.1. Highest-weight modules. A g-module M is weight if the action of h0 on M

is diagonalisable, i.e.

(2.1) M =!

!"h!0M!, h|M! = #(h) for all h % h0,# % h#0.

The decomposition (2.1) is called the weight-space decomposition of M ; the components

M! are called weight spaces. The support of a weight module M is the set

{# % h#0 | M! )= 0 } & h#0.


For any # % h#0, an element v % M! is a primitive vector of M if the submodule

U(g) · v & M is proper. Clearly M is reducible if and only if M has a non-zero primitive

vector. A non-zero vector v % M is a highest-weight vector if

i. g+ · v = 0;

ii. there exists % % h# such that h · v = %(h)v, for all h % h.

The unique functional % % h# is called the highest weight of the highest-weight vector

v. A weight g-module M is called highest weight (of highest weight %) if there exists a

highest-weight vector v % M (of highest weight %) that generates it.

Proposition 2.2. Suppose that M is a highest-weight g-module, generated by a highest-

weight vector v % M of highest weight % % h#. Then

i. the support of M is contained in %|h0 "Q+;

ii. M$|h0 = kv, and all weight spaces of M are finite-dimensional;

iii. M is indecomposable, and has a unique maximal submodule;

iv. if u % M is a highest-weight vector of highest-weight %$ % h#, and u generates

M , then %$ = % and u is proportional to v.

Let % % h#, and consider the one-dimensional vector space kv$ as an (h $ g+)-module

via

g+ · v$ = 0; h · v$ = %(h)v$, h % h.

The induced module

V(%) = U(g) #U(h%g+) kv$

is called the Verma module of highest-weight %.

Proposition 2.3. For any % % h#,

i. Up to scalar multiplication, there is a unique epimorphism from V(%) to any

highest-weight module of highest-weight %, i.e. V(%) is the universal highest-

weight module of highest-weight %;

ii. V(%) is a free rank one U(g!)-module.

2.2. The Shapovalov Form. The Shapovalov form is a contragredient symmetric

bilinear form on U(g) with values in U(h) = S(h). The evaluation of the Shapovalov form

at % % h# is a k-valued bilinear form, and is degenerate if and only if the Verma module


V(%) is reducible. By the Leibniz rule, U(g) is a weight g-module, with weight-space

decomposition

U(g) =!

("Q U(g)( .

The anti-involution 4 of g extends uniquely to an anti-involution of U(g) (denoted iden-

tically), and is such that

4 : U(g)( ! U(g)!( , - % Q.

It follows from the Poincare-Birkho"-Witt (PBW) Theorem that U(g) may be decom-

posed

U(g) = U(h) $ {g!U(g) + U(g)g+}

as a direct sum of vector spaces. Further, both summands are two-sided U(h)-modules

preserved by 4. Let q : U(g) ! U(h) denote the projection onto the first summand

parallel to the second; the restriction q|U(g)0 is an algebra homomorphism. Define

F : U(g) ( U(g) ! U(h) via F(x, y) = q(4(x)y), x, y % U(g).

The bilinear form F is called the Shapovalov form; we consider its restriction

F : U(g!) ( U(g!) ! U(h).

Distinct h0-weight spaces of U(g!) are orthogonal with respect to F, and so the study

of F on U(g!) reduces to the study of the restrictions

F! : U(g!)!! ( U(g!)!! ! U(h), # % Q+.

Any % % h# extends uniquely to a map U(h) ! k; write F!(%) for the composition

of F! with this extension, and write RadF!(%) for its radical. The importance of the

Shapovalov form stems from the following fact.

Proposition 2.4. Let # % Q+, % % h#. Then RadF!(%) & V(%)$|h0!! is the %|h0 " #

weight space of the maximal submodule of the Verma module V(%).

In particular, a Verma module V(%) is irreducible if and only if the forms F!(%) are

non-degenerate for every # % Q+. Thus an understanding of the forms F!, # % Q+, is an

understanding of the irreducibility criterion of the Verma modules of the highest-weight

theory.


2.3. Partitions and the Poincare-Birkho"-Witt Monomials. Let C be a set

parameterizing a root-basis (i.e. an h0-weight basis) of g+, via

C 4 - 5 x(-) % g+.

Define # : C ! #+ by declaring x(-) % g!(()+ , for all - % C. A partition is a finite

multiset with elements from C; write P for the set of all partitions. Set notation is used

for multisets throughout. The length |!| of a partition ! % P is the number of elements

of !, counting all repetition. Fix some ordering of the basis { x(-) | - % C } of g+; for

any ! % P, let

(2.5) x(!) = x(!1) · · · x(!k) % U(g+)

where k = |!| and (!i)1!i!k is an enumeration of the entries of ! such that (2.5) is

a PBW monomial with respect to the basis ordering. For any partition ! % P, write

y(!) = 4(x(!)). By the PBW Theorem, the spaces U(g+), U(g!) have bases

{ x(!) | ! % P } , { y(!) | ! % P } ,

respectively. For any partition ! % P and positive root * % #+, write

#(!) =$

(""

#(-); !& = { - % ! | #(-) = * } .

2.4. Shapovalov’s Lemma. The proof of the following useful lemma is elementary.

Lemma 2.6. Suppose that ! % P, that |!| = r, that (!i)1!i!r is an enumeration of !

and that . % Sym(r). Then

x(!1) · · · x(!r) = x(!)(1)) · · · x(!)(r)) + R

where R is a linear combination of terms x(%1) · · · x(%s) where %i % C for 1 # i # s and

s < r.

The following Lemma is due to Shapovalov [30]. Our proof follows that of an analogous

statement in [24].

Lemma 2.7. Let (g, h0, h, g+,4) be a Lie algebra with triangular decomposition. Sup-

pose that !, µ % P, that |!| = r and |µ| = s, and that (!i)1!i!r and (µi)1!i!s are

arbitrary enumerations of ! and µ, respectively. Let

Z = x(!r) · · · x(!1)y(µ1) · · · y(µs).

Then


i. deghq(Z) # r, s;

ii. if r = s, but |!&| )= |µ&| for some * % #+, then deghq(Z) < r = s;

iii. if r = s and |!&| = |µ&| =: m& for all * % #+, then the degree r = s term of

q(Z) is(

&"!+

$

)"Sym(m')

(

1!j!m'

[ x(!&)(j)), y(µ&

j ) ],

where for each * % #+, (!&j )1!j!m' , (µ&

j )1!j!m' are any fixed enumerations of

!& and µ& respectively.

Proof. The proof is by induction on |!| + |µ|. It is straightforward to show that all

three parts hold whenever |!| = 0 or |µ| = 0. Suppose then that all three parts hold for

all !$, µ$ % P such that |!$| + |µ$| < |!| + |µ|. Let 5 % Sym(r), . % Sym(s), and write

Z $ = x(!,(r)) · · · x(!,(1))y(µ)(1)) · · · y(µ)(s)).

By Lemma 2.6, Z = Z $ + R, where R is a linear combination of terms

x(%r%) · · · x(%1)y(&1) · · · y(&s%)

with r$ < r or s$ < s. Therefore, by inductive hypothesis, the Lemma holds for arbitrary

enumerations of ! and µ, if it holds for any particular pair of enumerations. Consider

#+ to carry some linearisation of its usual partial order, and choose any enumerations

of !, µ such that

#(!1) # · · · # #(!r) and #(µ1) # · · · # #(µs).

Moreover, as

q(x(!r) · · · x(!1)y(µ1) · · · y(µs)) = q(4(x(!r) · · · x(!1)y(µ1) · · · y(µs)) )

= q(x(µs) · · · x(µ1)y(!1) · · · y(!r)),

it may supposed without loss of generality that #(µ1) # #(!1). Now

q(Z) = q(x(!r) · · · x(!1)y(µ1) · · · y(µs))

= q([ x(!r) · · · x(!1), y(µ1) ]y(µ2) · · · y(µs))

=r$

i=1

q(Ai),

where, by the Leibniz rule,

Ai = x(!r) · · · x(!i+1)[ x(!i), y(µ1) ]x(!i!1) · · · x(!1)y(µ2) · · · y(µs),


for 1 # i # r. Let 0 # k # r be maximal such that #(!i) = #(µ1) for all 1 # i # k; the

terms Ai with 1 # i # k and k < i # r are to be considered separately. If 1 # i # k,

then [ x(!i), y(µ1) ] % h. Therefore, by the Leibniz rule,

Ai = Zi[ x(!i), y(µ1) ] + Ri,

where Zi = x(!r) · · · x(!i+1)x(!i!1) · · · x(!1)y(µ2) · · · y(µs) and Ri is a linear combina-

tion of terms

x(%r!1) · · · x(%1)y(&1) · · · y(&s!1), %1, . . . ,%r!1,&1, . . . ,&s!1 % C.

If instead k < i # r, then Ai is a linear combination of terms

x(!r) · · · x(!i+1)x(-)x(!i!1) · · · x(!1)y(µ2) · · · y(µs),

where #(-) = #(!i) " #(µ1) % #+, since #(µ1) # #(!1) # #(!i).

Note that deghq(Z) # max {deghq(Ai) }. Consider now each of the three parts of the

claim.

Part (i). For 1 # i # k,

(2.8) q(Ai) = q(Zi)[ x(!i), y(µ1) ] + q(Ri),

since q|U(g)0 is an algebra homomorphism. By part (i) of the inductive hypothesis,

deghq(Zi), deghq(Ri) # r " 1, s " 1,

and so deghq(Ai) # r, s. For k < i # r, again by part (i) of the inductive hypothesis,

deghq(Ai) # r, s " 1. Hence deghq(Z) # r, s, and so part (i) holds.

Part (ii). Suppose that r = s, and let * % #+ be such that |!&| )= |µ&|. For 1 # i # k,

consider q(Ai) by equation (2.8). By part (i) of the inductive hypothesis,

deghq(Ri) # r " 1 < r,

and so it remains only to consider q(Zi). Write !$ (respectively, µ$) for the parti-

tion consisting of the components of ! (respectively, µ) except for !i (respectively, µ1).

Then |!$| = |µ$| and |!$&| )= |µ$&|. Therefore, by part (ii) of the inductive hypothesis,

deghq(Zi) < r " 1; hence deghq(Ai) < r. For k < i # r, part (i) of the inductive

hypothesis implies that deghq(Ai) # s " 1 < r. Therefore deghq(Z) < r, as required.

Part (iii). Suppose that r = s and that |!&| = |µ&| for all * % #+. Observe that for

k < i # r, part (i) of the inductive hypothesis implies that

deghq(Ai) # s " 1 < r;


and that for 1 # i # k, by the same,

deghq(Ri) # r " 1 < r.

Therefore, the terms q(Ai) for k < i # r and the terms q(Ri) for 1 6 i 6 k can not

contribute to the degree-r component of q(Z); thus the degree-r component of q(Z) is

the degree-r component ofk$

i=1

q(Zi)[ x(!i), y(µ1) ].

As Zi satisfies the conditions of part (iii) of the inductive hypothesis, for 1 # i # k, the

formula follows. $

CHAPTER 4

Highest-Weight Theory for Truncated Current Lie

Algebras

In this chapter, the highest-weight theory for truncated current Lie algebras is extensively

studied, culminating in a reducibility criterion for the Verma modules. References to

material from Chapter 3 are distinguished by the specification of a page number in

parentheses. The notations of that chapter are used throughout. In particular, k is any

field of characteristic zero.

1. Truncated Current Lie Algebras

Let (g, h0, h, g+,4) be a Lie algebra with triangular decomposition, and let C denote a

set parameterizing a root-basis for g+. Fix a positive integer N, and let

g = g # k[t]/tN+1k[t]

denote the associated truncated current Lie algebra with the triangular decomposi-

tion of Example 1.6 (page 35). The integer N is the nilpotency index of g. Let

C = C ( { 0, . . . ,N }. Then C parameterises a basis for g+ consisting of h0-weight vectors

of homogeneous degree in t, via

C 4 - 5 x(-) % g+,

where x(-) = x(.) # td if - = (., d) % C. Define

# : C ! #+, degt : C ! { 0, . . . ,N }

via x(-) % g!(() # tdegt(() for all - % C. Order the basis { x(-) | - % C } of g+ by fixing

an arbitrary linearisation of the partial order by increasing homogeneous degree in t, i.e.

so that

degt(-) < degt(-$) 3 x(-) < x(-$), -, -$ % C.

As per Subsection 2.3 (page 38), the PBW basis monomials of U(g+) with respect to

this ordered basis are parameterised by a collection P of partitions. Partitions here are

43

44 4. HIGHEST-WEIGHT THEORY FOR TRUNCATED CURRENT LIE ALGEBRAS

(finite) multisets with elements from C. For any # % Q+, let

P! = {! % P | #(!) = # } .

For any 0 # d # N and ! % P, define

!d = { - % ! | degt- = d } ;

! is homogeneous of degree-d in t if ! = !d. The ordering of the basis of g+ is such that

for all ! % P,

x(!) = x(!0)x(!1) · · · x(!N), y(!) = y(!N) · · · y(!1)y(!0).

For any % % h# and 0 # d # N, let %d % h# be given by

/%d, h0 = /%, h # td0, h % h.

1.1. The Shapovalov form. As in Section 2 (page 35), there is a decomposition

U(g) = U(h) $ {g!U(g) + U(g)g+},

as a direct sum of two-sided U(g)0-modules. Denote by q : U(g) ! U(h) the projection

onto the first summand, parallel to the second. Let

F : U(g) ( U(g) ! U(h)

denote the Shapovalov form, and write F! for the restriction of F to the subspace

U(g!)!!, # % Q+.

The algebra U(g) =!

m"0 U(g)m is graded by total degree in the indeterminate t,

U(g)m = span { (x1 # td1) · · · (xk # tdk) |k$

i=1

di = m, k ! 0 } .

For any subspace V & U(g), let

Vm = U(g)m ' V, m ! 0,

and call V graded in t if V =!

m"0 Vm. The subalgebras U(g+), U(g!), and U(h) are

graded in t.

Lemma 1.1. For any m ! 0, q(U(g)m) & U(h)m.

Proof. The spaces g!U(g) and U(g)g+ are graded in t; hence so is the sum

{g!U(g) + U(g)g+}.

1. TRUNCATED CURRENT LIE ALGEBRAS 45

Therefore,

U(g)m = U(h)m $ {g!U(g) + U(g)g+}m,

for any m ! 0. $

Example 1.2. Let g = sl(3), and recall the notation of Example 1.2 (page 32). Let

N = 1, so that g = g $ (g # t). Write C = #+ and C = C ( { 0, 1 }. Then g+ has a basis

parameterised by C:

C 4 (*, d) 5 x(*) # td % g+.

Let # = !1 + !2. Then P! consists of the six partitions

(1.3){ (!1, 0), (!2, 0) } , { (!1 + !2, 0) } , { (!1, 0), (!2, 1) } ,

{ (!1, 1), (!2, 0) } , { (!1 + !2, 1) } , { (!1, 1), (!2, 1) } .

Order the set { x(-) | - % C } by the enumeration

x(!1)# t0, x(!1 + !2)# t0, x(!2)# t0, x(!1)# t1, x(!1 + !2)# t1, x(!2)# t1.

Then the PBW basis monomials of U(g!)!! corresponding to the partitions (1.3) are,

respectively,

(1.4)y(!2) # t0 · y(!1) # t0, y(!1 + !2) # t0, y(!2) # t1 · y(!1) # t0,

y(!1) # t1 · y(!2) # t0, y(!1 + !2) # t1, y(!2) # t1 · y(!1) # t1.

For notational convenience, write h!i,j = h(!i) # tj , for i = 1, 2 and j = 0, 1. The

restriction F! of the Shapovalov form, expressed as a matrix with respect to the ordered

basis (1.4), appears below.3

44444445

h!1+!2,0 + h!1,0 h!1,0 h!1,0h!2,1 + h!1,1 h!1,1(h!2,0 + 2) h!1,1 h!1,1h!2,1

h!1,0 h!1+!2,0 h!1,1 "h!2,1 h!1+!2,1 0

h!1,0h!2,1 + h!1,1 h!1,1 0 h!1,1h!2,1 0 0

h!1,1(h!2,0 + 2) "h!2,1 h!1,1h!2,1 0 0 0

h!1,1 h!1+!2,1 0 0 0 0

h!1,1h!2,1 0 0 0 0 0

6

77777778

This is an elementary calculation using the commutation relations. Observe that this

matrix is triangular, and that in particular the determinant (the Shapovalov determinant

at #) must be the product of the diagonal entries, viz.,

(1.5) (h(!1) # t1)4(h(!2) # t1)4(h(!1 + !2) # t1)2,

up to sign. This provides a criterion for the existence of primitive vectors in the weight

space %|h0 " # of a Verma module V(%), % % h#. We shall prove that the Shapovalov

determinant always lies in S(h # tN), and that for g a semisimple finite-dimensional Lie

algebra, the factors of the Shapovalov determinant are the analogues of those of (1.5).


Example 1.6. Let g be the Virasoro/Witt algebra, and adopt the notation of Example

1.4 (page 1.4). Let N = 1, and # = 2". Write C = #+ and C = C ( { 0, 1 }. Then P!

consists of the five partitions

(1.7) { (", 0), (", 0) } , { (2", 0) } , { (", 0), (", 1) } , { (2", 1) } , { (", 1), (", 1) } .

Order the basis

{ x(-) | - % C } = {Lm # td | m > 0, d = 0, 1 }

for g+ firstly by increasing degree d, and secondly by increasing index m. Then the

PBW basis monomials of U(g!)!! corresponding to (1.7) are, respectively,

(1.8) (L!1 # t0)2, L!2 # t0, L!1 # t1 · L!1 # t0, L!2 # t1, (L!1 # t1)2.

Write $m,i = (2mL0 + &(m)c) # ti, for m, i ! 0. The matrix of F!, expressed with

respect to the ordered basis (1.8), appears below.3

4444445

2$1,0($1,0 + 1) 3$1,0 2$1,1($1,0 + 1) 3$1,1 2($1,1)2

3$1,0 $2,0 3$1,1 $2,1 0

2$1,1($1,0 + 1) 3$1,1 $21,1 0 0

3$1,1 $2,1 0 0 0

2($1,1)2 0 0 0 0

6

7777778

Hence the Shapovalov determinant at # is given by detF! = 4$61,1$

22,1.

Example 1.9. Let g be the Virasoro/Witt algebra, and adopt the notation of Example

1.4 (page 1.4). Let N = 2, and # = 2". Write C = #+ and C = C ( { 0, 1, 2 }. Then P!

consists of the nine partitions

(1.10)

{ (", 0), (", 0) } , { (2", 0) } , { (", 0), (", 1) } ,

{ (", 0), (", 2) } , { (2", 1) } , { (", 1), (", 1) } ,

{ (", 1), (", 2) } , { (2", 2) } , { (", 2), (", 2) } .

Order the basis { x(-) | - % C } as per Example 1.6. Then the PBW basis monomials of

U(g!)!! corresponding to (1.10) are, respectively,

(1.11)

(L!1 # t0)2, L!2 # t0, L!1 # t1 · L!1 # t0,

L!1 # t2 · L!1 # t0, L!2 # t1, (L!1 # t1)2,

L!1 # t2 · L!1 # t1, L!2 # t2, (L!1 # t2)2.

The matrix of F! with respect to the ordered basis (1.11) appears on page 48. Notice

that the matrix has seven non-zero entries on the diagonal. Hence there is no reordering

of the basis (1.11) that will render the matrix triangular.

1. TRUNCATED CURRENT LIE ALGEBRAS 47

1.2. A modification of the Shapovalov Form. As observed in Example 1.9,

it is not always the case that the matrix for F!, # % Q+, can be made triangular by

an ordering of the chosen PBW monomial basis for U(g!)!!. A further permutation

of columns is necessary; this is performed by an involution - on the partitions, and

encapsulated in a modification B of the Shapovalov form F. For any - = (., d) % C,

write -- = (.,N " d) % C, and for any ! % P, write

!- = { -- | - % ! } .

So (!d)-

= (!-)N!d for all ! % P and all degrees d. For any # % Q+, let

B! : U(g!)!! ( U(g!)!! ! U(h)

be the bilinear form defined by

B!(y(!), y(µ)) = F!(y(!), y(µ-)), !, µ % P!.

Relative to any linear order of the basis { y(!) | ! % P! } of U(g!)!!, the matrices of

B! and F! are equal after a reordering of columns determined by the involution -. In

particular, the determinants detB! and detF! are equal up to sign.

484.

HIG

HE

ST

-WE

IGH

TT

HE

ORY

FO

RT

RU

NC

AT

ED

CU

RR

EN

TLIE

ALG

EB

RA

S

3

44444444444444445

2$1,0($1,0 + 1) 3$1,0 2$1,1($1,0 + 1) 2$1,2($1,0 + 1) 3$1,1 2($21,1 + $1,2) 2$1,1$1,2 3$1,2 2$2

1,2

3$1,0 $2,0 3$1,1 3$1,2 $2,1 3$1,2 0 $2,2 0

2$1,1($1,0 + 1) 3$1,1 $1,2($1,0 + 2) + $21,1 $1,1$1,2 3$1,2 2$1,1$1,2 $2

1,2 0 0

2$1,2($1,0 + 1) 3$1,2 $1,1$1,2 $21,2 0 0 0 0 0

3$1,1 $2,1 3$1,2 0 $2,2 0 0 0 0

2($21,1 + $1,2) 3$1,2 2$1,1$1,2 0 0 2$2

1,2 0 0 0

2$1,1$1,2 0 $21,2 0 0 0 0 0 0

3$1,2 $2,2 0 0 0 0 0 0 0

2$21,2 0 0 0 0 0 0 0 0

6

77777777777777778

Matrix of the Shapovalov form F! for the Virasoro/Witt truncated current Lie algebra.

# = 2", N = 2, $m,i := (2mL0 + &(m)c) # ti, m, i ! 0.

2. DECOMPOSITION OF THE SHAPOVALOV FORM 49

2. Decomposition of the Shapovalov Form

Throughout this section, let (g, h0, h, g+,4) denote a Lie algebra with triangular decom-

position, and let g denote the associated truncated current Lie algebra of nilpotency

index N. Let L denote the collection of all two-dimensional arrays of non-negative in-

tegers with rows indexed by #+ and columns indexed by { 0, . . . ,N }, with only a finite

number of non-zero entries. For any # % Q+, let

L! = {L % L | # =$

&"!+

$

0!d!N

L&,d * } .

The entries of an array in L! specify the multiplicity of each positive root in each

homogeneous degree component of a partition of #, i.e.

! % P! 23 ( |!&,d| )'&!+,

0!d!N

% L!.

Let

PL = {! % P | |!&,d| = L&,d, for all * % #+, 0 # d # N } .

Then for any L % L, PL is a non-empty finite set; if the root spaces of g are one-

dimensional, then PL is a singleton. The set L! parameterises a disjoint union decom-

position of the set P!:

(2.1) P! =2

L"L!PL.

For any S & P, let

span(S) = spank { y(!) | ! % S } ,

so that, for example, span(P) = U(g!) and span(P!) = U(g!)!! for any # % Q+.

For any # % Q+, the decomposition (2.1) of P! defines a decomposition of U(g!)!! =

span(P!):

U(g!)!! ="

L"L!

span(PL).

We construct an ordering of the set L! and show that, relative to this ordering, any

matrix expression of the modified Shapovalov form B! for g is block-upper-triangular

(cf. Theorem 2.14). The following Corollary, immediate from Theorem 2.14, provides a

multiplicative decomposition of the Shapovalov determinant, and is the most important

result of this section.

Corollary 2.2. Let # % Q+. Then

detB! =(

L"L!

detB!|span(PL).


2.1. An order on L!. Fix an arbitrary linearisation of the partial order on Q+. If

X is a set with a linear order, write X† for the set X with the reverse order, i.e. x # y in

X† if and only if x ! y in X. For example, the order on Z+† is such that 0 is maximal.

Suppose that (Xi)i"1 is a sequence of linearly ordered sets, and let

X = X1 ( X2 ( X3 ( · · ·

denote the ordered Cartesian product. The set X carries an order <X defined by declar-

ing, for all tuples (xi), (yi) % X, that (xi) <X (yi) if and only if there exists some m ! 1

such that xi = yi for all 1 # i < m, and xm < ym. This order on X is linear, and

is called the lexicographic order (or dictionary order). Fix an arbitrary enumeration of

the countable set #+ ( { 0, 1, . . . ,N }. Consider L as a subset of the ordered Cartesian

product of copies of the set Z+ indexed by this enumeration. Write L(#) for the set L

with the associated lexicographic order. For any L % L, write

#(L) = ($

&"!+

L&,0*,$

&"!+

L&,1*, . . . ,$

&"!+

L&,N*) % (Q+†)N+1,

|L| = ($

&"!+

L&,0,$

&"!+

L&,1, . . . ,$

&"!+

L&,N) % Z+† ( ZN

+.

For any # % Q+, define a map

$! : L! ! (Q+†)N+1 ( (Z+

† ( ZN+) ( L(#)

by

$!(L) = (#(L), |L|, L), L % L!.

Consider the sets (Q+†)N+1 and Z+

† (ZN+ to both carry lexicographic orders. Thus the

Cartesian product

(2.3) (Q+†)N+1 ( (Z+

† ( ZN+) ( L(#)

carries a lexicographic order, and this order is linear. For any # % Q+, we consider the

set L! to carry the linear order defined by the injective map $! and the linearly ordered

set (2.3).

2.2. Decomposition of the Shapovalov form.

Lemma 2.4. For any partitions !, µ % P,

x(!)y(µ) = x(!0)y(µN)x(!1)y(µN!1) · · · x(!N)y(µ0).


Proof. By choice of order for the basis { x(-) | - % C } of g+, and since y(µ) = 4(x(µ)),

by definition,

x(!)y(µ) = x(!0) · · · x(!N)y(µN) · · · y(µ0).

As [ x(!i), y(µj) ] = 0 if i + j > N, the claim follows. $

Proposition 2.5. Suppose that !, µ % P!, # % Q+, and further that #(!d) = #(µd),

for all 0 # d # k, for some 0 # k # N. Then

B0y(!), y(µ)

1=

(

0!d!k

B0y(!d), y(µd)

1· B

0y(!$), y(µ$)

1,

where !$ =#

k<d!N !d and µ$ =

#k<d!N µd.

Proof. Under the hypotheses of the claim,

B0y(!), y(µ)

1= q(x(!)y(µ-))

= q(x(!0)y((µ-)N)x(!1)y((µ-)N!1) · · · x(!N)y((µ-)0))

(by Lemma 2.4)

= q(x(!0)y((µ0)-)x(!1)y((µ1)

-) · · · x(!N)y((µN)

-))

=(

0!d!k

q(x(!d)y(µd-)) · q(x(!k+1)y((µk+1)

-) · · · x(!N)y((µN)

-))

(since q|U(g)0 is an algebra homomorphism)

=(

0!d!k

q(x(!d)y(µd-)) ·B

0y(!$), y(µ$)

1

(by Lemma 2.4)

= ((

0!d!k

B0y(!d), y(µd)

1) · B

0y(!$), y(µ$)

1. $

Lemma 2.6. Suppose that !, µ % P are partitions of homogeneous degree d.

i. If d = 0 and |!| < |µ|, or if d > 0 and |!| > |µ|, then B0y(!), y(µ)

1= 0.

ii. If |!| = |µ| and |!&| )= |µ&| for some * % #+, then B0y(!), y(µ)

1= 0.

Proof. This Lemma follows essentially from Lemma 2.7 (page 38), applied to the Lie

algebra with triangular decomposition (g, h0, h, g+,4). Let !, µ % P be partitions of

homogeneous degree d. Since

x(!)y(µ-) % U(g)|"|d+|µ|(N!d),


it follows from Lemma 1.1 that

(2.7) B0y(!), y(µ)

1% U(h)|"|d+|µ|(N!d).

On the other hand,

(2.8) deghB0y(!), y(µ)

1# |!|, |µ|

by Lemma 2.7 (page 38). Therefore, if B0y(!), y(µ)

1)= 0, and

B0y(!), y(µ)

1% U(h)m,

it must be that

(2.9) m # |!|N and m # |µ|N,

since the degree of h % h in t is at most N. Combining (2.7) and (2.9), it follows that if

B0y(!), y(µ)

1)= 0, then

(2.10) |!|d + |µ|(N " d) # |!|N,

and

(2.11) |!|d + |µ|(N " d) # |µ|N.

If d = 0, then inequality (2.10) becomes |µ| # |!|. Hence, if d = 0, and |!| < |µ|, then

B0y(!), y(µ)

1= 0. If d > 0, then inequality (2.11) yields |!| # |µ|. Hence, if d > 0 and

|!| > |µ|, it must be that B0y(!), y(µ)

1= 0. This proves part (i).

Suppose now that |!| = |µ| = r, and that |!&| )= |µ&| for some * % #+. Then, by

Lemma 2.7 (page 38), the inequality (2.8) becomes strict. Hence, if B0y(!), y(µ)

1)= 0,

then the inequalities (2.10) and (2.11) are also strict. These both yield rN < rN, which

is absurd. Hence it must be that B0y(!), y(µ)

1= 0, and part (ii) is proven. $

Lemma 2.12. Suppose that 1 % Q and 1 )% Q+. Then U(g)+ & g!U(g).

Proof. Because g = g! $ (h $ g+), we have U(g) = U(g!) # U(h $ g+) by the PBW

Theorem. The set of all weights of the h0-module U(h$ g+) is precisely Q+, and so, for

any 1 % Q,

(2.13) U(g)+ =.

!"Q+U(g!)+!! # U(h $ g+)

!.

Suppose that 1 % Q and 1 )% Q+. Then, in particular, 1 " # )= 0, for any # % Q+, and

so

U(g!)+!! & g!U(g).

Hence U(g)+ & g!U(g) by equation (2.13). $


Theorem 2.14. Suppose that # % Q+, and that L,M % L!. If L > M , then

B0y(!), y(µ)

1= 0,

for all ! % PL and µ % PM .

Proof. Suppose that L,M % L! and that L > M . Then one of the following hold:

• #(L) > #(M); or

• #(L) = #(M) and |L| > |M |; or

• #(L) = #(M), |L| = |M | and L > M in L(#).

Let ! % PL and let µ % PM .

Suppose that #(L) > #(M). Then there exists 0 # l # N such that #(!d) = #(µd)

for all 0 # d < l, and #(!l) > #(µl) in Q+†, i.e. #(!l) < #(µl) in Q+. If l > 0, then

Proposition 2.5 with k = l " 1 gives that

(2.15) B0y(!), y(µ)

1= $ ·B

0y(!$), y(µ$)

1

for some $ % S(h), where !$ =#

l!d!N !d and µ$ =

#l!d!N µd. In the remaining case

where l = 0, equation (2.15) holds with ! = !$, µ = µ$ and $ = 1. By Lemma 2.4,

(2.16) B0y(!$), y(µ$)

1= q(x(!l)y((µ-)N!l) · · · x(!N)y((µ-)0)).

Since #((µ-)N!l) = #(µl), the monomial x(!l)y((µ-)N!l) has weight 1 = #(!l)"#(µl).

Now 1 )% Q+, since #(!l) < #(µl) in Q+, and so

x(!l)y((µ-)N!l) % g!U(g)

by Lemma 2.12. Therefore

B0y(!$), y(µ$)

1= 0,

by equation (2.16) and the definition of the projection q. Hence B0y(!), y(µ)

1= 0 by

equation (2.15).

Suppose instead that #(L) = #(M). Then by Proposition 2.5,

B0y(!), y(µ)

1=

(

0!d!N

B0y(!d), y(µd)

1.

Suppose that |L| > |M |. Then either |!d| < |µd|, with d = 0, or |!d| > |µd| for some

0 < d # N. In either case,

B0y(!d), y(µd)

1= 0,


by Lemma 2.6 part (i), applied to the partitions !d, µd. Suppose that |L| = |M | and

that L > M in L(#). Then

L&,d )= M&,d for some * % #+, 0 # d # N,

so that |(!d)&| )= |(µd)&|. Therefore, Lemma 2.6 part (ii), applied to the partitions

!d, µd, implies that B0y(!d), y(µd)

1= 0. Hence B

0y(!), y(µ)

1= 0. $

3. Values of the Shapovalov Form

Throughout this section, let (g, h0, h, g+,4) denote a Lie algebra with triangular de-

composition, and let g denote the truncated current Lie algebra of nilpotency index N

associated to g. In Section 2, the space U(g!)!! is decomposed,

U(g!)!! ="

L"L!

span(PL)

and it is demonstrated that the determinant of the (modified) Shapovalov form B!

on U(g!)!! is the product of the determinants of the restrictions of B! to the spaces

span(PL), L % L!. In this section, the restrictions B|span(PL) are studied. Firstly, the

values of B|span(PL) with respect to the basis y(!), ! % PL, are calculated (cf. Proposition

3.3). This permits the recognition of B|span(PL), in the case where g carries a non-

degenerate pairing, as an S(h)-multiple of a non-degenerate bilinear form on span(PL)

(cf. Theorem 3.20). The form on span(PL) is constructed as a symmetric tensor power

of the non-degenerate form on g.

3.1. Values of the restrictions B|span(PL). Whenever !, µ % P and |!| = |µ| = n,

let

S0!, µ

1=

$

)"Sym(n)

(

1!i!n

[ x(!)(i)), y(µi) ] % S(h),

where (!i) and (µi), 1 # i # n are arbitrary enumerations of ! and µ, respectively.

Lemma 3.1. Suppose that !, µ % P and |!| = |µ|.

i. S0!, µ

1= S

0µ,!

1;

ii. if, in addition, ! and µ are homogeneous of degree-d in t, then

S0!-, µ

1= S

0!, µ-

1.

3. VALUES OF THE SHAPOVALOV FORM 55

Proof. Let n = |!| = |µ|, and choose some enumerations (!i), (µi), 1 # i # n of !

and µ. The anti-involution 4 point-wise fixes S(h), and so 4 fixes S0!, µ

1. On the other

hand,

4(S0!, µ

1) =

$

)"Sym(n)

(

1!i!n

4([ x(!)(i)), y(µi) ])

=$

)"Sym(n)

(

1!i!n

[ x(µi), y(!)(i)) ]

= S0µ,!

1

proving part (i). Suppose that !, µ are homogeneous of degree-d in t. For each 1 # i # n,

let 6i, -i % C be such that

!i = (6i, d), µi = (-i, d).

Then

(3.2) S0!-, µ

1=

$

)"Sym(n)

(

1!i!n

[ x(6)(i)) # tN!d, y(-i) # td ].

For any 1 # i # n and . % Sym(n),

[ x(6)(i)) # tN!d, y(-i) # td ] = [ x(6)(i)), y(-i) ] # tN

= [x(6)(i)) # td, y(-i) # tN!d ],

and hence S0!-, µ

1= S

0!, µ-

1by equation (3.2), proving part (ii). $

Proposition 3.3. Suppose that L % L, and that !, µ % PL. Then

(3.4) B0y(!), y(µ)

1=

(

0!d!N

(

&"!+

S0!&,d, (µ&,d)

-1

and B0y(!), y(µ)

1= B

0y(µ), y(!)

1.

Proof. Let !, µ % PL, L % L, and let 0 # d # N. Then

|!&,d| = |µ&,d| = L&,d, * % #+.

Write l = |!d| = |µd|. Then by Lemma 2.7 (page 38), applied to the Lie algebra with

triangular decomposition (g, h0, h, g+,4),

deghB0y(!d), y(µd)

1# l,

and the degree-l component of B0y(!d), y(µd)

1is given by

(3.5)(

&"!+

S0!&,d, (µ&,d)

-1,


since B0y(!d), y(µd)

1= q(x(!d)y(µd-)). By Lemma 1.1, and since ld + l(N " d) = lN,

B0y(!d), y(µd)

1% U(h)lN.

Therefore deghB0y(!d), y(µd)

1! l, since degt% # N for any % % h; and so B

0y(!d), y(µd)

1

is homogeneous of degree-l in h, and is equal to the expression (3.5). By Proposition

2.5,

B0y(!), y(µ)

1=

(

0!d!N

B0y(!d), y(µd)

1,

and so the equation (3.4) follows. The symmetry of B|span(PL) follows from equation

(3.4),

B0y(!), y(µ)

1=

(

0!d!N

(

&"!+

S0!&,d, (µ&,d)

-1

=(

0!d!N

(

&"!+

S0(µ&,d)

-,!&,d

1

=(

0!d!N

(

&"!+

S0µ&,d, (!&,d)

-1

= B0y(µ), y(!)

1

and parts (i) and (ii) of Lemma 3.1. $

3.2. Tensor powers of bilinear forms. If U , V are vector spaces, and % : U(V !

k is a bilinear map, write

(3.6) % : U&

V ! k

for the unique linear map such that %(u # v) = %(u, v) for all u % U , v % V .

Proposition 3.7. Suppose that U , V are vector spaces with bilinear forms. Then the

vector space U&

V carries a bilinear form defined by

(3.8) ( u1 # v1 | u2 # v2 ) = ( u1 | u2 )( v1 | v2 ),

for all u1, u2 % U and v1, v2 % V . Moreover, if the forms on U and V are non-degenerate,

then so is the form on U&

V .

Proof. Let % : U(U ! k, & : V (V ! k denote the bilinear forms on U , V , respectively.

Let

1 : (U&

U) ( (V&

V ) ! k

be given by

1(u1 # u2, v1 # v2) = %(u1 # u2)&(v1 # v2),


for all u1, u2 % U , v1, v2 % V , where the maps %, & are defined by (3.6). Then 1 is

bilinear, and so defines a linear map

1 : (U&

U)&

(V&

V ) ! k

by (3.6). Since

(U&

U)&

(V&

V ) += (U&

V )&

(U&

V ),

the map 1 may be considered as a bilinear form

( · | · ) : (U&

V ) ( (U&

V ) ! k.

Now if u1, u2 % U , v1, v2 % V , then

( u1 # v1 | u2 # v2 ) = 1(u1 # u2 # v1 # v2)

= 1(u1 # u2, v1 # v2)

= %(u1 # u2)&(v1 # v2)

= %(u1, u2)&(v1, v2),

and so this is the required bilinear form. The non-degeneracy claim follows immediately

from the definition (3.8) of the form. $

For any vector space U and non-negative integer n, write

Tn(U) = U&

· · ·&

U, (n times)

for the space of homogeneous degree-n tensors in U . For any ui % U , 1 # i # n, write&n

i=1 ui = u1 # · · ·# un,

so that

Tn(U) = span {&n

i=1 ui | ui % U, 1 # i # n } .

Write

u1 · · · un =n(

i=1

ui =1

n!

$

%"Sym(n)

&ni=1 u%(i)

for the symmetric tensor in ui % U , 1 # i # n, and let

Sn(U) = span {n(

i=1

ui | ui % U, 1 # i # n }

denote the space of degree-n symmetric tensors in U . Let

T(U) ="

n'0

Tn(U), S(U) ="

n'0

Sn(U),

denote the tensor and symmetric algebras over U , respectively.


Proposition 3.9. Suppose that U is a vector space endowed with a bilinear form, and

that n ! 0. Then Sn(U) carries a bilinear form defined by

(n(

i=1

ui |n(

i=1

vi ) =1

n!

$

)"Sym(n)

n(

i=1

( ui | v)(i) ),

for any ui, vi % U , 1 # i # n. Moreover, if the form on U is non-degenerate, then so is

the form on Sn(U).

Proof. Let A(U) denote the two-sided ideal of T(U) generated by the elements of the

set

{u1 # u2 " u2 # u1 | u1, u2 % U } .

Then T(U) = S(U)!

A(U) is a direct sum of graded vector spaces. Hence, for any

n ! 0,

(3.10) Tn(U) = Sn(U)!

An(U)

is a direct sum of vector spaces, where An(U) denotes the homogeneous degree-n com-

ponent of A(U). By Proposition 3.7, the tensor power Tn(U) carries a bilinear form

defined by

(3.11) (&n

i=1 ui |&n

i=1 vi ) =n(

i=1

( ui | vi ), ui, vi % U, 1 # i # n.

Observe that for any ui, vi % U , 1 # i # n,

$

%"Sym(n)

n(

i=1

( ui | v%(i) )

is independent of the enumeration of the elements v1, . . . , vn. It follows that the direct

sum (3.10) is orthogonal, with respect to the bilinear form (3.11). A form is defined on

Sn(U) by restriction of the form on Tn(U). For any ui, vi % U , 1 # i # n,

(n(

i=1

ui |n(

i=1

vi ) = (1

n!)2

$

%"Sym(n)

$

)"Sym(n)

n(

i=1

( u%(i) | v)(i) )

=1

n!

$

)"Sym(n)

n(

i=1

( ui | v)(i) ).

Hence Sn(U) carries the required bilinear form. If the form on U is non-degenerate, then

by Proposition 3.7 the form on Tn(U) is non-degenerate, and since the sum (3.10) is

orthogonal, the restriction of the form to Sn(U) is non-degenerate also. $


3.3. Lie algebras with non-degenerate pairing. A Lie algebra with triangular

decomposition (g, h0, h, g+,4) is said to have non-degenerate pairing if for all * % #,

there exists a non-zero h(*) % h, and a non-degenerate bilinear form

( · | · )& : g& ( g& ! k,

such that

(3.12) [x1,4(x2) ] = ( x1 | x2 )&h(*),

for all x1, x2 % g&.

If g has a non-degenerate pairing, then for any * % #, the space

[g&, g!& ] = [g!&, g& ]

is one-dimensional, and so the elements h(*) and h("*) can di"er only by a non-zero

scalar.

Example 3.13. Let g be a symmetrisable Kac-Moody Lie algebra over k (cf. Example

1.3, page 33), and let ( · | · ) denote a standard bilinear form on g (as per [20, page 20]).

The restriction of this form to h is non-degenerate. Therefore, for any # % h#, there

exists a unique h(#) % h such that

/#, h0 = ( h(#) | h ) h % h.

The map h : h# ! h is a linear isomorphism. For any * % #, let

( · | · )& : g& ( g& ! k,

be given by

( x1 | x2 )& = ( x1 | 4(x2) ), x1, x2 % g&.

Then for any * % #, the form ( · | · )& is non-degenerate, and is such that equation (3.12)

holds (see, for example, Theorem 2.2 of [20]). Hence g carries a non-degenerate pairing.

Example 3.14. Suppose that g is a Lie algebra with triangular decomposition, such

that for any root * % #,

dim g& = dim g!& = 1, and [g&, g!& ] )= 0.

Then for each * % #, we may choose an arbitrary non-zero

h(*) % [g&, g!& ]

and let the form ( · | · )& : g& ( g& ! k be defined by equation (3.12).


Example 3.15. Let g denote the Virasoro algebra (cf. Example 1.4, page 1.4). Let

* % #, and let m be the non-zero integer such that * = m". Then

g& = kLm, g!& = kL!m,

and [Lm,L!m ] = 2mL0 + &(m)c is non-zero. Therefore, by Example 3.14, g carries a

non-degenerate pairing, with

h(*) = 2mL0 + &(m)c, * = m".

Example 3.16. The Heisenberg Lie algebra a carries a non-degenerate pairing (cf.

Example 1.5, page 34). Let * % #, and let m be the non-zero integer such that * = m".

Then

a& = kam, a!& = ka!m,

and [ am, a!m ] = m" is non-zero. Therefore, by Example 3.14, a carries a non-degenerate

pairing, with

h(*) = m", * = m".

Suppose that (g, h0, h, g+,4) is a Lie algebra with triangular decomposition and non-

degenerate pairing, and let g denote the truncated current Lie algebra with nilpotency

index N associated to g. Non-degenerate bilinear forms are defined on the homogeneous

degree components of the roots spaces of g in the following manner. For all * % #+ and

0 # d # N, define a non-degenerate bilinear form ( · | · )&,d on g& # td by

(3.17) ( x1 # td | x2 # td )&,d = ( x1 | x2 )&, x1, x2 % g&.

For all * % #+ and 0 # d # N, let

C&,d = { - % C | #(-) = *, degt- = d } .

Lemma 3.18. Let * % #+ and let 0 # d # N. Then,

[ x(%), y(&-) ] = ( x(%) | x(&) )&,dh(*) # tN,

for all %,& % C&,d.

Proof. Let %$,&$ % C be such that % = (%$, d) and & = (&$, d). Then

y(&-) = y(&$) # tN!d = 4(x(&$)) # tN!d,


and x(%) = x(%$) # td. Therefore

[ x(%), y(&-) ] = [ x(%$),4(x(&$)) ] # tN

= ( x(%$) | x(&$) )&h(*) # tN

= ( x(%) | x(&) )&,dh(*) # tN,

by equation (3.17). $

3.4. Recognition of the restrictions B|span(PL). For any L % L, let

AL =9

0!d!N

9

&"!+

SL',d(g& # td).

The vector space AL has a basis parameterised by the partitions in PL:

PL 4 ! 5 x(!) % AL,

where, for all ! % P,

x(!) =9

0!d!N

9

&"!+

(

(""',d

x(-).

Proposition 3.19. Let * % #+ and 0 # d # N. If !, µ % P are partitions with

components in C&,d such that |!| = |µ| = k, then

S0!, µ-

1= k! (h(*) # tN)k ( x(!) | x(µ) )

where ( · | · ) is the form on Sk(g& # td) defined by the form on g& # td and Proposition

3.9.

Proof. The claim follows from Lemma 3.18 and the definition of the form on Sk(g&#td).

Let (!i) and (µi), 1 # i # k be any enumerations of ! and µ, respectively. Then:

S0!, µ-

1=

$

)"Sym(k)

(

1!i!k

[ x(!)(i)), y(µi-) ]

=$

)"Sym(k)

(

1!i!k

( x(!)(i)) | x(µi) )&,dh(*) # tN

= k! (h(*) # tN)k1

k!

$

)"Sym(k)

(

1!i!k

( x(!)(i)) | x(µi) )&,d

= k! (h(*) # tN)k ( x(!) | x(µ) ). $


For any L % L, the vector spaces span(PL) and AL are isomorphic by linear extension

of the correspondence

y(!) 5 x(!), ! % PL.

Let ( · | · ) denote the non-degenerate form on AL defined by the forms (3.17) on g& # td

and by Propositions 3.7 and 3.9. So

( x(!) | x(µ) ) =(

0!d!N

(

&"!+

( x(!&,d) | x(µ&,d) ),

for all !, µ % PL, where ( x(!&,d) | x(µ&,d) ) is defined by Proposition 3.9. Let JL be the

bilinear form on span(PL) given by bilinear extension of

JL( y(!) | y(µ) ) = ( x(!) | x(µ) ), !, µ % PL.

Then the form JL is non-degenerate.

Theorem 3.20. For any L % L,

B|span(PL) = h(L) · JL

where h(L) % S(h) is given by

h(L) =(

0!d!N

(

&"!+

(L&,d!) (h(*) # tN)L',d .

Proof. Let !, µ % PL. Then:

B0y(!), y(µ)

1=

(

0!d!N

(

&"!+

S0!&,d, (µ&,d)

-1

(by Proposition 3.3)

=(

0!d!N

(

&"!+

(L&,d!) (h(*) # tN)L',d( x(!&,d) | x(µ&,d) )


=: (

0!d!N

(

&"!+

(L&,d!) (h(*) # tN)L',d;· ( x(!) | x(µ) )

= h(L) · JL( y(!) | y(µ) ).

The set { y(!) | ! % PL } is a basis for span(PL), and so the equality follows. $

4. REDUCIBILITY OF VERMA MODULES 63

4. Reducibility of Verma Modules

Let (g, h0, h, g+,4) denote a Lie algebra with triangular decomposition and non-degen-

erate pairing, and let (g, h0, h, g+,4) denote the truncated current Lie algebra of nil-

potency index N associated to g. In this section we establish reducibility criterion for

a Verma module V(%) for g in terms of evaluations of the functional % % h#. We then

interpret this result separately for the semisimple finite-dimensional Lie algebras, for the

a!ne Kac-Moody Lie algebras, for the symmetrisable Kac-Moody Lie algebras, for the

Virasoro algebra and for the Heisenberg algebra.

Theorem 4.1. Let % % h# and let # % Q+.

i. The Verma module V(%) for g contains a non-zero primitive vector of weight

%|h0 " # if and only if

(4.2) /%N,h(*)0 = 0

for some * % #+ such that #" * % Q+;

ii. V(%) is reducible if and only if equation (4.2) holds for some * % #.

Proof. Let % % h# and let * % #+. By Proposition 2.4 (page 37), the Verma module

V(%) has a non-zero primitive vector of weight %|h0 "# if and only if the form F!(%) is

degenerate. The determinants detF! and detB! can di"er only in sign, and

detF!(%) = /%,detF!0.

Hence such a primitive vector exists if and only if /%,detB!0 vanishes. Now

/%,detB!0 = /%,(

L"L!

detB|span(PL)0

(by Corollary 2.2)

= /%,(

L"L!

detJL · h(L)|PL|0

(by Theorem 3.20)

=(

L"L!

detJL · /%,h(L)0|PL|.

For any L % L!, the form JL is non-degenerate, and so detJL is a non-zero scalar. Hence

/%,detB!0 vanishes if and only if /%,h(L)0 vanishes for some L % L!. As

h(L) =(

0!d!N

(

&"!+

(L&,d!) (h(*) # tN)L',d ,


/%,detB!0 vanishes if and only if /%,h(*) # tN0 is zero for some * % #+ for which

there exists L % L! and 0 # d # N with L&,d > 0. This condition on * is equivalent

to requiring that there exist some partition µ % P! for which |µ&| > 0, which occurs

precisely when # " * % Q+. Hence the first part is proven; as h(*) and h("*) are

proportional, for any * % #, the second part follows. $

It is apparent from Theorem 4.1 that the reducibility of a Verma module V(%) for g

depends only upon %N.

4.1. Symmetrisable Kac-Moody Lie algebras. Let g be a symmetrisable Kac-

Moody Lie algebra as per Examples 1.3 (page 33) and 3.13. The map

h : h# ! h,

from Example 3.13 transports the non-degenerate form ( · | · ) on h to the space h# via

( # | - ) = ( h(#) | h(-) ), #, - % h#.

Hence, for any % % h# and * % #,

/%N,h(*)0 = ( h(%N) | h(*) ) = ( %N | * ),

by definition of the map h. The following Corollary of Theorem 4.1 may be viewed as a

generalisation of Corollary 4.4.

Corollary 4.3. Let g be a symmetrisable Kac-Moody Lie algebra, and let g denote

the truncated current Lie algebra of nilpotency index N associated to g. Then, for any

% % h#, the Verma module V(%) for g is reducible if and only if %N is orthogonal to

some root of g with respect to the symmetric bilinear form.

4.2. Finite-dimensional semisimple Lie algebras. The following Corollary is

a special case of Corollary 4.3.

Corollary 4.4. Let g be a finite-dimensional semisimple Lie algebra, and let g denote

the truncated current Lie algebra of nilpotency index N associated to g. Then, for any

% % h#, the Verma module V(%) for g is reducible if and only if %N is orthogonal to

some root of g in the geometry defined by the Killing form.

Hence the reducibility criterion for Verma modules for g can be described by a finite

union of hyperplanes in h#.


Figure 1: Reducibility criterion for Verma modules of g, where g is of type G2

Example 4.5. Figure 1(a) of page 12 and Figure 1 of page 65 illustrate the reducibility

criterion for the Lie algebras g over R with root systems A2 and G2, respectively. Roots

are drawn as arrows. A Verma module V(%) for g is reducible if and only if %N belongs

to the union of hyperplanes indicated.

4.3. A!ne Kac-Moody Lie algebras. We refine the criterion of Corollary 4.3

for the a!ne Kac-Moody Lie algebras. Let g denote a finite-dimensional semisimple Lie

algebra over the field k with Cartan subalgebra h, root system # and Killing form ( · | · ).

Let g denote the a!nisation of g,

g = g # k[s, s!1] $ kc $ kd,

with Lie bracket relations

[x # sm, y # sn ] = [x, y ] # sm+n + m(m,!n( x | y )c, [ c, g ] = 0,

[ d, x # sm ] = mx # sm,

for all x, y % g and m,n % Z. Let # denote the root system of g, and let

h = h $ kc $ kd,


denote the Cartan subalgebra. Consider any % % h# as a functional on h by declaring

%(c) = %(d) = 0.

This identifies h# with a subspace of h#. Let ", % % h# be given by

/", h0 = 0, /", c0 = 0, /",d0 = 1,

/%, h0 = 0, /%, c0 = 1, /%,d0 = 0,

so that

(4.6) h# = h# $ k" $ k%.

The symmetric bilinear form ( · | · ) on h# may be obtained as an extension of the Killing

form on h#, via

(4.7) ( " | h# ) = ( % | h# ) = 0, ( " | " ) = ( % | % ) = 0, ( " | % ) = 1.

The sum (4.6) is orthogonal with respect to this form. For any % % h#, let )% % h#

denote the projection of % on to h# defined by the decomposition (4.6). The root system

# =# re - #im of g is given by,

(4.8) #re = {*+ m" | * % #, m % Z } , #im = {m" | m % Z, m )= 0 } .

Corollary 4.9. Let g denote an a!ne Kac-Moody Lie algebra, and let g denote the

truncated current Lie algebra of nilpotency index N associated to g. Then, for any

% % h#, the Verma module V(%) for g is reducible if and only if /%N, c0 = 0 or ( *%N | * ) =

m /%N, c0 for some * % # and m % Z.

Proof. It is immediate from (4.6) and (4.7) that

% = )% + ( % | % )" + ( % | " )%.

Hence

/%, c0 = ( % | " )/%, c0 = ( % | " ).

Therefore /%, c0 = 0 if and only if ( % | 7 ) = 0 for some 7 % #im. For * % # and m % Z,

( % | *+ m" ) = ( % | * ) + m( % | " ) = ( )% | * ) + m/%, c0

and so ( % | *+ m" ) = 0 if and only if ( )% | * ) = "m/%, c0. The claim now follows from

(4.8) and Corollary 4.3. $


(a) g of type A(1)2 (b) g of type B(1)

2

Figure 2: Reducibility criterion for the Verma modules of g, where g is of type A(1)2 or

B(1)2

Example 4.10. Figures 2(a) and 2(b) of page 67 and Figure 1(b) of page 12 illustrate

the reducibility criterion of Corollary 4.9 in the case where k = R and g is the Lie

algebra with root systems A2, B2 and G2, respectively. Thus, respectively, g is the a!ne

Kac-Moody Lie algebra of type A(1)2 , B(1)

2 and G(1)2 . A Verma module V(%) for g is

reducible if and only if *%N belongs to the described infinite union of hyperplanes, where

the length of the dashed line segment is |/%N, c0| times the length of a short root for g.

4.4. The Virasoro Algebra. The following Corollary is immediate from Theorem

4.1 and Examples 1.4 (page 1.4) and 3.15.

Corollary 4.11. Let g denote the Virasoro algebra, and let g denote the truncated

current Lie algebra of nilpotency index N associated to g. Then, for any % % h#, the

Verma module V(%) for g is reducible if and only if

2m/%N,L00 + &(m)/%N, c0 = 0,

for some non-zero integer m.

Hence, if & is defined by &(m) = m3!m12 and k = R, a Verma module V(%) for g

is reducible if and only if %N belongs to the infinite union of hyperplanes indicated

in Figure 3. The extension of a functional in the horizontal and vertical directions is

determined by evaluations at c and L0, respectively.


Figure 3: Reducibility criterion for Verma modules of g, where g is the Virasoro algebra

4.5. The Heisenberg Algebra. The following Corollary is immediate from The-

orem 4.1 and Examples 1.5 (page 34) and 3.16.

Corollary 4.12. Let a denote the truncated current Lie algebra of nilpotency index N

associated to the Heisenberg algebra a. Then, for any % % h#, a Verma module V(%) for

a is reducible if and only if /%N, "0 = 0.

4.A. Characters of Irreducible Highest-Weight Modules

Let g denote a Lie algebra with triangular decomposition and non-degenerate pairing,

and let g denote the truncated current Lie algebra of nilpotency index N associated to

g. Theorem 4.1 describes a reducibility criterion for Verma modules for g, but provides

little information on the size of the maximal submodule. This appendix describes the

characters of the irreducible highest-weight g-modules under the assumption that h is

one-dimensional (and hence h0 = h). For example, g may be the Lie algebra sl(2), the

Witt algebra, or a modified Heisenberg algebra.

For any - % h#, let L(-) denote the irreducible highest-weight g-module of highest-weight

-, and for any % % h#, let L(%) denote the irreducible highest-weight g-module of highest

weight %. Let

{ e! | # % h# }

4.B. IMAGINARY HIGHEST-WEIGHT THEORY FOR TRUNCATED CURRENT LIE ALGEBRAS 69

denote a multiplicative copy of the additive group h#, so that

e! · e( = e!+( , #, - % h#.

If M is a vector space graded by h#, M = $!"h!M!, such that all components M! are

finite-dimensional, write

charM =$

!"h!

(dimM!) e!.

Proposition 4.A.1. Let g, g be as above, and let % % h#. Let 0 # m # N be minimal

such that %n = 0 for all m < n # N. Then, if m > 0,

charL(%) = e$0 · (charU(g!))m,

and if m = 0, then L(%) is a g-module isomorphic to L(%0).

Proof. Suppose that m = 0. Since g is the quotient of g by the ideal $0<i!Ng# ti, the

g-module L(%0) is a natural g-module. Moreover, L(%0) is an irreducible highest-weight

g-module of highest-weight %, and so L(%0) += L(%).

Suppose instead that m > 0. Then it must be that %m )= 0. Let g$ denote the truncated

current Lie algebra of nilpotency index m associated to g. Let

%$ = (%0, . . . ,%m) % (h$)#,

and let V(%$) denote the Verma module for g$ of highest-weight %$. Since g$ is the

quotient of g by the ideal $m<i!Ng # ti, the g$-module V(%$) is a natural g-module.

Moreover, V(%$) is of highest-weight % as a g-module. Since h is one-dimensional, V(%$)

is an irreducible g$-module, by Theorem 4.1. Hence V(%$) is the irreducible g-module

of highest-weight %, i.e. L(%) += V(%$) as g-modules. In particular, L(%) and V(%$) are

isomorphic as h#-graded vector spaces. Now

charV(%$) = e$0 · (charU(g!))m

by Proposition 2.3 part (ii) (page 36), and so the claim follows. $

4.B. Imaginary Highest-Weight Theory for Truncated Current Lie Algebras

Let g denote the finite-dimensional Lie algebra sl(2) over the field k, with root system

# = {±! }, and let g denote the a!nisation of g (cf. Subsection 4.3). Let h denote the


Cartan subalgebra of g, let # denote the root system, and let " denote the fundamental

imaginary root. Let

#+ = {! + m" | m % Z } - {m" | m % Z, m > 0 } ,

so that # = #+ - "#+. Let

g+ = $."!+g., g! = $."!+g!.,

so that

(4.B.1) g = g! $ h $ g+.

The subset #+ & # is the imaginary partition of the root system (cf. Section 2 of

Chapter 1). The decomposition (4.B.1) defined by #+ does not satisfy the axioms of

a triangular decomposition in the sense of Chapter 3, nor in the sense of [24]: the

additive semigroup Q+ generated by #+ is not generated by any linearly independent

subset of Q+. Let g denote the truncated current Lie algebra of nilpotency index N

associated to g. We investigate the di!culty inherent in employing our techniques to

derive reducibility criterion for the Verma modules V(%) for g, % % h#. The Theorem

3.20 holds in this setting. However, as we shall see, the degeneracy of an evaluation

/%,B!0 of the modified Shapovalov form may not be deduced from the degeneracy of

the evaluations /%,B|span(PL)0, where L % P!.

Let N = 1, and as per Chapter 3 and Section 1, let

C = #+, C = C ( { 0, 1 } .

All root spaces of g are one-dimensional, so the choice of basis for g+

C 4 7 5 x(7) % g.

is unique up to scalar multiples. Fix the order of the basis elements { x(-) | - % C } by

firstly comparing degree in the indeterminate t, and secondly by the following order of

{ x(7) | 7 % C }:

· · · x(! " 2"), x(! " "), x(!), x(! + "), x(! + 2"), · · · · · · x("), x(2"), · · ·

For any integer m > 0, define partitions

µm = { (! " m", 0) } - { (", 0) (m times) } ,

-m = { (! " m", 1) } - { (", 1) (m times) } ,

and ! = { (!, 0) }. Let # = ! % Q+. Then

{! } - {µm, -m | m > 0 } & P!.

4.B. IMAGINARY HIGHEST-WEIGHT THEORY FOR TRUNCATED CURRENT LIE ALGEBRAS 71

Elementary computation using the Lie bracket relations shows that, for any m > 0,

B0y(!), y(-m)

1= ("2)mh(! " m") # t0, B

0y(µm), y(!)

1= ("2)mh(! " m") # t1.

Hence, if the basis { y(µ) | µ % P! } is to be linearly ordered so that the matrix repre-

sentation of B! is upper triangular, then it must be that both

y(µm) < y(!), y(!) < y(-m),

for all m > 0. Thus the matrix of B! would be bilaterally infinite.

The degeneracy of a bilaterally-infinite upper-triangular matrix can not be determined

from its diagonal entries, as the following simple example demonstrates. Let V denote

the vector space with basis the symbols

(4.B.2) { vm | m % Z }

and let ) : V ! V be defined by linear extension of the rule

) : vm *! vm+1, m % Z.

Then ) is an automorphism of V . Order the basis elements (4.B.2) by their indices.

Then the matrix representation M of ) with respect to this ordered basis will be upper

triangular, in the sense that Mi,j = 0 whenever i > j. However, the diagonal entries

Mi,i are all identically zero.

CHAPTER 5

Characters of Exponential-Polynomial Modules

1. Preliminaries

For any positive integer r, denote by Zr the additive group of integers considered modulo

r, by 7(r) the set of primitive roots of unity of order r, and by ,r some fixed element of

7(r). Denote by ord3 the order of a finite-order automorphism 3. If 3 is an endomor-

phism of a vector space V , write

V |/" = { v % V | 3(v) = !v }

for the eigenspace of eigenvalue !, for any ! % k. Let A = k[t, t!1].

1.1. Ramanujan sums. Let µ denote the Mobius function, i.e. the function

µ : N ! {"1, 0, 1 }

such that µ(d) = ("1)l if d is the product of l distinct primes, l ! 0, and µ(d) = 0

otherwise. For any r > 0, the function µ satisfies the fundamental property

(1.1)$

d|r

µ(d) = (r,1,

where ( denotes the Kronecker function. A summation.

d|r ad is to be understood as

the sum of all the ad where d is a positive divisor of r. Let # : N ! N denote Euler’s

totient function, so that

#(d) = # { 0 < k # d | gcd(k, d) = 1 } , d > 0.

For any positive integer d and n % Z, the quantity cd(n) defined by (4.3) (page 13) is

called a Ramanujan sum, a von Sterneck function, or a modified Euler number. These

quantities have extensive applications in number theory (see, for example, [29], [25]),

although we require only the most basic properties, such as those described in [11]. In

particular, we note the identities

(1.2) cr(n) =$

'"((r)

,n =$

d|gcd(r,n)

dµ(r

d)

73

74 5. CHARACTERS OF EXPONENTIAL-POLYNOMIAL MODULES

where n % Z and r > 0. The function cd(·) : Z ! k is a d-even arithmetic function, i.e.

cd(n) = cd(gcd(d, n)), n % Z.

Any r-even arithmetic function may be expressed as a linear combination of the functions

cd(·), where d is a divisor of r [10]; such an expression is called a Ramanujan-Fourier

transform.

1.2. Exponential-polynomial functions. Define an endomorphism . of the vec-

tor space F via

(. · ")(m) = "(m + 1), m % Z, " % F .

The rule t *! . endows F with the structure of an A-module. For any " % F , the action

of h on H("), via ", may be equivalently defined by

(h # a) · b = (a · ")(0)(ab), a % A, b % im " & A.

Define E & F by

(1.3) E = {" % F | c · " = 0 for some c % k[t] }

For any " % E , the annihilator ann(") & k[t] is a non-zero ideal of k[t]; the unique monic

generator c# % ann(") is called the characteristic polynomial of ". The equivalence of

these definitions and those given in Section 1 of Chapter 1 is demonstrated by Proposition

1.8. The definition (1.3) implies that E is a submodule of the A-module F .

The exponential-polynomial functions are those whose values solve a homogeneous linear

recurrence relation with constant coe!cients. Indeed, suppose that c(t) % k[t] is a non-

zero polynomial of degree q, and write c(t) =.q

k=0 cktk. Then c · " = 0 if and only

if

(1.4) 0 = (c · ")(m) = c0"(m) + c1"(m + 1) + · · · + cq"(m + q),

for all m % Z, i.e. precisely when the values of " satisfy the recurrence relation (1.4)

defined by c. In particular, a solution " to c · " = 0 is determined by any q of its

consecutive values. Therefore, if " % E is non-zero, then the support of " is not wholly

contained in any of the infinite subsets of consecutive integers N,"N & Z. It follows

from Lemma 1.5 that the submonoid of Z generated by the support of " is of the form

rZ, for some r > 0. Equivalently, im" = k[tr, t!r], and so " % F $. Thus E \ { 0 } & F $.

Lemma 1.5. Suppose that A is a submonoid of Z such that N,"N )& A. Then A = rZ,

where r % A is any non-zero element of minimal absolute value.

1. PRELIMINARIES 75

Proof. Let r % A'N be of minimal absolute value. For any m % A'"N, we have that

m + kr % A where k is the unique positive integer such that

0 # m + k0r < r.

Thus m+kr = 0 by the minimality of r; it follows that r divides m, for any m % A'"N.

Moreover,

"r = m + (k " 1)r % A

since k " 1 is non-negative. It follows therefore that "r is the element of minimal

absolute value in A ' "N. The argument above with inequalities reversed shows that

"r divides all positive elements of A, and so A & rZ. The opposite inclusion is obvious

since r,"r % A and A is closed under addition. $

For any k ! 0 and ! % k&, define the function &",k % F by

&",k(m) = mk!m, m % Z.

Lemma 1.6. For any !, µ % k& and k ! 0,

i. (t " µ) · &",k = (!" µ)&",k + !.k!1

j=0

0kj

1&",j ;

ii. (t " !)k · &",k = k!!k&",0.

Proof. For any m % Z,

(t · &",k)(m) = &",k(m + 1) = (m + 1)k!m+1

= !.k

j=0

0kj

1mj!m

= !.k

j=0

0kj

1&",j(m).

Therefore,

(t " µ) · &",k = !.k

j=0

0kj

1&",j " µ&",k,

and so part (i) is proven. Part (ii) is proven by induction. The claim is trivial if k = 0,

so suppose that the claim holds for some k ! 0. Then

(t " !)k+1 · &",k+1 = (t " !)k · !.k

j=0

0k+1j

1&",j

= !0k+1

k

1k!!k&",0 (by inductive hypothesis)

= (k + 1)!!k+1&",0,

where part (i) is used in obtaining the first and second equalities. Therefore the claim

holds for all k ! 0 by induction. $

Proposition 1.7. The set {&",k | ! % k&, k ! 0 } & F is linearly independent.


Proof. Suppose that -",k % k, ! % k&, k ! 0, are scalars such that the sum

" =$

""k"

$

k"0

-",k&",k,

is finite and equal to zero. Write Z = {! % k& | -",k )= 0 for some k ! 0 }, and let

n" = max { k | -",k )= 0 } , ! % Z.

Then, for any ! % Z,

0 =(

µ"Z

(t " µ)nµ+1!0$,µ · "

=(

µ"Z

(t " µ)nµ+1!0$,µ · (-",n$&",n$

)

= -",n$!n$n"!

(

µ"Z,µ)="

(!" µ)nµ+1 · &",0,

by Lemma 1.6. Therefore -",n$= 0 for all ! % Z, which is absurd, unless Z is the empty

set. $

Proposition 1.8. Suppose that " % E . Then " has a unique expression

(1.9) " =$

""k"

""exp(!),

as a finite sum of products of polynomials functions "" and exponential functions exp(!),

! % k&. Moreover,

(1.10) c#(t) =(

""Z

(t " !)deg #$+1,

where Z = {! % k& | "" )= 0 }.

Proof. Let c % k[t] be of degree q. The equation c · " = 0 is equivalent to the relation

(1.4), and so the space consisting of all solutions " is at most q-dimensional. Now write

Z & k& for the set of all roots of c. The field k is algebraically closed, and so

(1.11) c(t) +k"

(

""Z

(t " !)m$ ,

where m" is the multiplicity of the root ! % Z. Lemma 1.6 shows that the set

{&",k | ! % Z, 0 # k < m" } ,

which is of size.

""Z m" = q, consists of solutions to c ·" = 0. By Proposition 1.7, this

set is linearly independent, and hence is a basis for the solution space. Therefore any

" % E has a unique expression (1.9).

2. LOOP-MODULE REALISATION OF N(#) 77

Now suppose that " has the form (1.9), let c % k[t] be non-zero, and write c in the

form (1.11). By Lemma 1.6 part (i), c · " = 0 if and only if m" > deg"" whenever

"" )= 0. The polynomial (1.10) is the minimal degree monic polynomial that satisfies

this condition, and hence is the characteristic polynomial. $

2. Loop-Module Realisation of N(")

For " % F , let kv# be the one-dimensional h-module defined by

h # a · v# = (a · ")(0)v#, a % A.

Let g+ · v# = 0, and denote by

V(") = Indg

h%g+kv#

the induced g-module. This definition is equivalent to the definition (1.5) of Chapter

1. The module V(") and its unique irreducible quotient L(") are not Z-graded. In this

section, it is shown that if " % F $, then N(") is isomorphic to an irreducible constituent

of the loop module 'L("), and moreover that this constituent may be described in terms

of the semi-invariants of an action of the cyclic group Zr on L("), r = deg". The results

of this section are due to Chari and Pressley [9] (see also [7]).

2.1. Cyclic group action on L(").

Lemma 2.1. Suppose that " % F $, that r = deg", and that , % k& is such that ,r = 1.

Then for all a % A,

(a(,t) · ")(0) = (a · ")(0).

Proof. The support of " is contained in rZ. Therefore, if a(t) =.

i aiti, then

(a(,t) · ")(0) =$

i*0 (mod r)

ai ,i"(i) =

$

i*0 (mod r)

ai "(i) = (a · ")(0). $

Proposition 2.2. Suppose that " % F $, and that r = deg". Then there exists an

order-r automorphism ' = '# of the vector space L(") defined by '(v#) = v# and

'(x # a · w) = x # a(,!1t) · '(w), x % g, a % A, w % L("),

where , = ,r. Moreover, ' decomposes L(") as a direct sum of eigenspaces

L(") ="

i"Zr

L(")|''i


in a manner compatible with the weight-space decomposition induced by h # t0.

Proof. The rule t *! ,!1t extends to an automorphism of A, which defines an auto-

morphism of the loop algebra g. This automorphism in turn defines an automorphism

' of U(g). The universal module V(") may be realised as the quotient of U(g) by the

left ideal I generated by g+ and by the elements of the set

{h # a " (a · ")(0) | a % A} .

The map ' preserves this set by Lemma 2.1:

'(h # a " (a · ")(0)) = h # a(,!1t) " (a · ")(0)

= h # a(,!1t) " (a(,!1t) · ")(0).

Clearly ' preserves g+, and so '(I) = I. Therefore ' is well-defined on the quotient

V(") of U(g). The monomial

f # tn1 · · · f # tnk · v# % V(")

is an eigenvector of eigenvalue ,!m where m =.k

i=1 ni, and so the Poincare-Birkho"-

Witt Theorem guarantees a decomposition

(2.3) V(") ="

i"Zr

V(")|''i

of V(") into eigenspaces for '. It is easy to check that ' commutes with the action of

h # t0, and that if U is a submodule of V("), then so is '(U). Thus, if U is a proper

submodule, then so is '(U). Hence ' preserves the maximal submodule of V("), and so

is defined on the quotient L("). This induced map is of order r, by construction, and

decomposes L(") in the manner claimed by (2.3). $

2.2. Irreducible constituents of the loop module. For any " % F $, define an

automorphism '# of the vector space 'L(") via

'#(u # a) = '#(u) # a(,rt), u % L("), a % A,

where r = deg".

Theorem 2.4. Suppose that " % F $, and that r = deg". Let , = ,r and ' = '#. Then:

i. ' is automorphism of the Z-graded g-module 'L(") of order r;

2. LOOP-MODULE REALISATION OF N(#) 79

ii. ' decomposes 'L(") as a direct sum of eigenspaces

'L(") ="

i"Zr

'L(")|''i ,

where

'L(")|''i ="

m"Z

L(")|'"

'i$m # tm, i % Zr;

iii. For any i % Zr, the Z-graded g-modules 'L(")|''i and N(") are isomorphic.

Proof. For any x % g, u % L("), a % A and m % Z,

'(x # tm · u # a) = '((x # tm · u) # tma)

= ,m'(x # tm · u) # tma(,t)

= ,m,!m(x # tm · '(u)) # tma(,t)

= x # tm · ('(u) # a(,t))

= x # tm · '(u # a),

where ' = '#. The map ' is of order r by definition, and so part (i) is proven. Part

(ii) follows immediately from Proposition 2.2. Let i % Zr, and write U = 'L(")|''i . The

generating weight spaces U#(0)! & U and H(") & N(") are isomorphic as Z-graded

h-modules, via

v# # tmr+i *! tmr, m % Z.

This map extends uniquely to an epimorphism of Z-graded g-modules U ! N("). There-

fore it is su!cient to prove that U is an irreducible Z-graded g-module. Suppose that

W is a graded submodule of U . Then W contains a non-zero homogeneous maximal

vector v # tn. The g-module epimorphism U ! L(") that is induced by t *! 1 maps

this element to a non-zero maximal vector of L("). Therefore v = !v# is a non-zero

scalar multiple of the highest-weight vector. Hence W has non-trivial intersection with

the generating weight space U#(0)! of U . The Z-graded h-module U#(0)! is irreducible,

so U#(0)! & W , and thus W = U . Therefore U is irreducible. $

2.3. Characters and semi-invariants. Theorem 2.4 describes the modules N(")

in terms of the semi-invariants of L(") with respect to the action of the cyclic group Zr

defined by ', where r = deg". In particular, we have the following description of the

character of an exponential-polynomial module.


Corollary 2.5. Suppose that " % E is non-zero and that deg" = r. Then

charN(") =$

k"0

$

n"Z

dim L(")k|''n XkZn,

where , = ,r.

3. Semi-invariants of Actions of Finite Cyclic Groups

A Z+-graded vector-space is a vector space V over k with a decomposition V =!

k"0 V (k)

of V into finite-dimensional subspaces indexed by Z+. If V is a Z+-graded vector space

and r is a positive integer, then the tensor power

V r := V # · · · # V (r times)

is also a Z+-graded vector space, with the decomposition

V r =!

k"0 V r(k), V r(k) =!

k1+···+kr=k V (k1) # · · ·# V (kr).

The finite cyclic group Zr acts on V r by cycling homogeneous tensors; the generator

1 % Zr acts via the vector space automorphism

(r : v1 # · · · # vr *! vr # v1 · · ·# vr!1, vi % V,

and this action preserves the grading, so that (r(V r(k)) = V r(k), for any k ! 0. For

any U & V r, let

Un = U |(r

'nr, n % Z.

The automorphism (r decomposes V r as a direct sum of Z+-graded vector spaces

V r =!

n"ZrV r

n , V rn =

!k"0 V r

n (k), V rn (k) = (V r(k))n.

Associated to any Z+-graded vector space U is the generating function

PU (X) =$

k"0

dim Uk Xk % Z+[[X]].

Theorem. For any Z+-graded vector space V , r > 0 and n % Z,

PV rn(X) =

1

r

$

d|r

cd(n)+PV (Xd)

, rd.

In this section, we describe an elementary proof of this statement. In the particular case

where U is the regular representation of Zr and V = S(U) is the symmetric algebra, the

statement follows from Molien’s Theorem and the identity (1.2).

3. SEMI-INVARIANTS OF ACTIONS OF FINITE CYCLIC GROUPS 81

Fix a Z+-graded vector space V , let

B = { (k, s) % Z2+ | 1 # s # dim V (k) }

and for each k ! 0, choose a basis { vks }1!s!dimV (k) for V (k). For any r > 0 and k ! 0,

let

Dr,k = { ((k1, s1), . . . , (kr, sr)) % Br |.r

i=1 ki = k } .

The elements of Dr,k parameterise a graded basis of V r(k):

Dr,k 4 ((k1, s1), . . . , (kr, sr)) = I 5 vI = vk1s1

# · · ·# vkrsr

% V (k).

Define an automorphism %r of the sets Dr,k via the rule

%r : ((k1, s1), . . . , (kr, sr)) *! ((kr, sr), (k1, s1), . . . , (kr!1, sr!1)).

The automorphisms (r and %r are compatible in the sense that

(r(vI) = v%r(I), I % Dr,k, k ! 0.

For I % Dr,k, write ordI = d for the minimal positive integer such that (%r)d(I) = I.

For any positive divisor d of r, let

Or,d(k) = # { I % Dr,k | ordI = d } , k ! 0,

and write Or,d(X) =.

k"0 Or,d(k)Xk for the generating function. It is apparent that

(3.1) PV r(X) = (PV (X))r =$

d|r

Or,d(X)

Lemma 3.2. Suppose that l, r are positive integers and that l | r. Then

{ d | d > 0, r/l | d and d | r } = { r/d$ | d$ > 0, d$ | l } .

Proof. If d > 0 and rl | d, then there exists some positive integer s such that

d =r

ls =

r

(l/s);

if in addition d | r, then d$ := l/s is a positive integer, and so d = r/d$ with d$ | l.

Conversely, if d$ | l, then r/l | r/d$, and it is obvious that r/d$ | r. $

Proposition 3.3. For any Z+-graded vector space V and any r > 0,

PV rn(X) =

1

r

$

d|gcd(r,n)

dOr, rd(X).

for all n % Z.


Proof. Suppose that k ! 0, and write Dr,k =2

O"P O for the decomposition of Dr,k

into a disjoint union of orbits for the action of Zr defined by %r. Then

V r(k) ="

O"P

UO, UO = span { vI | I % O } ,

and moreover (r(UO) = UO. For any orbit O % P, the action of (r on UO defines the

regular representation of Zd, where d = #O is the size of the orbit; in particular, the

eigenvalues of (r on UO are precisely the roots of unity , such that ,d = 1, each with

multiplicity 1. Now ,nr is of order r

gcd(r,n) . Therefore,

dim V rn (k) = # {O % P | r

gcd(r,n) | #O }

= # {O % P | #O = rd for some d | gcd(r, n) }

where the last equality follows from Lemma 3.2 with l = gcd(r, n). The number of orbits

O % P of size r/d is precisely d/r · Or,r/d(k). It follows therefore that

dimV rn (k) =

$

d|gcd(r,n)

dr Or, r

d(k),

which yields the required equality of generating functions. $

Proposition 3.4. For any Z+-graded vector space V and positive integers r, d with

d | r,

Or,d(X) = Od,d(Xrd )

Proof. Suppose that k ! 0, that

I = ((k1, s1), . . . , (kr, sr)) % Dr,k

and that ordI = d. Then

I $ = ((k1, s1), . . . , (kd, sd)) % Dd, kdr

.

and ordI $ = d. This establishes a bijection between order-d elements of the sets Dr,k

and Dd, kdr

, and so Or,d(k) = Od,d(kdr ). Therefore

Or,d(X) =$

k"0

Od,d(kdr )Xk

=$

k"0

Od,d(k)(Xrd )k

= Od,d(Xrd ). $

3. SEMI-INVARIANTS OF ACTIONS OF FINITE CYCLIC GROUPS 83

It follows immediately from Proposition 3.4 and equation (3.1) that

(3.5) (PV (X))r =$

d|r

Od,d(Xrd ).

Proposition 3.6. For any Z+-graded vector space V and r > 0,

Or,r(X) =$

d|r

µ(d)+PV (Xd)

, rd

.

Proof. The claim is trivial if r = 1, so suppose that s > 1 and that the claim holds for

all 0 < r < s. Then:

Os,s(X) = (PV (X))s "$

d|s,d)=s

Od,d(Xrd ) (by equation (3.5))

= (PV (X))s "$

d|s,d)=s

$

d%|d

µ(d$)+PV (X

sd%

d ), d

d%

(by inductive hypothesis)

= (PV (X))s "$

e|s,e )=1

0 $

d|e,d)=e

µ(d)1

(PV (Xe))se

(write e = sd!

dand use Lemma 3.2)

= (PV (X))s "$

e|s,e )=1

("µ(e)) (PV (Xe))se (by equation (1.1))

=$

e|s

µ(e) (PV (Xe))se ,

and so the claim holds for s also. $

Theorem 3.7. For any Z+-graded vector space V , r > 0 and n % Z,

PV rn(X) =

1

r

$

d|r

cd(n)+PV (Xd)

, rd.


Proof. For any n % Z,

PV rn(X) =

1

r

$

d|gcd(r,n)

dOr, rd(X) (by Proposition 3.3)

=1

r

$

d|gcd(r,n)

dO rd, rd(Xd) (by Proposition 3.4)

=1

r

$

d|gcd(r,n)

d$

d%| rd

µ(d$)+PV (Xdd%)

, rdd%


=1

r

$

e|r

0 $

d|gcd(e,n)

dµ(e

d)1(PV (Xe))

re

=1

r

$

e|r

ce(n) (PV (Xe))re ,

where the last equality follows from equation (1.2). $

4. Exponential-Polynomial Modules

In this section, we show that if " % E , then the module L(") is an irreducible highest-

weight module for the truncated current Lie algebra g("). An explicit formula for the

character of such a module was obtained in Chapter 4. Therefore, we are able to derive

an explicit formula for charN(") by employing the results of Sections 2 and 3.

4.1. Modules for truncated current Lie algebras.

Proposition 4.1. Suppose that " % E . Then the defining ideal g # c#A & g acts

trivially on the g-module L("), and so L(") is a g(")-module.

Proof. Let kv+ denote the one-dimensional h-module defined by

h # a · v+ = (a · ")(0)v+, a % A.

Then by definition of the characteristic polynomial c#, the subalgebra h# c#A & h acts

trivially upon v+, and so kv+ may be considered as an h(")-module. Let g+(") ·v+ = 0,

and let

M = Indg(#)h(#)%g+(#) kv+

denote the induced g(")-module. Denote by L the unique irreducible quotient of M .

Then L is a g-module, via the canonical epimorphism g " g("), and is irreducible with

4. EXPONENTIAL-POLYNOMIAL MODULES 85

highest-weight defined by the function ". Hence L(") += L as g-modules, and the claim

follows from the construction of L. $

4.2. Tensor products.

Proposition 4.2. Let "1,"2 % E . Then

L("1 + "2) += L("1)&

L("2),

as g-modules if c#1 and c#2 are co-prime.

Proof. Let " = "1 + "2. Then c# = c#1c#2 since c#1 and c#2 are co-prime. By

Proposition 4.1, L(") is an irreducible module for g("), and by the Chinese Remainder

Theorem,

(4.3) g(") += g("1) $ g("2).

By Proposition, 4.1 L("i) is a module for g("i), i = 1, 2. The Lie algebra g("i) is finite-

dimensional, and k is algebraically closed, and so U(g("i)) is Schurian [27], i = 1, 2.

Thus U(g("i)) is tensor-simple [2], and so L("1)&

L("2) is an irreducible module for

U(g("1))&

U(g("2)). The decomposition (4.3) and the Poincare-Birkho"-Witt Theorem

imply that

U(g("1))&

U(g("2)) += U(g(")),

and so L("1)&

L("2) is an irreducible module for g("). The irreducible highest-weight

modules L(") and L("1)&

L("2) are of equal highest weight, by the Leibniz rule, and

hence are isomorphic. $

4.3. Semi-invariants of the modules L(").

Lemma 4.4. Suppose that " % E is non-zero. Then " % F $, and "" = "'" whenever

!, , % k& and ,r = 1, r = deg". Moreover, there exists & % E such that

i. " = +r&, and

ii. c# =-

i"Zrc1(,i

rt) is a decomposition of c# into co-prime factors.

Proof. According to the discussion of subsection 1.2, " % F $ and r = deg" > 0. Thus

the support of " is contained in the support rZ of +r and so " = 1r+r". Hence

"" = (1

r+r")" =

1

r

$

i"Zr

"('ir"),


for any ! % k&. If ,r = 1, then the expression on the right-hand side is invariant under

the substitution ! *! ,!, and so the first claim is proven.

Multiplication by ,r decomposes k& into a disjoint union of orbits for the cyclic group Zr,

and all orbits are of size r. Choose any set B of representatives, so that k& =2

i"Zr,irB.

Then & =.

""B ""exp(!) has the required property, by Proposition 1.8. $

Remark 4.5. The function & % E of Lemma 4.4 is not unique. Indeed, if

& =$

i

aiexp(µi)

has the required property, then so does &$ =.

i aiexp(,nir µi) for any ni % Zr.

For any " % F , consider L(") as a Z+-graded vector space via

L(") ="

k"0

L(")(k), L(")(k) = L(")(#(0)!k)!, k ! 0.

Proposition 4.6. Suppose that " % E is non-zero and that " = +r&, where r = deg"

and & % E , as per Lemma 4.4. Then there exists an isomorphism

$ : L(") ! L(&)r

of Z+-graded vector spaces such that (r = $ 1 '# 1 $!1.

Proof. For j % Zr, write &j = exp(,!jr )&. Then c# =

-j"Zr

c1j is a decomposition of

c# into co-prime factors, and " =.

j"Zr&j . By the Chinese Remainder Theorem, there

exists a finite linearly independent set { ai | i % I } & A such that { ai + c1A | i % I } is

a basis for A/c1A and

ai 8 0 (mod c1j ), j )8 0 (mod r), i % I, j % Zr.

Write ai,j(t) = ai(,!jr t), i % I, j % Zr. Then by symmetry, { ai,j + c1jA | i % I } is a

basis for A/c1jA and

ai,j 8 0 (mod c1k), j )8 k (mod r), i % I, j, k % Zr.

For any i % I and j % Zr,

(4.7) '#(f # ai,j · w) = f # ai,j(,!1r t) · '#(w) = f # ai,j+1 · '#(w), w % L(").

By Proposition 4.2, there exists an isomorphism

+ : L(") !&

j"ZrL(&j)


of g-modules, and we may assume that +(v#) = #j"Zrv1j . For any k % Zr, identify

(4.8) L(&k) = 1 # · · ·# L(&k) # · · · # 1 &&

j"ZrL(&j).

Then L(&k) is generated by the action of the basis { f # ai,k | i % I } of g+(&k) on the

highest-weight vector +(v#). Therefore, modulo the identification (4.8),

(+ 1 '# 1 +!1)(L(&k)) & L(&k+1), k % Zr,

by equation (4.7). Since '# is an automorphism of the Z+-graded vector space L("), the

restriction

(+ 1 '# 1+!1) : L(&k) ! L(&k+1),

is an isomorphism of the Z+-graded vector spaces. These isomorphisms obviously induce

isomorphisms 6j : L(&j) ! L(&0) = L(&), and

6j :-

i"I(f # ai,j)ki · v1j *!-

i"I(f # ai)ki · v1,

by equation (4.7). Let 6 =&

j"Zr6j, and write $ for the composition

6 1+ : L(") ! L(&)r.

The vector space L(&)r is spanned by the homogeneous tensors

&j"Zr

-i"I(f # ai)ki,jv1, ki,j ! 0.

For any homogeneous tensor of this form

($ 1 '# 1 $!1) · (&

j"Zr

-i"I(f # ai)ki,jv1)

= 6 1 (+ 1 '# 1 +!1)(&

j"Zr

-i"I(f # ai,j)ki,jv1j )

= 6 1 (+ 1 '# 1 +!1)(-

j"Zr

-i"I(f # ai,j)ki,j ·#j"Zrv1j )

= 6(-

j"Zr

-i"I(f # ai,j)ki,j$1 ·#j"Zrv1j )

= 6(&

j"Zr

-i"I(f # ai,j)ki,j$1v1j )

=&

j"Zr

-i"I(f # ai)ki,j$1v1

= (r(&

j"Zr

-i"I(f # ai)ki,jv1),

where the second and fourth equalities are by construction of the polynomials ai,j and

the Leibniz rule. Therefore $ 1 '# 1$!1 = (r as required. $


4.4. Character Formulae.

Theorem 4.9. Suppose that a % F is a polynomial function and that " = aexp(!) for

some ! % k&. Then

PL(#)(X) =

<=

>

1!Xa+1

1!X if a % Z+,

(1 " X)!(deg a)!1 otherwise.

Proof. Let N = deg a, and write " =.N

k=0 ak&",k. By Proposition 4.1, L(") is a module

for the truncated current Lie algebra g("). The Cartan subalgebra of g(") has a basis

{h # (t " !)k | 0 # k # N } .

By Lemma 1.6, h # (t " !)N acts on the highest-weight vector v# by the scalar

(4.10) ((t " !)N · ")(0) = N!!NaN.

If N = 0, then (4.10) takes the value a % k, and so L(") is the irreducible g-module of

highest weight a. Therefore

PL(#)(X) =

<=

>

1!Xa+1

1!X if a % Z+,

11!X otherwise.

If N > 0, then aN is non-zero; thus (4.10) is non-zero and the claim follows from Propo-

sition 4.A.1 (page 69). $

Suppose that " % E is non-zero, deg" = r, and that & % E is given by Lemma 4.4. Then

(4.11) & =$

i

aiexp(!i)

for some finite collection of polynomial functions ai % F and distinct !i % k&, such that

if (!i/!j)r = 1, then i = j.

Theorem 4.12. Suppose that " % E is non-zero, deg" = r, and that

" = +r

$

i

aiexp(!i)

where the ai % F and !i % k& are given by (4.11). Let

P#(X) =

-ai"Z+

(1 " Xai+1)

(1 " X)M,

where M =.

i(deg ai + 1) and the product is over those indices i such that ai % Z+.

Then

charN(") =1

r

$

n"Z

$

d|r

cd(n)+P#(Xd)

, rd

Zn.


Proof. By Corollary 2.5 and Proposition 4.6,

charN(") =$

n"Z

PL(1)rn(X)Zn,

and by Theorem 3.7

(4.13) PL(1)rn(X) =

1

r

$

d|r

cd(n)+PL(1)(X

d), r

d.

By Proposition 4.2, there is an isomorphism of g-modules

L(&) +=&

i L(&i), &i = aiexp(!i),

since the !i are distinct. In particular, PL(1) =-

i PL(1i), and so

PL(1) = P#

by Theorem 4.9. Therefore the claim follows from equation (4.13). $

Index of Symbols

N { 1, 2, . . .}

Z+ { 0, 1, 2, . . .}

Z ring of integers

Zr ring of integers modulo r

R, C fields of real and complex numbers

k& non-zero elements of a field k

7(r) primitive roots of unity of order r

,r fixed element of 7(r)

gcd(m,n) greatest common divisor

Sym(n) symmetric group on n symbols

sgn()) sign of ) % Sym(n)

#S size of a finite set S

( Kronecker function

/%, v0 evaluation of a functional % at v

V # dual of the vector space V

EndV endomorphism algebra of V

A ring of Laurent polynomials

U(g) universal enveloping algebra

PBW Poincare-Birkho"-Witt

S(V ) symmetric algebra of V

T(V ) tensor algebra of V

ad x adjoint operator of x % g

g& root space of g

M! weight space of a module M

V |/" space of eigenvectors of eigenvalue ! for 3 % EndV

e,h, f basis elements of sl(2) 1

! positive root of sl(2) 2

91

92 INDEX OF SYMBOLS

a loop algebra associated to a Lie algebra a 1

%M loop module associated to M 4

F vector space of all functions " : Z ! k 2

F $ set of " % F such that H(") is irreducible and not one-dimensional 2

E vector space of exponential-polynomial functions 3, 74

deg" degree of " % F # 2

exp(!) exponential map exp(!)(m) = !m 3

"" coe!cient of exp(!) in the expression of " % E 3

c# characteristic polynomial of " % E 3, 74

O Chari’s category O 2

H(") Z-graded h-module defined by " % F 2

N(") irreducible Z-graded g-module defined by " % F # 3

V(") universal g-module defined by " % F # (not graded) 3, 77

L(") irreducible g-module defined by " % F # (not graded) 3, 77

g(") truncation of the loop algebra g 4

V(!) imaginary Verma module of highest-weight ! 7

M(0) quotient of the imaginary Verma module V(0) 7

M(n) weight space of M(0) 8, 18

x(k) x # tk 17

An ring of symmetric Laurent polynomials in n variables 8

$i elementary symmetric function 19

p(k) sum of k-powers of the indeterminants 19

m(-) spanning set element of An 19

$n discriminant function 8

w(#) singular vector 9, 23

V (&) universal g-module generated by the h-module & 29

L (&) final g-module generated by the h-module & 29

E(!) negative, even sums of exponential functions 9

INDEX OF SYMBOLS 93

E(!n) all " % E(") such that coe!cients sum to "2n 27

h0 diagonal subalgebra of g 31

4 involution on g 31

V(%) Verma module of highest-weight % % h 36

%i component of a functional % % h$ 11, 44

F Shapovalov form 37, 44

q projection defining Shapovalov form 37

C set that parameterises a root basis of g 38, 43

P set of partitions in C or C 38, 43

|!| length of ! % P 38

#(!) weight of ! % P 38, 43

x(!), y(!) PBW monomials defined by ! % P 32, 38

g truncated current Lie algebra 9

N nilpotency index of g 9

C C ( { 0, . . . ,N } 43

!- dual of ! % P 47

B modified Shapovalov form 47

L set of multiplicity arrays 49

( · | · )& non-degenerate bilinear form on g# ( g"# 11, 59

h(*) element of h given by non-degenerate pairing 11, 59

charN(") formal character of an exponential-polynomial module 13

N(")k,n homogeneous component an exponential-polynomial module 13

+r function with constant value r on its support rZ 13

&",k elementary exponential-polynomial function 75

cd(n) Ramanujan sum 13, 73

# Euler’s totient function 73

µ Mobius function 73

PV Poincare series of a Z+-graded vector space V 80

'# cyclic automorphism of L(") 77

'# cyclic automorphism the g-module 'L(") 78

Bibliography

[1] M. F. Atiyah and I. G. Macdonald. Introduction to commutative algebra. Addison-Wesley Publishing

Co., Reading, Mass.-London-Don Mills, Ont., 1969.

[2] V. Bavula. Each Schurian algebra is tensor-simple. Comm. Algebra, 23(4):1363–1367, 1995.

[3] Y. Billig and K. Zhao. Weight modules over exp-polynomial Lie algebras. J. Pure Appl. Algebra,

191(1-2):23–42, 2004.

[4] P. Casati and G. Ortenzi. New integrable hierarchies from vertex operator representations of poly-

nomial Lie algebras. arXiv:nlin.SI/0405040v1.

[5] P. Casati and G. Ortenzi. New integrable hierarchies from vertex operator representations of poly-

nomial Lie algebras. J. Geom. Phys., 56(3):418–449, 2006.

[6] V. Chari. Integrable representations of a!ne Lie-algebras. Invent. Math., 85(2):317–335, 1986.

[7] V. Chari and J. Greenstein. Graded level zero integrable representations of a!ne lie algebras.

arXiv:math.RT/ 0602514v2.

[8] V. Chari and J. Greenstein. Quantum loop modules. Represent. Theory, 7:56–80 (electronic), 2003.

[9] V. Chari and A. Pressley. New unitary representations of loop groups. Math. Ann., 275(1):87–104,

1986.

[10] E. Cohen. A class of arithmetical functions. Proc. Nat. Acad. Sci. U.S.A., 41:939–944, 1955.

[11] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher transcendental functions. Vol.

III. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. Based, in part, on notes

left by Harry Bateman.

[12] S. Eswara Rao. Classification of loop modules with finite-dimensional weight spaces. Math. Ann.,

305(4):651–663, 1996.

[13] V. M. Futorny. The parabolic subsets of root system and corresponding representations of a!ne

Lie algebras. In Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989),

volume 131 of Contemp. Math., pages 45–52, Providence, RI, 1992. Amer. Math. Soc.

[14] V. M. Futorny. Imaginary Verma modules for a!ne Lie algebras. Canad. Math. Bull., 37(2):213–218,

1994.

[15] V. M. Futorny. Representations of a!ne Lie algebras, volume 106 of Queen’s Papers in Pure and

Applied Mathematics. Queen’s University, Kingston, ON, 1997.

[16] F. Geo"riau. Sur le centre de l’algebre enveloppante d’une algebre de Taki". Ann. Math. Blaise

Pascal, 1(2):15–31 (1995), 1994.

[17] F. Geo"riau. Homomorphisme de Harish-Chandra pour les algebres de Taki" generalisees. J. Alge-

bra, 171(2):444–456, 1995.

[18] J. Greenstein. Characters of simple bounded modules over an untwisted a!ne Lie algebra. Algebr.

Represent. Theory, 6(2):119–137, 2003.

95

96 BIBLIOGRAPHY

[19] H. P. Jakobsen and V. G. Kac. A new class of unitarizable highest weight representations of

infinite-dimensional Lie algebras. In Nonlinear equations in classical and quantum field theory

(Meudon/Paris, 1983/1984), volume 226 of Lecture Notes in Phys., pages 1–20. Springer, Berlin,

1985.

[20] V. G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition,

1990.

[21] V. G. Kac and A. K. Raina. Bombay lectures on highest weight representations of infinite-

dimensional Lie algebras, volume 2 of Advanced Series in Mathematical Physics. World Scientific

Publishing Co. Inc., Teaneck, NJ, 1987.

[22] H. Li. On certain categories of modules for a!ne Lie algebras. Math. Z., 248(3):635–664, 2004.

[23] G. Magnano and F. Magri. Poisson-Nijenhuis structures and Sato hierarchy. Rev. Math. Phys.,

3(4):403–466, 1991.

[24] R. V. Moody and A. Pianzola. Lie algebras with triangular decompositions. Canadian Mathematical

Society Series of Monographs and Advanced Texts. John Wiley & Sons Inc., New York, 1995. , A

Wiley-Interscience Publication.

[25] C. A. Nicol and H. S. Vandiver. A von Sterneck arithmetical function and restricted partitions with

respect to a modulus. Proc. Nat. Acad. Sci. U. S. A., 40:825–835, 1954.

[26] M. Pedroni and P. Vanhaecke. A Lie algebraic generalization of the Mumford system, its symme-

tries and its multi-Hamiltonian structure. Regul. Chaotic Dyn., 3(3):132–160, 1998. J. Moser at 70

(Russian).

[27] D. Quillen. On the endormorphism ring of a simple module over an enveloping algebra. Proc. Amer.

Math. Soc., 21:171–172, 1969.

[28] M. Raıs and P. Tauvel. Indice et polynomes invariants pour certaines algebres de Lie. J. Reine

Angew. Math., 425:123–140, 1992.

[29] S. Ramanujan. On certain trigonometrical sums and their applications in the theory of numbers

[Trans. Cambridge Philos. Soc. 22 (1918), no. 13, 259–276]. In Collected papers of Srinivasa Ra-

manujan, pages 179–199. AMS Chelsea Publ., Providence, RI, 2000.

[30] N. Shapovalov. On a bilinear form on the universal enveloping algebra of a complex semisimple Lie

algebra. Funct. Anal. Appl., 6:65–70, 1972.

[31] S. J. Taki". Rings of invariant polynomials for a class of Lie algebras. Trans. Amer. Math. Soc.,

160:249–262, 1971.

REPRESENTATIONS OF INFINITE-DIMENSIONAL...

Documents

Transcript of REPRESENTATIONS OF INFINITE-DIMENSIONAL...