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Geometric Realizations of the Basic Representation of the
Affine General Linear Lie Algebra
Joel Lemay
Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial
fulfillment of the requirements for the degree of
Doctor of Philosophy in Mathematics 1
Department of Mathematics and Statistics
Faculty of Science
University of Ottawa
© Joel Lemay, Ottawa, Canada, 2015
1The Ph.D. program is a joint program with Carleton University, administered by the Ottawa-Carleton Institute of Mathematics and Statistics
Abstract
The realizations of the basic representation of glr are well-known to be parametrized
by partitions of r and have an explicit description in terms of vertex operators on
the bosonic/fermionic Fock space. In this thesis, we give a geometric interpreta-
tion of these realizations in terms of geometric operators acting on the equivariant
cohomology of certain Nakajima quiver varieties.
ii
Resume
Il est bien connu que les realizations de la representation de base de glr sont parametrises
par les partitions de r et que chacune de ces realization possede une description ex-
plicite en termes d’operateurs de sommet qui agissent sur l’espace de Fock bosonique
et fermionique. Dans cette these, nous donnons une interpretation geometrique de
ces realizations en termes d’operateurs geometriques qui agissent sur la cohomologie
equivariante des varietees de carquois de Nakajima.
iii
Acknowledgements
I would like to thank my mother and father for always being there for me and sup-
porting me in all my endeavours. All my successes in life begin with you.
Moreover, I would like to thank Alistair Savage not only for his invaluable help
throughout the writing of this thesis, but also for his guidance and encouragement
throughout my graduate studies. I would also like to thank Yuly Billig for his helpful
advice.
Finally, I thank the University of Ottawa, NSERC and OGS for their financial
support during my Ph.D. studies.
iv
Contents
1 The Basic Representation of the Affine General Linear Lie Alge-
bra 4
2 Geometric Invariant Theory 28
3 Quiver Varieties 42
4 Vector Bundles and Geometric Operators 63
5 Geometric Realizations of the Basic Representation 83
v
Introduction
Let g be a semisimple Lie algebra and denote the corresponding untwisted affine Lie
algebra by g. The basic representation of g, which we denote by Vbasic = Vbasic(g), is
the irreducible highest weight representation whose highest weight is the fundamental
weight corresponding to the additional node of the affine Dynkin diagram (compared
to the corresponding finite Dynkin diagram). The basic representation is so-named
since it is, in a sense, the simplest nontrivial representation of g. In the late 70’s and
early 80’s mathematicians began constructing the first explicit realizations of Vbasic.
The first such realization was given by Lepowsky and Wilson in [14] for Vbasic(sl2).
Their construction was later generalized to arbitrary simply-laced affine Lie algebras
and twisted affine Lie algebras in [10], and this construction became known as the
principal realization of Vbasic. However, Frenkel and Kac in [6], and Segal in [25],
gave an entirely different realization of Vbasic; this construction was referred to as the
homogeneous realization. While the principal and homogeneous realizations seemed
completely unrelated, it was discovered by Kac and Peterson in [12], and by Lepowsky
in [13], that the two realizations depend implicitly on the choice of a so-called maximal
Heisenberg subalgebra of g. Indeed, one can associate a realization of Vbasic to each
maximal Heisenberg subalgebra of g.
In this thesis, we will focus on the case where g = glr. While glr is not semisimple,
it is a one-dimensional central extension of the semisimple Lie algebra slr, and thus has
a similar representation theory. Up to conjugacy under the adjoint action of the Kac-
1
CONTENTS 2
Moody group, the maximal Heisenberg subalgebras of an affine Lie algebra are known
to be parametrized by the conjugacy classes of the Weyl group of the corresponding
finite-dimensional Lie algebra (see [12, Proposition of Section 9]). The Weyl group of
glr is the symmetric group on r elements, Sr, and the conjugacy classes of Sr are in
one-to-one correspondence with partitions of r, i.e. s-tuples (r1, . . . , rs) ∈ (N+)s such
that
r = r1 + · · ·+ rs, and r1 ≤ · · · ≤ rs.
Thus, there exists a realization of Vbasic(glr) for each partition of r, the principal
and homogeneous realizations corresponding to the two extreme partitions (r) and
(1, . . . , 1), respectively. The realizations for every partition of r were described by
ten Kroode and van de Leur in [26] using vertex operators acting on bosonic Fock
space (a representation of the Heisenberg algebra) and fermionic Fock space (a repre-
sentation of the Clifford algebra). More precisely, for each partition of r, there exists
a precise vector space isomorphism between bosonic Fock space and fermionic Fock
space (known as the boson-fermion correspondence), and thus the Heisenberg and
Clifford algebras may be thought of as operators acting on a common space. The
construction in [26] defines a representation of glr on bosonic/fermionic Fock space in
terms of vertex operators (i.e. formal power series) of Heisenberg and Clifford algebra
operators. The so-called “zero-charge” subspace of bosonic/fermionic Fock space is
then shown to be isomorphic, as a glr-representation, to Vbasic.
In this thesis, we give a geometric interpretation of these algebraic realizations
of Vbasic(glr). Our general strategy is as follows. We fix a partition (r1, . . . , rs) of r
and consider the moduli space of framed torsion-free sheaves of rank r and second
Chern class n, M(s, n). In [15], Licata and Savage showed that, under a suitable
torus action, the localized equivariant cohomology of (a disjoint union of infinitely-
many copies) ofM(s, n) provides a suitable geometric analogue of bosonic/fermionic
Fock space. This is accomplished by defining an action of the Heisenberg and Clifford
CONTENTS 3
algebras on this cohomology in terms of the top Chern classes of certain equivariant
vector bundles on M(s, n). The construction given in [15] naturally corresponds to
the homogeneous realization in [26], and thus we generalize their construction to an
arbitrary partition. With this framework in place, we define a new set of operators
using vector bundles on certain subvarieties of M(s, n). We then show that these
operators may be expressed as vertex operators of our “geometric” Heisenberg and
Clifford algebra operators, and that the formulas we obtain exactly match those
found in the algebraic realizations of the representation of glr on Vbasic, thus giving
us geometric realizations of Vbasic.
The paper is organized into 5 chapters. In Chapter 1, we review the Heisenberg
and Clifford algebra representations on bosonic and fermionic Fock space. We also
briefly summarize the various algebraic realizations of Vbasic found in [26]. In Chapter
2, we recall some of the basic concepts from algebraic geometry, especially geometric
invariant theory, that we will use in subsequent chapters. In Chapter 3, we will
discuss our main geometric object of interest: Nakajima quiver varieties (of which the
aforementioned moduli space M(s, n) is a special case). Chapter 4 will describe our
method of constructing geometric operators on the localized equivariant cohomology
of quiver varieties from equivariant vector bundles. Finally, in Chapter 5, we define
our geometric analogues of the action of the Heisenberg algebra, Clifford algebra, and
glr on bosonic and fermionic Fock space. We also present our main theorem (Theorem
5.21), which is a geometric analogue of Theorem 1.20 (the main theorem of [26]).
Chapter 1
The Basic Representation of the
Affine General Linear Lie Algebra
In this first chapter, we will summarize the known algebraic realizations of the basic
representation of glr. In particular, the inequivalent realizations are parametrized by
the different partitions of r. For a more in-depth treatment of this topic, the reader
is encouraged to see [26] or [11]. The goal in the subsequent chapters will be to give
a geometric construction of the representations presented here.
We begin with a description of the s-coloured oscillator algebra and s-coloured
Clifford algebra, along with the associated s-coloured bosonic and fermionic Fock
spaces.
Definition 1.1. (s-coloured oscillator algebra) For s ∈ N+, the s-coloured oscillator
algebra, s, is the complex Lie algebra
s :=s⊕`=1
(⊕n∈Z
CP`(n)
)⊕ Cc,
with the Lie bracket determined by
[s, P`(0)] = 0, [P`(n), Pk(m)] =1
nδ`,kδn+m,0c, n 6= 0,
for all `, k = 1, . . . , s and m,n ∈ Z.
4
1. The Basic Representation of the Affine General Linear Lie Algebra 5
Remark 1.2. Our presentation of the s-coloured oscillator algebra differs slightly
from the one sometimes found elsewhere in the literature. In particular, in [26,
Section 1], s is defined as the complex Lie algebra with basis α`(n)n∈Z,`=1,...,s ∪ c
and Lie bracket given by
[α`(n), αk(m)] = nδ`,kδn+m,0c.
The connection between the two presentations is made by setting
P`(n) =1
|n|α`(n), n 6= 0, and P`(0) = α`(0).
We favour the presentation given in Definition 1.1 since this one turns out to be more
natural from a geometric point of view.
Definition 1.3. (s-coloured Heisenberg algebra) The subalgebra
s0 =s⊕`=1
⊕n∈Z−0
CP`(n)
⊕ Cc,
of s is the so-called s-coloured Heisenberg algebra and, as we will see later, it serves
as the main ingredient in the various realizations of the basic representation of glr.
Let Λ ⊆ C[t1, t2, . . . ] denote the ring of symmetric functions in infinitely many
variables. It is well-known that
Λ = C[p1, p2, . . . ],
where pn is the n-th power sum
pn =∞∑i=1
tni .
We define the s-coloured bosonic Fock space to be the space
B := B⊗s, where B := Λ⊗C C[q, q−1].
1. The Basic Representation of the Affine General Linear Lie Algebra 6
We define a Z-grading on B given by
B =⊕c∈Z
Bc, Bc := Λ⊗ Cqc.
This induces a Zs-grading on B given by
B =⊕c∈Zs
Bc, Bc := Bc1 ⊗ · · · ⊗Bcs .
For c = (c1, . . . , cs) ∈ Zs, we use the notation |c| = c1 + · · · + cs. We then have a
Z-grading on B given by
B =⊕c∈Z
B(c), B(c) =⊕|c|=c
Bc.
One can easily verify that the mapping
P`(n) 7→ 1⊗`−1 ⊗ ∂
∂pn⊗ 1⊗s−`, n > 0,
P`(−n) 7→ 1⊗`−1 ⊗ 1
npn ⊗ 1⊗s−`, n > 0,
P`(0) 7→ 1⊗`−1 ⊗ q ∂∂q⊗ 1⊗s−`, c 7→ 1,
defines an irreducible representation of s on B.
Remark 1.4. Again, our definition of s-coloured bosonic Fock space differs slightly
from the definition sometimes found elsewhere in the literature. In particular, s-
coloured bosonic Fock space is often only considered as a representation of the s-
coloured Heisenberg algebra (rather than of the full oscillator algebra). The s-coloured
Heisenberg subalgebra acts trivially on the C[q, q−1] factors of bosonic Fock space,
and so, as an s0-module, B is isomorphic to a direct sum of infintely many copies of
Λ⊗s. In [26, Section 1], bosonic Fock space is defined as
C[x1, x2, . . . ]⊗s,
1. The Basic Representation of the Affine General Linear Lie Algebra 7
with the action of the Heisenberg algebra given by
α`(n) 7→ 1⊗`−1 ⊗ ∂
∂xn⊗ 1⊗s−`, n > 0,
α`(−n) 7→ 1⊗`−1 ⊗ nxn ⊗ 1⊗s−`, n > 0,
c 7→ 1.
However, one has an isomorphism (a priori, only of rings) Λ → C[x1, x2, . . . ], de-
termined by the mapping pn 7→ nxn. This in turn induces an isomorphism Λ⊗s →
C[x1, x2, . . . ]⊗s. One easily checks that the diagram
Λ⊗s C[x1, x2, . . . ]⊗s
Λ⊗s C[x1, x2, . . . ]⊗s
|n|P`(n) α`(n)
commutes for all ` = 1, . . . , s and n ∈ Z − 0 (this justifies the statement from
Remark 1.2 that P`(n) = 1|n|α`(n)). Thus, the map Λ⊗s → C[x1, x2, . . . ]
⊗s is in fact
an isomorphism of s-coloured Heisenberg algebra representations.
Remark 1.5. Notice that the sets 1, . . . , s × N+ and N+ are countable. Pick a
bijection ϕ : 1, . . . , s × N+ → N+. Then the mapping
P`(k) 7→ P (ϕ(`, k)), P`(−k) 7→ ϕ(`, k)
kP (−ϕ(`, k)), c 7→ c,
for all ` ∈ 1, . . . , s and k ∈ N+, is an isomorphism from the s-coloured Heisen-
berg algebra to the 1-coloured Heisenberg algebra. Thus, we will often use the term
“Heisenberg algebra” without specifying the number of colours. Moreover, one can
show (see, for instance, [11, Lemma 14.4(a)]) that Λ⊗s and Λ are isomorphic as Heisen-
berg algebra modules. The reason for the s-coloured presentation of the Heisenberg
algebra will become clear when we describe the various realizations of the basic rep-
resentation of glr.
1. The Basic Representation of the Affine General Linear Lie Algebra 8
Definition 1.6. (s-coloured Clifford algebra) The s-coloured Clifford algebra, Cl, is
the complex associative algebra generated by elements ψ`(i), ψ∗` (i), ` = 1, . . . , s, i ∈ Z,
and relations
ψ`(i), ψ∗k(j) = δijδ`k, ψ`(i), ψk(j) = ψ∗` (i), ψ∗k(j) = 0.
where x, y = xy + yx.
A semi-infinite monomial is an expression of the form
I = i1 ∧ i2 ∧ i3 ∧ · · · ,
where the in are integers such that
i1 > i2 > i3 > . . . , in+1 = in − 1, for n 0.
The charge of a semi-infinite monomial I is the integer c = c(I) such that
in = c− n+ 1, for all n 0.
The energy of a semi-infinite monomial is∑n∈N+
in − (c− n+ 1).
Let F be the complex vector space with basis the set of all semi-infinite monomials.
The charge induces a grading on F :
F =⊕c∈Z
Fc,
where Fc is the subspace with basis the set of semi-infinite monomials of charge c.
Define wedging and contracting operators ψ(i) and ψ∗(i), i ∈ Z, on F by
ψ(i)(i1 ∧ i2 ∧ · · · ) =
(−1)k(i1 ∧ · · · ∧ ik ∧ i ∧ ik+1 ∧ · · · ), if ik > i > ik+1,
0, if i = in, for some n,
1. The Basic Representation of the Affine General Linear Lie Algebra 9
ψ∗(i)(i1 ∧ i2 ∧ · · · ) =
(−1)k−1(i1 ∧ · · · ∧ ik−1 ∧ ik+1 ∧ · · · ), if i = ik,
0, if i 6= in, for all n.
It is easy to see that, when they do not act as zero, the wedging and contracting
operators raise and lower the charge of a semi-infinite monomial by 1, and so
ψ(i) : Fc → Fc+1, ψ∗(i) : Fc → Fc−1,
for all i, c ∈ Z.
We define s-coloured fermionic Fock space to be
F := F⊗s.
By our decomposition of F in terms of the charge, we have a Zs-grading
F =⊕c∈Zs
Fc, Fc := Fc1 ⊗ · · · ⊗ Fcs .
This induces a Z-grading on F given by
F =⊕c∈Z
F(c), F(c) :=⊕|c|=c
Fc.
One can show that we have a representation of Cl on F given by:
ψ`(i)|Fc 7→ (−1)c1+···+c`−1(1⊗`−1 ⊗ ψ(i)⊗ 1⊗s−`
),
ψ∗` (i)|Fc 7→ (−1)c1+···+c`−1(1⊗`−1 ⊗ ψ∗(i)⊗ 1⊗s−`
).
One can also show that F is a faithful, irreducible representation of Cl. In fact, F is
generated by the so-called vacuum vector, |0〉⊗s, where
|0〉 = 0 ∧ −1 ∧ −2 ∧ · · · .
Note that F is referred to as the spin module in [26].
1. The Basic Representation of the Affine General Linear Lie Algebra 10
Remark 1.7. We have used here the convention that Clifford algebra generators of
different colours anti-commute. However, the Clifford algebra is sometimes defined
by letting generators of different colours commute (see for example [15, Section 1.2]).
In this case it is necessary to modify the action of Cl on F to
ψ`(i)|Fc 7→ 1⊗`−1 ⊗ ψ(i)⊗ 1⊗s−`,
ψ∗` (i)|Fc 7→ 1⊗`−1 ⊗ ψ∗(i)⊗ 1⊗s−`.
Remark 1.8. By an argument completely analogous to Remark 1.5, the s-coloured
Clifford algebra is isomorphic to the 1-coloured algebra, and, by identifying the two
algebras, s-coloured fermionic Fock space is isomorphic to 1-coloured fermionic Fock
space. Thus, we will often refer to the Clifford algebra and fermionic Fock space
without specifying the number of colours.
In the sequel, it will be useful to think of fermionic Fock space not only in terms
of semi-infinite monomials, but also in terms of Young diagrams. Recall that a Young
diagram is a finite collection of boxes arranged in rows and columns such that the
number of boxes in the i-th row is greater than or equal to the number of boxes in
the (i+ 1)-st row. There are different conventions regarding the way to draw Young
diagrams, but we will choose the English notation. That is, our rows will be left-
justified and subsequent rows are placed underneath the previous one, as illustrated
in Figure 1.1. We say that a box is in the (i, j)-th position if it is in the i-th row and
j-th column of the diagram. For example, in Figure 1.1, the box labelled with a “?”
is in the (3, 2)-th position. We may thus view a Young diagram λ as being a subset
of (N+)2, where (i, j) ∈ λ if and only if λ has a box in the (i, j)-th position. For any
Young diagram λ and any k ∈ N+, let λk denote the number of boxes in its k-th row
(λk = 0 if there are no boxes in the k-th row). For any (i, j) ∈ (N+)2, define the arm
and leg length of (i, j) to be
aλ(i, j) := λi − j and `λ(i, j) := maxi′ ∈ N+ | (i′, j) ∈ λ − i,
1. The Basic Representation of the Affine General Linear Lie Algebra 11
?
Figure 1.1: Young diagram
respectively. Intuitively, for each (i, j) ∈ λ, aλ(i, j) and `λ(i, j) count the number of
boxes to the right of and below (i, j), respectively. For two Young diagrams λ and µ,
we define the relative hook length of (i, j) to be
hλ,µ(i, j) := aλ(i, j) + `µ(i, j) + 1. (1.1)
If λ = µ, we will simply write hλ(i, j) := hλ,λ(i, j). Finally, we define the residue of
box in the (i, j)-th position to be j − i.
Clearly, every weakly decreasing sequence of non-negative integers
(λ1, λ2, λ3, . . . ),
such that λk = 0 for k 0 uniquely defines a Young diagram. Thus, one has a
bijection between the set of semi-infinite monomials of a fixed charge and the set of
Young diagrams. Indeed, suppose
I = i1 ∧ i2 ∧ i3 ∧ · · · ,
is a semi-infinite monomial of charge c. Define λ(I) to be the Young diagram deter-
mined by
λ(I)k = ik − c+ k − 1. (1.2)
Lemma 1.9. The mapping I 7→ λ(I) is a bijection from the set of semi-infinite
monomials of charge c to the set of Young diagrams. Moreover, if I is a semi-infinite
monomial of energy n, then λ(I) consists of n boxes.
1. The Basic Representation of the Affine General Linear Lie Algebra 12
Hence, by Lemma 1.9, we may think of Fc as the complex vector space with
basis the set of all Young diagrams, thus giving us a Young diagram interpretation of
fermionic Fock space.
The description of the realizations of the basic representation of glr will rely
on the precise relationship between bosonic and fermionic Fock spaces known as the
boson-fermion correspondence. It is well-known that the set of Schur functions sλ,
where λ runs over all Young diagrams, is a basis of Λ (see statement (3.3) of [17]).
We then have a vector space isomorphism
Bc → Fc,
sλ ⊗ qc 7→ λ,(1.3)
for each c ∈ Z. This induces isomorphisms B(c)∼=−→ F(c) and B
∼=−→ F.
We turn our attention now to the representation theory of Lie algebras. Let g
be a matrix Lie algebra and let g = g⊗C[t, t−1] be the loop algebra on g. The affine
Lie algebra of g is
g := g⊕ Cc,
with Lie bracket given by
[g, c] = 0, [x⊗ tn, y ⊗ tm] = [x, y]⊗ tn+m + nδn+m,0 tr(xy)c. (1.4)
Here we follow the conventions in [26, Equation (4.2.5)] (note that the two-cocycle
µ in [26] simplifies to a multiple of the trace map when restricted to elements of the
form x⊗ tn).
Remark 1.10. In the literature, the Lie bracket on g is usually defined as
[x⊗ tn, y ⊗ tm] = [x, y]⊗ tn+m + nδn+m,0〈x|y〉c,
where 〈x|y〉 = tr(adx ad y) is the Killing form on g. For each of the classical Lie
algebras, the Killing form is just a (nonzero) multiple of the trace map. In particular,
1. The Basic Representation of the Affine General Linear Lie Algebra 13
for slr,
〈x|y〉 = 2r tr(xy).
Thus, up to a scaling factor on c, there is no difference between using the Killing form
and the trace map. Meanwhile, on glr, the Killing form is degenerate since
〈I|x〉 = 0,
for all x ∈ glr, whereas the trace map is nondegenerate, making it a more appropriate
choice. This is our motivation for defining the Lie bracket on g by (1.4).
Remark 1.11. The loop algebra g is sometimes defined as
g =⊕n∈Z
einθg,
where θ is a formal parameter. Of course, one has the obvious isomorphism of Lie
algebras ⊕n∈Z
einθg→ g⊗ C[t, t−1],
einθx 7→ x⊗ tn,
and so both definitions are equivalent. We will use this alternative description when
it is more convenient.
Next, we describe the basic representation, Vbasic = Vbasic(g), of g when g = slr
or glr, r ∈ N+ (although the description applies to other Lie algebras as well). Let
Ei,j1≤i,j≤r be the standard basis of glr. That is Ei,j is the r×r matrix with (i, j)-th
entry 1 and zeros elsewhere. Recall that, for r ≥ 2, slr is generated by its Chevalley
generators Ei, Fi, Hi, i = 0, 1, . . . , r − 1, which are given by
Ei = Ei,i+1 ⊗ 1, Fi = Ei+1,i ⊗ 1, Hi = (Ei,i − Ei+1,i+1)⊗ 1, i = 1, . . . , r − 1,
E0 = Er,1 ⊗ t, F0 = E1,r ⊗ t−1, H0 = ((Er,r − E1,1)⊗ 1) + c.
The Chevalley generators of slr satisfy the well-known Kac-Moody relations (for slr):
1. The Basic Representation of the Affine General Linear Lie Algebra 14
1. [Hi, Hj] = 0,
2. [Ei, Fj] = δijHi,
3. [Hi, Ej] = aijEj,
4. [Hi, Fj] = −aijFj,
5. (adEi)(1−aij)(Ej) = 0, for i 6= j,
6. (adFi)(1−aij)(Fj) = 0, for i 6= j,
where
aij =
2, if i = j,
−2, if i 6= j,
for r = 2,
aij =
2, if i = j,
−1, if |i− j| = 1,
0, otherwise,
for r > 2.
Note that the r× r matrix (aij) is called the generalized Cartan matrix associated to
slr. We define the triangular decomposition of slr:
slr = sl+
r ⊕ h⊕ sl−r ,
where sl+
r (resp. sl−r ) is the subalgebra generated by the Ei (resp. the Fi) and h is the
abelian subalgebra with basis Hi. The subalgebra h is called a Cartan subalgebra
of slr. Given γ = (γ0, γ1, . . . , γr−1) ∈ Zr, a highest weight representation, V , of slr is
a representation such that there exists a vector v0 ∈ V satisfying
Ei · v0 = 0, and Hi · v0 = γiv0, (1.5)
for all i = 0, 1, . . . , r − 1, and
U(slr) · v0 = V,
1. The Basic Representation of the Affine General Linear Lie Algebra 15
where U(slr) is the universal enveloping algebra of slr. Note that, in terms of the
triangular decomposition of slr, (1.5) is equivalent to
sl+
r · v0 = 0, and Hi · v0 = γiv0.
The r-tuple γ is called the highest weight of V and the vector v0 is called a highest
weight vector of V . If V is irreducible, one can show that any two highest weight
vectors are proportional, thus, in this case, we will often abuse terminology and call
v0 the highest weight vector of V . Given a highest weight representation V of highest
weight γ, for each β ∈ Zr, define
Vβ :=
v ∈ V
∣∣∣∣∣ Hi · v =
(γi −
r−1∑j=0
aijβj
)v
.
Clearly, v0 ∈ V0. Let 1i denote the r-tuple with 1 in its i-th entry and zeros elsewhere.
It follows from the Kac-Moody relations that, restricted to Vβ,
Ei, Fi : Vβ → Vβ∓1i ,
and so
V =⊕β∈Zr
Vβ. (1.6)
We call the decomposition in (1.6) the weight space decomposition of V .
Remark 1.12. In our treatment of weights above, we are slightly abusing terminol-
ogy. Given a semisimple Lie algebra g and a Cartan subalgebra h ⊆ g, a weight is
defined to be a linear map
β : h→ C,
and the associated weight space decomposition of a representation V is
V =⊕β
Vβ, Vβ = v ∈ V | h · v = β(h)v for all h ∈ h.
However, the Chevalley generators H0, H1, · · ·Hr−1 form a basis of our Cartan sub-
algebra h and thus induce a basis of the vector space of weights. Our description
merely identifies the weight with its coefficients with respect to this basis.
1. The Basic Representation of the Affine General Linear Lie Algebra 16
Definition 1.13 (Basic representation of slr). The basic representation of slr, which
we denote Vbasic(slr), is the (unique) irreducible highest weight representation with
highest weight γ = (1, 0, . . . , 0).
Remark 1.14. See [11, Proposition 9.3] for a proof that there is a unique irreducible
highest weight representation of slr with highest weight γ = (1, 0, . . . , 0).
There is another way to describe Vbasic(slr) that amounts to the same thing.
Namely, one can define Vbasic(slr) to be the unique irreducible representation that
admits a vector v0 satisfying
(slr ⊗ C[t]) · v0 = 0, and c · v0 = v0.
It is easy to check that this alternate description is equivalent to Definition 1.13
and that v0 here plays the role of the highest weight vector. The advantage to this
construction is that it naturally generalizes to glr.
Definition 1.15 (Basic representation of glr). The basic representation of glr, which
we denote Vbasic(glr), is the (unique) irreducible representation that admits a vector
v0 (highest weight vector) such that
(glr ⊗ C[t]) · v0 = 0, and c · v0 = v0.
From now on, unless otherwise specified, Vbasic will always mean Vbasic(glr).
Remark 1.16. The proof that Vbasic is unique is essentially the same as [11, Propo-
sition 9.3].
Remark 1.17. There is a way to describe Vbasic that is more akin to Definition 1.13
by introducing a triangular decomposition of glr. Firstly, we observe that
glr = slr ⊕ CI,
and so
glr = slr ⊕(I ⊗ C[t, t−1]
).
1. The Basic Representation of the Affine General Linear Lie Algebra 17
Thus, we can define a triangular decomposition of glr by extending the triangular
decomposition of slr. Namely, we write
glr = gl+
r ⊕ h⊕ gl−r ,
where
gl±r := sl
±r ⊕
(⊕k∈N+
C(I ⊗ t±k)
),
and h is the Cartan subalgebra with basis H0, H1, . . . , Hr−1, (I ⊗ 1). We can then
define Vbasic to be the unique irreducible representation that admits a vector v0 (high-
est weight vector) satisfying
gl+
r · v0 = 0, Hi · v0 = δi,0v0, (I ⊗ 1) · v0 = 0.
Our next task is to describe the various vertex operator realizations of Vbasic as
given in [26]. The motivation for these different realizations comes from the following
proposition.
Proposition 1.18. Let V be a representation of the 1-coloured Heisenberg algebra
such that c acts as the identity and there exists an N ∈ N such that
α(n1) · · ·α(nk) · v = 0,
for all v ∈ V and ni ∈ N+ such that n1 + · · · + nk > N . Then V is isomorphic to a
direct sum of copies of Λ.
Proof: This is [11, Lemma 14.4(b)] (where the fact that c acts as the identity
implies a = 1).
Suppose that g is an affine Lie algebra with underlying finite dimensional Lie
algebra g and V is a representation of g. Furthermore, suppose that we have a (1-
coloured) Heisenberg subalgebra s0 ⊆ g such that the restriction of V to s0 satisfies
1. The Basic Representation of the Affine General Linear Lie Algebra 18
the hypotheses of Proposition 1.18. Then, by Proposition 1.18, one can, in principle,
give a realization of V completely in terms of (a direct sum of copies of) Λ. That is,
V ∼=⊕i∈I
Λ,
as s0-modules, for some indexing set I. Define the vacuum space, Ω = Ω(V ) to be
the subspace
Ω :=v ∈ V | α(k) · v = 0 for all k ∈ N+
.
It is an easy exercise to see that 1ii∈I is a basis of Ω(⊕
i∈I Λ). Therefore,
V ∼= Ω(V )⊗ Λ,
as s0-modules, where the action of s0 on Ω(V )⊗ Λ is given by
α(k) · (v ⊗ f) = v ⊗ α(k) · f,
for all k ∈ Z − 0 and v ⊗ f ∈ Ω(V ) ⊗ Λ. Of course, in general, one has many
Heisenberg subalgebras of g, and so we naturally have several realizations of V . When
g is semisimple, a complete classification of the Heisenberg subalgebras of g is given
in [12, Section 9]. We briefly summarize the classification below.
Let W be the Weyl group of g. Fix a Cartan subalgebra h ⊆ g and an element
w ∈ W . Let m be the order of w. Recall that there is an action of W on h induced
by the action of W on the root space of g. Write
h =m−1⊕k=0
hk, hk := h ∈ h | w · h = e−2πik/mh.
There exists an x ∈ g such that Ad(exp 2πix)|h = w and [x, h0] = 0 (see, for instance,
[8, Section 14.3] and [4, Theorem 2.5.5]). For each h = (h0, h1, . . . , hm−1) ∈ h and
k ∈ Z, define h(k) to be the loop
h(k) = eiθk/m Ad(exp iθx)hk ∈ g, (1.7)
1. The Basic Representation of the Affine General Linear Lie Algebra 19
where k is the unique integer 0 ≤ k ≤ m− 1 such that k ≡ k mod m. Then
sw0 :=
(⊕k∈Z
h(k)
)⊕ Cc, (1.8)
is a maximal Heisenberg subalgebra of g. Let W ′ be a set of representatives of
the conjugacy classes of W . Then the subalgebras sw0 , w ∈ W ′, form a complete
nonredundant list of maximal Heisenberg subalgebras of g, up to conjugacy under
the adjoint action of the Lie group of g (see [12, Proposition of Section 9]). Hence,
the maximal Heisenberg subalgebras of g are parametrized by the conjugacy classes
of W .
Recall that the Weyl group of slr is the symmetric group Sr. Define a partition
of r of length s to be an s-tuple of positive integers (r1, . . . , rs) such that
r1 ≥ · · · ≥ rs, r1 + · · ·+ rs = r.
Every σ ∈ Sr can be written as a product of disjoint cycles, σ = c1 · · · cs, and so σ
defines a partition of r by setting ri equal to the length of the cycle ci (reordering
the ci if necessary). Two elements in Sr are conjugate if and only if they define the
same partition of r. Thus, conjugacy classes in Sr, and hence Heisenberg subalgebras
of slr, are parametrized by partitions of r.
We can extend these ideas to glr (since the Weyl group of glr is also Sr). That is,
to each partition of r we associate a Heisenberg subalgebra of glr as in (1.7) and (1.8).
The most well-known Heisenberg subalgebras of glr are the so-called principal Heisen-
berg subalgebra and the homogeneous Heisenberg subalgebra, which correspond to the
partitions (r) and (1, 1, . . . , 1), respectively. The principal Heisenberg subalgebra is
given by
α(n) =
(E0 + E1 + · · ·+ Er−1)n, for n > 0,
(F0 + F1 + · · ·+ Fr−1)−n for n < 0,
where the Ek and Fk are the Chevalley generators of slr (which, of course, is a sub-
algebra of glr) and exponentiation is meant to be taken in the underlying associative
1. The Basic Representation of the Affine General Linear Lie Algebra 20
algebra of glr. Setting
α(0) = I ⊗ 1,
the α(n), n ∈ Z, define a 1-coloured oscillator algebra. The homogeneous Heisenberg
subalgebra, presented as an r-coloured algebra, is given by
α`(n) = E`,` ⊗ tn,
for n ∈ Z− 0 and ` = 1, . . . , r. Setting
α`(0) = E`,` ⊗ 1,
defines an r-coloured oscillator algebra. The Heisenberg subalgebras associated to
the other partitions of r are “blends” of these two extremes. Namely, for an arbitrary
partition (r1, . . . , rs), we divide matrices in glr into s2 blocks, with the (i, j)-th block
of size ri × rj for all i, j = 1, . . . , s. The (`, `)-th diagonal block corresponds to the
subalgebra glr` ⊆ glr, and on the level of affine algebras, glr` ⊆ glr. Let E`k, F
`k , H
`k be
the Chevalley generators for slr` ⊆ glr` . The Heisenberg subalgebra corresponding to
the partition (r1, . . . , rs), presented as an s-coloured algebra, is given by
α`(n) =
(E`0 + E`
1 + · · ·+ E`r`−1)n, for n > 0,
(F `0 + F `
1 + · · ·+ F `r`−1)−n, for n < 0,
for ` = 1, . . . , s. It is worth noting that, restricted to the `-th colour, the α`(n)
determine the principal Heisenberg subalgebra of glr` . Set
α`(0) = I` ⊗ 1,
where I` is the identity matrix in glr` . Then the α`(n) define an s-coloured oscillator
algebra.
To each partition of r, we have associated a Heisenberg (and oscillator) subal-
gebra of glr, and, as mentioned before, each Heisenberg subalgebra gives rise to a
1. The Basic Representation of the Affine General Linear Lie Algebra 21
realization of Vbasic. These various realizations were described explicitly in [26] in
terms of vertex operators on bosonic and fermionic Fock space. We summarize the
main theorem of that paper here.
Conceptually, it is easier to begin with the principal realization (corresponding
to the partition (r)) before generalizing to the realizations corresponding to arbitrary
partitions of r. One begins by defining formal fermionic fields
ψ(z) :=∑k∈Z
ψ(k)zk, ψ∗(z) :=∑k∈Z
ψ∗(k)z−k,
where z is a formal variable. These are power series with coefficients in the (1-
coloured) Clifford algebra. Define the normal ordering on the Clifford algebra gener-
ators
: ψ(i)ψ∗(j) : =
ψ(i)ψ∗(j), if j > 0,
−ψ∗(j)ψ(i), if j ≤ 0.
We also define formal bosonic fields
α(z) :=: ψ(z)ψ(z) : .
Expanded as a Laurent series α(z) =∑
k∈Z α(k)z−k, one finds
α(k) =∑i∈Z
: ψ(i)ψ∗(i+ k) : .
Theorem 1.19. Let ω = e2πi/r. Then the homogeneous components of
: ψ(ωpz)ψ∗(ωqz) : − ωp−q
1− ωp−q, 1 ≤ p, q ≤ r, p 6= q,
: ψ(z)ψ∗(z) :,
together with the identity operator, span a Lie algebra of operators on F (1-coloured
fermionic Fock space) that is isomorphic to glr. Moreover, via this identification with
glr,
F(0) ∼= Vbasic,
1. The Basic Representation of the Affine General Linear Lie Algebra 22
as glr-modules and the α(k), k ∈ Z−0 (resp. k ∈ Z), together with the identity op-
erator, form a basis of the principal Heisenberg subalgebra (resp. oscillator subalgebra)
of glr.
Proof: This is a special case of Theorem 1.20 below.
Note that the isomorphism F(0)∼=−→ Vbasic in Theorem 1.19 is determined by
|0〉 7→ v0. Also, recall that the idea behind the realizations of Vbasic was to describe
Vbasic in terms of Λ (or bosonic Fock space), whereas Theorem 1.19 describes Vbasic
in terms of fermionic Fock space. However, via the boson-fermion correspondence
(see (1.3)), F(0) ∼= B(0), and so Theorem 1.19 satisfies our goal of describing Vbasic in
terms of bosonic Fock space.
We would now like to generalize Theorem 1.19 to an arbitrary partition of r. We
fix, once and for all, a partition,
r = (r1, . . . , rs),
of r. As with our construction of the Heisenberg subalgebra associated to r, we divide
matrices in glr into s2 blocks of size ri × rj. The operators associated to the (`, `)-th
diagonal blocks correspond to the subalgebra glr` and the `-th colour of the Heisenberg
subalgebra is the principal Heisenberg subalgebra of glr` . Thus, we can simply take s-
copies of the principal realization above, which amounts to taking s-coloured versions
of the Heisenberg and Clifford algebras as well as s-coloured versions of bosonic and
fermionic Fock space. The operators associated to the off-diagonal blocks can be
obtained by “mixing” Clifford algebra generators of different colours.
Let R′ = lcmr1, . . . , rs and define
R =
R′, if R′
(1ri
+ 1rj
)∈ 2Z for all i, j,
2R′, if R′(
1ri
+ 1rj
)/∈ 2Z for some i, j.
1. The Basic Representation of the Affine General Linear Lie Algebra 23
Define the normal ordering on the s-coloured Clifford algebra generators
: ψ`(i)ψ∗k(j) : =
ψ`(i)ψ∗k(j), if j > 0,
−ψ∗k(j)ψ`(i), if j ≤ 0.
We introduce the formal s-coloured fermionic and bosonic fields:
ψ`(z) :=∑k∈Z
ψ`(i)z(R/r`)k, ψ∗` (z) :=
∑k∈Z
ψ∗` (k)z−(R/r`)k,
α`(z) =∑k∈Z
α`(k)z−(R/r`)k := : ψ`(z)ψ∗` (z) :,
This brings us to the main theorem of [26] (as stated at the end of the introduction).
Theorem 1.20. Let ω = e2πi/R. Then the homogeneous components of
: ψ`(ωpz)ψ∗k(ω
qz) : −δk`ω(R/r`)(p−q)
1− ω(R/r`)(p−q),
(` 6= k, 1 ≤ p ≤ r`, 1 ≤ q ≤ rk)
or (` = k, 1 ≤ p 6= q ≤ r`),
: ψ`(z)ψ∗` (z) :, 1 ≤ ` ≤ s,
together with the identity operator, span a Lie algebra of operators on F (s-coloured
fermionic Fock space) that is isomorphic to glr. Moreover, via this identification with
glr,
F(0) ∼= Vbasic,
as glr-modules and the α`(k), k ∈ Z − 0 (resp. k ∈ Z), together with the identity
operator, give a basis of the Heisenberg subalgebra (resp. oscillator subalgebra) of glr
associated to the partition r.
As was the case with Theorem 1.19, the isomorphism F(0)∼=−→ Vbasic in Theorem
1.20 is determined by |0〉⊗s 7→ v0.
As stated at the outset, our goal is to give a geometric version of the various
realizations of Vbasic. Our strategy will be as follows. We first construct geometric
1. The Basic Representation of the Affine General Linear Lie Algebra 24
analogues the oscillator and Clifford algebra representations on bosonic and fermionic
Fock space. Our method for doing this will be similar to [15, Section 3]. Next, we
would like to construct an action of glr on our geometric fermionic Fock space in the
spirit of Theorem 1.20. To do this we recall a previous observation:
glr = slr ⊕(I ⊗ C[t, t−1]
),
and so glr is generated by the Chevalley generators Ek, Fk, Hk, k = 0, 1, . . . , r − 1
of slr and loops on the identity, I ⊗ tn, n ∈ Z. Thus, to define an action of glr on
our geometric Fock space, it is enough to define the action of Ek, Fk, Hk and I ⊗ tn.
Algebraically, the action of glr on fermionic Fock space is given by the homogeneous
components of vertex operators as in Theorem 1.20. We explicitly describe the action
of Ek, Fk and I⊗tn in the following Lemma 1.21 below (the action of Hk is determined
by Ek and Fk since Hk = [Ek, Fk]). First, we observe that for each k = 0, 1, . . . , r−1,
we can write
k = r1 + · · ·+ r`−1 + k′,
for unique 1 ≤ ` ≤ s and 0 ≤ k′ < r`−1.
Lemma 1.21. The action of glr on F from Theorem 1.20 is given by the map ρ :
glr → gl(F), described below. For each k = 0, 1, . . . , r−1, write k = r1+· · ·+r`−1+k′.
If k′ 6= 0, then
ρ(Ek) =∑i∈Z
ψ`(k′ + ir`)ψ
∗` (k′ + ir` + 1), (1.9)
ρ(Fk) =∑i∈Z
ψ`(k′ + ir` + 1)ψ∗` (k
′ + ir`). (1.10)
If k′ = 0 and ` 6= 1,
ρ(Ek) =∑i∈Z
ψ`−1((i+ 1)r`−1)ψ∗` (ir` + 1), (1.11)
ρ(Fk) =∑i∈Z
ψ`(ir` + 1)ψ∗`−1((i+ 1)r`−1). (1.12)
1. The Basic Representation of the Affine General Linear Lie Algebra 25
If k′ = 0 and ` = 1, (i.e. k = 0)
ρ(E0) =∑i∈Z
ψs(irs)ψ∗1(ir1 + 1), (1.13)
ρ(F0) =∑i∈Z
ψ1(ir1 + 1)ψ∗s(irs). (1.14)
Finally,
ρ(I ⊗ tn) =s∑`=1
α`(nr`) =
|n|∑s
`=1 r`P`(nr`), if n 6= 0,∑s`=1 P`(0), if n = 0.
(1.15)
Proof: We will prove Equation (1.9) and leave the remaining equations up to
the reader since they can be derived in a similar manner. The proof simply requires
picking out the appropriate components from the vertex operators constructed in [26].
As before, we decompose the r × r matrices of glr into s2 blocks of size ri × rj. For
1 ≤ i, j ≤ s and 1 ≤ p ≤ ri, 1 ≤ q ≤ rj, define Eijpq to be the matrix with 1 in the
(p, q)-th entry of the (i, j)-th block and zeros elsewhere. Let ωj = e2πi/rj be an rj-th
root of unity. As in [26, Equation (2.3.3)], define
Aijpq :=1√rirj
ri∑a=1
rj∑b=1
ωpai ω−qbj Eij
ab.
The Aijpq’s form a basis of gln. Thus, one can express the Eijpq’s in terms of the Aijpq’s
by [26, Equation (2.3.7)],
Eijpq =
1√rirj
ri∑a=1
rj∑b=1
ω−pai ωqbj Aijab. (1.16)
Now, as in [26, Equation (3.3.4)], define
hr =s∑`=1
r∑p=1
r` − 2p+ 1
2r`E``pp.
The element hr induces a ZR-gradation of glr,
glr =⊕n∈ZR
gl(n)r , gl(n)
r := x ∈ glr | [hr, x] =n+mR
Rx for some m ∈ Z,
1. The Basic Representation of the Affine General Linear Lie Algebra 26
where we use the notation n = n mod R. Then the elements
x(n) := e−iθ adhrei(n/R)θxn, x = (xn) ∈ glr, n ∈ Z, (1.17)
form a spanning set of glr (see [26, Equation (4.2.4)]). For each x ∈ glr and n ∈ Z,
define
x(n) := x(n)− δn0 tr(hrx)c, (1.18)
as in [26, Equation (4.2.9)]. The map ρ : glr → gl(F) is given explicitly in terms of
the elements Aijpq(n). Namely, we define the formal power series
Aijpq(z) :=∑n∈Z
ρ(Aijpq(n))z−n,
and let ρ be the map determined by setting
Aijpq(z) =z(R/2)((1/rj)−(1/ri))
√rirj
: ψi(ωpz)ψ∗j (ω
qz) : − 1
riδij
ω(R/ri)(p−q)
1− ω(R/ri)(p−q), if p 6= q,
(1.19)
Aijpp(z) :=z(R/2)((1/rj)−(1/ri))
√rirj
: ψi(ωpz)ψ∗j (ω
pz) : . (1.20)
Note that our equations vary slightly from [26, Equation 5.5.4] since we do not “ab-
sorb” the zR/2ri terms into our definition of the fermionic fields.
Let k = r1 + · · ·+ r`−1 + k′ ∈ 0, 1, . . . , r − 1 such that k′ 6= 0. Then
Ek = Ek,k+1 = E``k′,k′+1.
By [26, Equation (3.4.5)],
[hr, E``k′,k′+1] =
(k′ + 1
r`− k′
r`+
1
2r`− 1
2r`
)E``k′,k′+1 =
1
r`E``k′,k′+1,
and so E``k′,k′+1 ∈ gl(R/r`)r and, by Equations (1.17), (1.18) and (1.16),
Ek = E``k′,k′+1(R/r`) = E``
k′,k′+1(R/r`) =1
r`
r∑p,q=1
ω−(p−q)k′+q` A``pq(R/r`).
1. The Basic Representation of the Affine General Linear Lie Algebra 27
Now, ρ(A``pq(R/r`)) is equal to the coefficient of z−R/r` in the vertex operators in
Equations (1.19) and (1.20). A quick calculation then reveals that
ρ(A``pq(R/r`)) =1
r`
∑i∈Z
ω(p−q)i−q` ψ`(i)ψ
∗` (i+ 1).
Thus,
ρ(Ek) =1
r2`
r∑p,q=1
∑i∈Z
ω(i−k′)(p−q)` ψ`(i)ψ
∗` (i+1) =
1
r`
∑i∈Z
r`−1∑(p−q)=0
ω(i−k′)(p−q)` ψ`(i)ψ
∗` (i+1).
Since,
r`−1∑(p−q)=0
ω(i−k′)(p−q)` =
r`, if r` | (i− k′),
0, otherwise,
one finds,
ρ(Ek) =∑
r`|(i−k′)
ψ`(i)ψ∗` (i+ 1) =
∑i∈Z
ψ`(k′ + r`i)ψ
∗` (k′ + r`i+ 1).
Chapter 2
Geometric Invariant Theory
Our main geometric objects of interest will be the so-called Nakajima quiver varieties,
which are examples of geometric (invariant) quotients. The purpose of this chapter
will be to review some of the basic results from geometric invariant theory that we
will need before studying quiver varieties in the following chapter. The material in
this chapter is based largely on [19]. However, while [19] deals in full generality with
schemes over an arbitrary field, we will deal exclusively with algebraic varieties over
C.
Throughout this chapter and all subsequent chapters, whenever X is a set and
G is a group acting on X, we will denote the set of G-fixed points of X by XG.
Let X be an algebraic variety and let G an algebraic group acting on X, i.e. the
group action
σ : G×X → X,
(g, x) 7→ g · x,
is a morphism of varieties. Then for any G-invariant open set U ⊆ X, we have an
induced action of G on OX(U) given by
(g · f)(x) = f(g−1 · x),
28
2. Geometric Invariant Theory 29
for all g ∈ G, f ∈ OX(U) and x ∈ U . Note that if f ∈ OX(U)G, then f is constant
on the orbit G · x for each x ∈ U . Thus, f induces a well-defined function on the
orbit space U/G given by
f : U/G→ C,
G · x 7→ f(x).(2.1)
This leads us to the definition of a geometric quotient. Note that our definition is a
slightly simplified version of [19, Definition 0.6].
Definition 2.1 (Geometric quotient). Let X be an algebraic variety and let G an
algebraic group acting on X with group action σ : G × X → X. A pair (Y, φ),
where Y is an algebraic variety and φ : X → Y is a morphism of varieties, is called a
geometric quotient of X by G if the following conditions hold.
1. The diagram
G×X X
X Y
σ
p
φ
φ,
commutes, where the map p is the projection onto the second factor.
2. The map φ is surjective and the fibres of φ are precisely the G-orbits of X (i.e.
for all y ∈ Y , there exists an x ∈ X such that φ−1(y) = G · x).
3. The variety Y has the quotient topology induced by φ.
4. The structure sheaf of Y is given by
OY (U) =f : U → C | f ∈ OX(φ−1(U))G
,
where f is defined as in Equation (2.1).
2. Geometric Invariant Theory 30
Remark 2.2. Geometric quotients are not guaranteed to exist. However, if a geo-
metric quotient (Y, φ) of X by G does exist, then by (2) and (3) of Definition 2.1, it
is clear that, topologically, Y ∼= X/G. Hence, whenever a geometric quotient (Y, φ)
exists, we will identify it with (X/G, π), where π : X X/G is the canonical pro-
jection. Moreover, by a slight abuse of notation, we will simply refer to X/G as the
geometric quotient of X by G, leaving the map π implied.
We will, of course, also be interested in morphisms between geometric quotients.
The next lemma will prove useful in constructing such morphisms.
Lemma 2.3. Let X and Y be algebraic varieties and let GX and GY be algebraic
groups acting on X and Y , respectively, such that X/GX and Y/GY are geometric
quotients. If ϕ : X → Y is a morphism of varieties and ψ : GX → GY is a group
homomorphism such that
GY × Y Y
GX ×X X
ψ × ϕ ϕ , (2.2)
commutes (where the horizontal arrows represent the group action), then the induced
map
ϕ : X/GX → Y/GY ,
GX · x 7→ GY · ϕ(x),
is a well-defined morphism of varieties. Moreover, if ϕ and ψ are isomorphisms of
varieties and groups, respectively, then ϕ is an isomorphism of varieties.
Proof: The fact that ϕ is well-defined follows directly from the commutativity
of Diagram (2.2). Clearly, ϕ is continuous. Thus, to show that ϕ is a morphism of
varieties, it remains only to show that the mapping f 7→ f ϕ is a pullback
OY/GY (U)→ OX/GX (ϕ−1(U)),
2. Geometric Invariant Theory 31
for all open U ⊆ Y . Let πX : X X/GX and πY : Y Y/GY denote the canonical
projections. By definition of geometric quotients, this is equivalent to showing that
the mapping f 7→ f ϕ is a map
OY (π−1Y (U))GY → OX(π−1
X (ϕ−1(U))GX = OX(ϕ−1(π−1Y (U))GX ,
for all open U ⊆ Y . For any f ∈ OY (π−1Y (U))GY , x ∈ π−1
X (ϕ−1(U)) and g ∈ GX ,
(g · (f ϕ))(x) = f(ϕ(g−1 · x)) = f(ψ(g−1) · ϕ(x)) = f(ϕ(x)) = (f ϕ)(x),
where the second equality follows from Diagram (2.2) and the third equality follows
from the fact that f ∈ OY (π−1Y (U))GY . Therefore, we conclude that ϕ is a morphism
of varieties.
In the case that ϕ and ψ are isomorphisms, we have the following commutative
diagram:
GY × Y Y
GX ×X X
ψ−1 × ϕ−1 ϕ−1.
By repeating all the same arguments, we have a morphism of varieties
ϕ−1 : Y/GY → X/GX ,
GY · y 7→ GX · ϕ−1(y),
which is the inverse of ϕ. Thus, ϕ is an isomorphism.
We now turn our attention to tangent spaces, as they offer a very useful means
for studying the local behaviour of a variety. We begin by recalling the definition.
Let X be an algebraic variety. Recall that the stalk of x ∈ X is
OX,x := lim−→OX(U) (for open sets U 3 x)
2. Geometric Invariant Theory 32
= [f, U ] | open U 3 x and f ∈ OX(U),
where [f, U ] is the equivalence class of (f, U) under the equivalence relation (f, U) ∼
(f ′, U ′) if and only if there exists an open V ⊆ U∩U ′ such that x ∈ V and f |V = f ′|V .
By a slight abuse of notation, we will denote [f, U ] simply by f , leaving the open set
U implied. A derivation of OX,x is a linear map δ : OX,x → C such that
δ(fg) = δ(f)g(x) + f(x)δ(g),
for all f, g ∈ OX,x.
Definition 2.4. (Tangent space) Let X be an algebraic variety. For x ∈ X, the
tangent space of X at x is the set of derivations of OX,x, i.e.
Tx(X) := δ ∈ HomC(OX,x,C) | δ(fg) = δ(f)g(x) + f(x)δ(g), for all f, g ∈ OX,x.
If X is an affine variety, we may define the tangent space in more global terms
by replacing the stalk OX,x with the coordinate ring C[X] of X, i.e.
Tx(X) = δ ∈ HomC(C[X],C) | δ(fg) = δ(f)g(x) + f(x)δ(g), for all f, g ∈ C[X].
If X and Y are algebraic varieties and ϕ : X → Y is a morphism of varieties,
then we have an induced map on the stalks
ϕ∗ : OY,ϕ(x) → OX,x,
f 7→ f ϕ,
for each x ∈ X (in the affine case, we may replace the stalks OX,x and OY,ϕ(x) with
the coordinate rings C[X] and C[Y ]). This induces a map on the tangent spaces
dϕ : Tx(X)→ Tϕ(x)(Y ),
δ 7→ δ ϕ∗,
2. Geometric Invariant Theory 33
called the differential of ϕ (at x). If X is smooth, then we say that ϕ is etale if dϕ
is an isomorphism. Note that this is not the original definition of an etale morphism
(see [7, Definition 17.3.1]), but rather an equivalent characterization in the setting of
smooth varieties (see [7, Corollary 17.11.2]).
Let G be an algebraic group acting on X and let σ : G × X → X denote the
group action. Define σg := σ(g,−) : X → X and σx := σ(−, x) : G → X for all
g ∈ G and x ∈ X. We then have induced maps on the tangent spaces
dσg : Tx(X)→ Tg·x(X) and dσx : T1(G)→ Tx(X).
For sufficiently nice varieties, the following lemma allows us to compute the tangent
space of X/G.
Lemma 2.5. Let X be a smooth algebraic variety and let G be a reductive algebraic
group acting freely on X such that the geometric quotient X/G exists. Let π : X
X/G denote the canonical projection. Then, for all x ∈ X,
0→ T1(G)dσx−−→ Tx(X)
dπ−→ TG·x(X/G)→ 0,
is a short exact sequence.
Proof: Luna’s etale slice theorem (see [16, Section III, Theoreme du slice etale])
implies that, for all x ∈ X, there exists a subvariety V ⊆ X, which we may assume to
be smooth (see Remark 1 following the slice theorem in [16]), containing x such that
σ|G×V is etale (note that since G acts freely on X, the stabilizer Gx is trivial and the
orbit G · x is closed). Therefore,
dσ : T(1,x)(G× V )→ Tx(X),
is an isomorphism. Luna’s etale slice theorem also implies that the morphism σ,
induced from the following commutative diagram
2. Geometric Invariant Theory 34
G× V X
(G× V )/G ∼= V X/G,
σ
σ
where the vertical maps are the canonical projections, is etale. Thus,
dσ : TG·(1,x)((G× V )/G ∼= Tx(V )→ TG·x(X/G),
is an isomorphism. Notice that we have a commutative diagram:
Gi1
−−−→ G× Vπσ−−−− X/G
id
y σ
y id
yG
σx
−−−→ Xπ−− X/G,
where i1(g) = (g, x) for all g ∈ G. This induces the following commutative diagram
on the tangent spaces:
0 → T1(G)di1
−−−−→T(1,x)(G× V )
∼= T1(G)⊕ Tx(V )
dπdσ−−−−− TG·x(X/G) → 0
id
y dσ
y id
y0 → T1(G)
dσx
−−−−→ Tx(X)dπ−−− TG·x(X/G) → 0.
(2.3)
Clearly, the first row of Diagram (2.3) is a short exact sequence. Since the vertical
arrows of Diagram (2.3) are all isomorphisms, it follows that the second row is also a
short exact sequence.
Corollary 2.6. Let X and G be as in Lemma 2.5. Then, for all x ∈ X,
TG·x(X/G) ∼= Tx(X)/ im dσx.
Proof: By Lemma 2.5 and the first isomorphism theorem, TG·x(X/G) = im dπ ∼=
Tx(X)/ ker dπ = Tx(X)/ im dσx.
2. Geometric Invariant Theory 35
Let us keep the assumptions of Lemma 2.5. Suppose ϕ : X → X is a G-
equivariant morphism of varieties. Then we have a well-defined morphism
ϕ : X/G→ X/G,
G · x 7→ G · ϕ(x).
Let x ∈ X such that G · x ∈ X/G is a fixed point of ϕ. Then, there exists a unique
g ∈ G such that ϕ(x) = g−1 · x and the map ϕ induces a map on TG·x(X/G):
dϕ : TG·x(X/G)→ TG·x(X/G).
By Corollary 2.6, we may identify TG·x(X/G) with Tx(X)/ im dσx. Via this iden-
tification, dϕ induces a map on Tx(X)/ im dσx, which we compute in the following
lemma.
Lemma 2.7. Let X, G, ϕ, x and g be as above. Let
ψ : TG·x(X/G)→ Tx(X)/ im dσx,
be the isomorphism described in Corollary 2.6. Then we have well-defined maps
dϕ : Tx(X)/ im dσx → Tϕ(x)(X)/ im dσϕ(x),
δ + im dσx 7→ dϕ(δ) + im dσϕ(x),
and
dσg : Tϕ(x)(X)/ im dσϕ(x) → Tx(X)/ im dσx,
δ + im dσϕ(x) 7→ dσg(δ) + im dσx.
2. Geometric Invariant Theory 36
Moreover, the following diagram commutes.
TG·x(X/G) TG·x(X/G)
Tx(X)/ im dσx
Tϕ(x)(X)/ im dσϕ(x)
Tx(X)/ im dσx
dϕ
ψ ψ
dϕ dσg
(2.4)
In particular, the induced action of dϕ on Tx(X)/ im dσx is given by the dashed line
in Diagram (2.4)
Proof: Let τg : G→ G be the map τg(h) = ghg−1. Note that τg is a morphism of
varieties. We have the following commutative digram:
Gσx
−−−→ Xπ−− X/G
id
y ϕ
y ϕ
yG
σϕ(x)
−−−−→ Xπ−− X/G
τg
y σg
y id
yG
σx
−−−→ Xπ−− X/G,
which induces the following commutative diagram on the tangent spaces:
0 → T1(G)dσx
−−−−→ Tx(X)dπ−−− TG·x(X/G) → 0
id
y dϕ
y dϕ
y0 → T1(G)
dσϕ(x)
−−−−→ Tϕ(x)(X)dπ−−− TG·x(X/G) → 0
dτg
y dσg
y id
y0 → T1(G)
dσx
−−−−→ Tx(X)dπ−−− TG·x(X/G) → 0,
where every row is a short exact sequence by Lemma 2.5. Therefore, dϕ(im dσx) ⊆
im dσϕ(x) and dσg(im dσϕ(x)) ⊆ im dσx, thus proving that dϕ and dσg are well-defined.
2. Geometric Invariant Theory 37
One then easily verifies the commutativity of Diagram (2.4).
Remark 2.8. Let X and G be as in Lemma 2.7 and let H be an algebraic group
acting on X such that the G and H actions commute. Then each h ∈ H induces a
G-equivariant morphism of varieties ϕh : X → X, given by x 7→ h · x. Let x ∈ X
such that G · x ∈ (X/G)H . Then H acts on TG·x(X/G) by
h · δ = dϕh(δ),
for all h ∈ H and δ ∈ TG·x(X/G). By Corollary 2.6, we may identify TG·x(X/G) with
Tx(X)/ im dσx. For each h ∈ H, let g(h) ∈ G be the element such that h·x = g(h)−1·x.
By Lemma 2.7, the induced action of H on Tx(X)/ im dσx is given by
h · (δ + im dσx) = (dσg(h) dϕh)(δ + im dσx) =(dσg(h) dϕh
)(δ) + im dσx,
for all h ∈ H and δ + im dσx ∈ Tx(X)/ im dσx.
Let X be an algebraic variety (not necessarily smooth) and G an algebraic group
(not necessarily reductive) acting on X. Let g ∈ G and consider the morphism
φg : X → X ×X,
x 7→ (x, g · x).
By definition of a variety, the diagonal
∆(X) = (x, x) ∈ X ×X,
is a closed subset X ×X. Therefore,
φ−1g (∆(X)) = Xg,
is closed in X. Hence,
XG =⋂g∈G
Xg,
2. Geometric Invariant Theory 38
is a closed subvariety of X. For each g ∈ G, let
ϕg : X → X,
x 7→ g · x.
Recall that, for each x ∈ XG, the group G acts on Tx(X) via
g · δ = dϕg(δ),
for all g ∈ G and δ ∈ Tx(X). The following lemma shows that we may naturally
identify Tx(XG) with Tx(X)G.
Lemma 2.9. Let X be an algebraic variety and let G be an algebraic group acting on
X. Let
i : XG → X,
be the inclusion map. Then, for all x ∈ X, the differential di maps Tx(XG) isomor-
phically onto Tx(X)G.
Proof: Without loss of generality, we may assume that X is affine. Suppose
X ⊆ An and let C[X] and C[XG] be the coordinate rings of X and XG, respectively.
Both C[X] and C[XG] are quotients of the polynomial ring C[t1, . . . , tn]. For each
g ∈ G, we may consider the morphism ϕg to be an n-tuple (ϕ1g, . . . , ϕ
ng ) of polynomials
in C[X]. Then
C[XG] ∼= C[X]/J,
where J is the radical of the ideal generated by the set ϕig− ti | g ∈ G, i = 1, . . . , n.
In particular, ϕig+J = ti+J in C[XG] and so, ϕ∗g(f)+J = f+J for all f+J ∈ C[XG].
The pullback of i is the map
i∗ : C[X] C[XG],
f 7→ f + J.
2. Geometric Invariant Theory 39
Now, we consider the differential di : Tx(XG) → Tx(X). Since i∗ is surjective, it is
clear that di is injective. Thus, it remains only to show that di(Tx(XG)) = Tx(X)G.
Let δ ∈ Tx(XG). Then for any g ∈ G and f ∈ C[X],
(g · (di(δ)))(f) = δ(ϕ∗g(f) + J) = δ(f + J) = (di(δ))(f).
Thus, di(δ) ∈ Tx(X)G and so, di(Tx(XG)) ⊆ Tx(X)G. Conversely, let ε ∈ Tx(X)G.
Notice that, for all g ∈ G and i ∈ 1, . . . , n,
ε(ϕig − ti) = ε(ϕig)− ε(ti) = (g · ε)(ti)− ε(ti) = ε(ti)− ε(ti) = 0.
Thus, ε vanishes on the generators of J . Moreover, the generators of J vanish on x,
and so ε(J) = 0. Hence, we have a well-defined derivation
δ : C[XG]→ C,
f + J 7→ ε(f),
and di(δ) = ε. Therefore, Tx(X)G ⊆ di(Tx(XG)), completing the proof.
In the following chapters, we will commonly identify the tangent space of varieties
with the middle cohomology of certain complexes. In light of the previous lemma,
it will be useful for us to compute the fixed points of the middle cohomology of
complexes.
Lemma 2.10. Let G be a finite-dimensional torus or a finite group and let
Xα
−−−→ Yβ−− Z
be a complex of of G-modules such that α is injective and β is surjective. Let αG =
α|XG and βG = β|Y G. Then ker βG/ imαG ∼= (ker β/ imα)G as G-modules.
2. Geometric Invariant Theory 40
Proof: We have the following commutative diagram
XG αG−−−−→ Y G
βG−−− ZG
→
i
→ →
Xα
−−−→ Yβ−− Z,
where the vertical arrows represent the inclusion maps. Note that the restrictions
αG and βG remain injective and surjective, respectively. By commutativity of the
diagram, i induces a G-module morphism
i∗ : ker βG/ imαG → ker β/ imα,
y + imαG 7→ i(y) + imα = y + imα.
Suppose y+imαG ∈ ker i∗. This implies that y ∈ imα, and so we have a (unique)
x ∈ X such that α(x) = y. Because y ∈ Y G, for every g ∈ G we have
α(g · x) = g · α(x) = g · y = y = α(x).
Since α is injective, we have that g · x = x. Therefore, x ∈ XG and so y ∈ imαG.
Thus ker i∗ = 0 and i∗ is injective. Hence, by the First Isomorphism Theorem,
ker βG/ imαG ∼= im i∗.
We claim that im i∗ = (ker β/ imα)G, which would complete the proof of the
lemma. Clearly, im i∗ ⊆ (ker β/ imα)G, thus it remains only to check the reverse
inclusion. If G = (C∗)d is a d-dimensional torus, then we have a weight space decom-
position of X:
X =⊕k∈Zd
Xk, Xk := x ∈ X | (g1, . . . , gd)·x = gk11 · · · g
kdd x, for all (g1, . . . , gd) ∈ G.
We also have a weight space decomposition Y =⊕
k∈Zd Yk, where the Yk are defined in
the same way as the Xk. Suppose y = (yk) ∈ ker β such that y+imα ∈ (ker β/ imα)G.
Thus, for every g ∈ G, we have (g · y) − y ∈ imα. Moreover, since α is a morphism
of G-modules, α|Xk: Xk → Yk, and so
((g · y)− y)k = (gk11 · · · g
kdd − 1)yk ∈ imα,
2. Geometric Invariant Theory 41
for all k ∈ Zd and g = (g1, . . . , gd) ∈ G. For each k 6= 0, we may choose a g =
(g1, . . . , gd) ∈ G such that gk1 · · · gkd 6= 1, and thus yk ∈ imα. Therefore,
y + imα = y0 + imα ∈ im i∗,
since y0 ∈ Y G.
IfG is a finite group, we again suppose y ∈ ker β such that y+imα ∈ (ker β/ imα)G,
i.e. (g · y)− y ∈ imα for all g ∈ G. Thus,
1
|G|∑g∈G
(g · y − y) =
(1
|G|∑g∈G
g · y
)− y ∈ imα,
and so
y + imα =1
|G|∑g∈G
g · y + imα ∈ im i∗,
since 1|G|∑
g∈G g · y ∈ Y G.
We end this chapter with the following lemma, which gives a sufficient set of
conditions to determine when a morphism of varieties is an isomorphism.
Lemma 2.11. Let X and Y be smooth algebraic varieties and let ϕ : X → Y be a
bijective etale morphism. Then ϕ is an isomorphism (of varieties).
Proof: The Inverse Function Theorem found in [18, Theorem 5.31] implies that ϕ
is birational. Then, since ϕ is bijective and Y is smooth (in particular, Y is normal),
by a version of Zariski’s Main Theorem (see [20, Chapter III, Section 9, Theorem I]),
it follows that ϕ is an isomorphism.
Chapter 3
Quiver Varieties
The key result of this chapter is Theorem 3.20, in which we show that the Zm-fixed
point set of certain Nakajima quiver varieties is isomorphic to a disjoint union of
Nakajima quiver varieties of type Am−1. In Chapter 5, we will see that the equivariant
cohomology of these varieties will provide us with a suitable geometric analogue of
Vbasic.
Let Q = (Q0, Q1) be a quiver. For all ρ ∈ Q1, write t(ρ) and h(ρ) for the tail
and the head of ρ, respectively. Let Q = (Q0, Q1) be the double quiver of Q. That is,
Q0 = Q0, and Q1 = Q1 ∪Q1,
where Q1 is the set of arrows in Q1 with orientation reversed. We then have a natural
involution ¯ : Q1 → Q1, which sends every arrow in Q1 to its corresponding reverse
arrow in Q1 and vice versa. We define the function ε : Q1 → ±1 by
ε(ρ) =
1, if ρ ∈ Q1,
−1, if ρ ∈ Q1.
Let V =⊕
k∈Q0Vk and W =
⊕k∈Q0
Wk be Q0-graded complex vector spaces and let
42
3. Quiver Varieties 43
n = dimV and s = dimW . Define
EV =⊕ρ∈Q1
Hom(Vt(ρ), Vh(ρ)), LV,W =⊕k∈Q0
Hom(Vk,Wk).
We define a “multiplication” EV × EV → LV,V given by
(AB)k =∑
ρ∈Q1,t(ρ)=k
AρBρ,
for all A = (Aρ), B = (Bρ) ∈ EV . We then define
M = M(V,W ) = EV ⊕ LW,V ⊕ LV,W .
The function ε : Q1 → ±1 induces a function ε : EV → EV given by
ε(C)ρ = ε(ρ)Cρ.
We have a symplectic form ω on M given by
ω((C1, i1, j1), (C2, i2, j2)) = tr(ε(C1)C2) + tr(i1j2 − i2j1).
Let GV =∏
k∈Q0GL(Vk). Then GV acts on M via
g · (C, i, j) = (gCg−1, gi, jg−1),
for all g ∈ GV and (C, i, j) ∈M. Then GV is an algebraic group whose action on M
preserves the symplectic form ω. The moment map vanishing at the origin is given
by
µ : M→ LV,V ,
(C, i, j) 7→ ε(C)C + ij,
where the Lie algebra LV,V of GV is identified with its dual via the trace.
Remark 3.1. Note that the set M and the map µ do not depend on the orientation
of the quiver Q. Thus, our construction above, as well as the definition of quiver
varieties below, depend only on the underlying (undirected) graph of Q.
3. Quiver Varieties 44
Definition 3.2 (Invariant, Stable). Let S =⊕
k∈Q0Sk, where each Sk is a subspace
of Vk. For C ∈ EV , we say that S is C-invariant if Cρ(St(ρ)) ⊆ Sh(ρ) for all ρ ∈ Q1.
An element (C, i, j) ∈M is called stable if the following condition holds: if S is
a Q0-graded subspace of V such that S is C-invariant and im(i) ⊆ S, then S = V .
We denote by Mst the set of stable points of M.
Remark 3.3. Recall that a path (of length `) in a quiver is a sequence of arrows
ρ` · · · ρ2ρ1 such that h(ρk) = t(ρk+1). For any path p = ρ` · · · ρ2ρ1 and any C ∈ EV ,
let C(p) := Cρ` · · ·Cρ2 Cρ1 , which is a linear map Vt(ρ1) → Vh(ρ`). For (C, i, j) ∈M,
define
S(C, i, j) := spanC(p)i(xk) | p is a path starting at k, xk ∈ Wk, k ∈ Q0
.
Notice that (C, i, j) is stable if and only if S(C, i, j) = V . Indeed, suppose (C, i, j) is
stable. We have that S(C, i, j) is a Q0-graded subspace with k-th component Sk :=
S(C, i, j)∩Vk for all k ∈ Q0. Moreover, S(C, i, j) is C-invariant and im(i) ⊆ S(C, i, j),
thus S(C, i, j) = V .
Conversely, suppose S(C, i, j) = V . Let T be a C-invariant Q0-graded subspace
of V with im(i) ⊆ T . Then we have S(C, i, j) ⊆ T , and so T = V . Thus, (C, i, j) is
stable.
Lemma 3.4. The set Mst0 := µ−1(0) ∩Mst is a quasi-affine algebraic variety.
Proof: We will show that Mst (resp. µ−1(0)) is a Zariski open (resp. closed) subset
of M. By choosing bases for Vk and Wk for each k ∈ Q0, we may identify M with a
direct sum of matrix vector spaces, which can then be identified with affine space. Via
this identification, it is clear that µ−1(0) is the vanishing set of a set of polynomials
whose variables are the entries of the matrices defined by C, i and j, and thus µ−1(0)
is a Zariski closed set.
By Remark 3.3,
Mst = (C, i, j) ∈M | S(C, i, j) = V .
3. Quiver Varieties 45
Suppose x ∈ Wk, for some k ∈ Q0, such that i(x) 6= 0, and p = ρm · · · ρ2ρ1 is a path in
Q starting at k with m ≥ n = dimV . Let p` = ρ` · · · ρ2ρ1, for all 1 ≤ ` ≤ m. Choose
` minimal such that i(x), C(p1)i(x), . . . , C(p`)i(x) is linearly dependent. Then
C(p`)i(x) = a0i(x) + a1C(p1)i(x) + · · ·+ a`−1C
(p`−1)i(x),
for some aq ∈ C, q = 0, 1, . . . , ` − 1, where we may choose aq = 0 if h(ρq) 6= h(ρ`).
Therefore,
C(p)i(x) = (Cρm · · · Cρ`+1)(a0i(x) + a1C
(p1)i(x) + · · ·+ a`−1C(p`−1)i(x)),
and so C(p)i(x) ∈ spanC(p′)i(x) | p′ is a path of length smaller than m. Thus, by
induction,
S(C, i, j) = spanC(p)i(xk) | p is a path starting at k
of length less than n, xk ∈ Wk, k ∈ Q0.
For each k ∈ Q0 let i(k)` , 1 ≤ ` ≤ dimWk, denote the images under i of the basis
elements of Wk. Let X(C, i, j) be the matrix whose columns are C(pk)i(k)`k
, where we
take all the paths pk starting at k of length less that n, 1 ≤ `k ≤ dimWk, k ∈ Q0. By
the above argument, S(C, i, j) is the column space of X(C, i, j), and so, by basic linear
algebra, S(C, i, j) = V if and only if rk(X(C, i, j)) = n. The rank of a matrix may
be defined as the size of the largest (square) submatrix with nonzero determinant.
Hence, Mst may be described as
Mst = (C, i, j) ∈M | there is an n× n submatrix of X(C, i, j) with det 6= 0.
The complement of Mst, which we denote Munst (the set of unstable points), is thus
Munst = (C, i, j) ∈M | every n× n submtarix of X(C, i, j) has det = 0.
Under our identification with affine space, the determinant of a submatrix yields a
polynomial whose variables are the entries of the submatrix, hence Munst is a vanishing
3. Quiver Varieties 46
set of polynomials, and thus a Zariski closed set. Therefore, Mst is a Zariski open
set.
Therefore, since Mst0 is the intersection of an open subset and a closed subset of
M, it is a quasi-affine variety.
Lemma 3.5. The group GV acts freely on Mst.
Proof: Let (C, i, j) ∈Mst. Suppose g = (gk) ∈ GV such that g ·(C, i, j) = (C, i, j)
(i.e. gCg−1 = C, gi = i, and jg−1 = j) and let v ∈ V . By Remark 3.3, S(C, i, j) = V ,
thus we may write v as a finite sum:
v =∑q
cqC(pq)i(xkq),
where cq ∈ C, pq is a path in Q starting at kq, xkq ∈ Wkq , and kq ∈ Q0, for each q.
Applying g to v we get
g(v) = g
(∑q
cqC(pq)i(xkq)
)=∑q
cqgC(pq)i(xkq) =
∑q
cqC(pq)gi(xkq)
=∑q
cqC(pq)i(xkq) = v.
Therefore, g = 1 and thus GV acts freely on Mst.
Definition 3.6 (Nakajima Quiver Variety). The Nakajima quiver variety (associated
to Q) is the geometric quotient
M = M(V,W ) := Mst0 (V,W )/GV .
We will write [C, i, j]GV (or simply [C, i, j] when there is no risk of confusion) to
denote the GV -orbit of a point (C, i, j) ∈Mst0 .
3. Quiver Varieties 47
As stated in Remark 2.2, geometric quotients are not guaranteed to exist, thus
Defintion 3.6 requires some justification. The fact that there exists a geometric quo-
tient of Mst0 by GV is essentially proved by Nakajima in [21, Corollary 3.12], though
our definition of quiver varieties in Definition 3.6 is seemingly different from Naka-
jima’s definition in [21, Section 3.ii]. For convenience, we make precise the relation
between the two definitions.
Definition 3.7 (Costable). An element (C, i, j) ∈ M is costable if whenever a Q0-
graded C-invariant subspace S ⊆ V satisfies j(S) = 0, then S = 0. Denote by Mcost
the set of costable points of M and by Mcost0 the intersection Mcost ∩ µ−1(0).
Remark 3.8. Using arguments completely analogous to Lemmas 3.4 and Lemma
3.5, one can easily show that Mcost0 is a quasi-affine variety and GV acts freely on
Mcost0 .
Definition 3.9 (Dual Nakajima quiver variety). The dual Nakajima quiver variety
(associated to Q) is the quotient
M∗ = M∗(V,W ) := Mcost0 (V,W )/GV .
Remark 3.10. Note that in [21, Definition 3.9], Definition 3.7 is used as the definition
of stability. By [21, Corollary 3.12], the Nakajima quiver variety defined in [21, Section
3.ii] corresponds to our definition of the dual Nakajima quiver variety in Definition 3.9.
However the conditions of stability (in Definition 3.2) and costability (in Definition
3.7) are merely duals of each other, which yields an isomorphism of varieties between
Mst0 and Mcost
0 . This in turn will allow us to naturally identify M with M∗.
Lemma 3.11. There exists an isomorphism of varieties Mst0 (V,W )→Mcost
0 (V ∗,W ∗)
and a group isomorphism GV → GV ∗ such that the following diagram commutes:
3. Quiver Varieties 48
GV ∗ ×Mcost0 (V ∗,W ∗) Mcost
0 (V ∗,W ∗)
GV ×Mst0 (V,W ) Mst
0 (V,W )
,
where the horizontal arrows represent the group action.
Proof: Let (C, i, j) ∈M. For each Cρ ∈ Hom(Vt(ρ), Vh(ρ)), we have the transpose
map (Cρ)∗ ∈ Hom(V ∗h(ρ), V
∗t(ρ)), so we have a mapping EV → EV ∗ given by C 7→ C∗,
where
(C∗)ρ = (Cρ)∗,
for every ρ ∈ Q1. Moreover, since i ∈ LW,V and j ∈ LV,W , we have i∗ ∈ LV ∗,W ∗ and
j∗ ∈ LW ∗,V ∗ . Hence, we have a map
M(V,W )→M(V ∗,W ∗),
(C, i, j) 7→ (C∗, j∗, i∗).(3.1)
We claim that above mapping induces a map M(V,W ) → M∗(V ∗,W ∗). First, let
(C, i, j) ∈Mst0 . Note that ε(C∗)C∗ = C∗ε(C)∗. Indeed, for every ρ ∈ Q1,
ε(C∗)ρ(C∗)ρ = ε(ρ)(C∗)ρ(C
∗)ρ = ε(ρ)(Cρ)∗(Cρ)
∗ = (Cρ)∗(ε(C)ρ)
∗ = (C∗)ρ(ε(C)∗)ρ.
Thus, for each k ∈ Q0, the Hom(V ∗k , V∗k ) component of ε(C∗)C∗ is
(ε(C∗)C∗)k =∑
ρ∈Q1, t(ρ)=k
ε(C∗)ρ(C∗)ρ =
∑ρ∈Q1, t(ρ)=k
(C∗)ρ(ε(C)∗)ρ = (C∗ε(C)∗)k.
Hence, applying the moment map to (C∗, j∗, i∗) gives
µ(C∗, j∗, i∗) = ε(C∗)C∗ + j∗i∗ = C∗ε(C)∗ + j∗i∗ = (ε(C)C + ij)∗ = µ(C, i, j)∗ = 0,
where the final equality holds because (C, i, j) ∈ µ−1(0). Next we show that (C∗, j∗, i∗)
is costable. Indeed, suppose S =⊕
k∈Q0Sk ⊆ V ∗ such that S is C∗-invariant and
3. Quiver Varieties 49
i∗(S) = 0. For each k ∈ Q0, let Uk = v ∈ Vk | ϕ(v) = 0 for all ϕ ∈ Sk and set
U =⊕
k∈Q0Uk. For any ρ ∈ Q1, u ∈ Ut(ρ) and ϕ ∈ Sh(ρ) we have
ϕ(Cρ(u)) = (Cρ)∗(ϕ)(u) = (C∗)ρ(ϕ)︸ ︷︷ ︸
∈St(ρ)
(u) = 0.
Therefore, Cρ(u) ∈ Uh(ρ), and so U is C-invariant. Moreover, for any i(x) ∈ im(i)
and ϕ ∈ S,
ϕ(i(x)) = i∗(ϕ)︸ ︷︷ ︸=0
(x) = 0.
Thus, i(x) ∈ U , and so im(i) ⊆ U . Because (C, i, j) is stable, we have that U = V .
By construction of U , ϕ(V ) = ϕ(U) = 0 for every ϕ ∈ S, which means S = 0. Hence,
(C∗, j∗, i∗) is costable.
Therefore, the mapping (C, i, j) 7→ (C∗, j∗, i∗) is a mapping
Mst0 (V,W )→Mcost
0 (V ∗,W ∗).
Note that this mapping is a morphism of varieties (since by identifying M with affine
space, this map corresponds to permuting coordinates and is thus a regular map).
Furthermore, by a similar argument, one can show that the mapping
Mcost0 (V ∗,W ∗)→Mst
0 (V ∗∗,W ∗∗),
(C, i, j) 7→ (C∗, j∗, i∗),(3.2)
is also a morphism of varieties. Via the canonical isomorphisms V ∗∗ ∼= V and W ∗∗ ∼=
W , one has a natural isomorphism of varieties
Mst0 (V ∗∗,W ∗∗)
∼=−→Mst0 (V,W ). (3.3)
The composition of the maps (3.2) and (3.3) is the inverse of map (3.1). Thus, the
map (3.1) is an isomorphism of varieties Mst0 (V,W )
∼=−→Mcost0 (V ∗,W ∗). Note also that
under this mapping
g · (C, i, j) = (gCg−1, gi, jg−1) 7→((gCg−1)∗, (jg−1)∗, (gi)∗)
3. Quiver Varieties 50
= ((g∗)−1C∗g∗, (g∗)−1j∗, i∗g∗)
= (g∗)−1 · (C∗, j∗, i∗)
for all g ∈ GV and (C, i, j) ∈Mst0 (V,W ). Hence, we have a commutative diagram
GV ∗ ×Mcost0 (V ∗,W ∗) Mcost
0 (V ∗,W ∗)
GV ×Mst0 (V,W ) Mst
0 (V,W )
,
where the horizontal arrows represent the group action and the mapping GV ×
Mst0 (V,W )→ GV ∗ ×Mcost
0 (V ∗,W ∗) is given by (g, (C, i, j)) 7→ ((g∗)−1, (C∗, j∗, i∗)).
Corollary 3.12. The varieties M(V,W ) and M∗(V ∗,W ∗) are isomorphic (as vari-
eties).
Proof: This follows from Lemma 3.11 and Lemma 2.3.
Corollary 3.13. The varietes Mst0 and M are smooth.
Proof: By Corollary [21, Lemma 3.10, Corollary 3.12], we have that Mcost0 and
M∗ are smooth varieties. Thus, by Lemma 3.11 and Corollary 3.12, Mst0 and M are
also smooth varieties.
Let (C, i, j) ∈Mst0 . The tangent space of Mst
0 at (C, i, j) is
T(C,i,j)(Mst0 ) = T(C,i,j)(µ
−1(0)) = ker dµ,
3. Quiver Varieties 51
and one can easily check that the differential of the moment map µ at (C, i, j) is
dµ : M→ LV,V ,
(D, a, b) 7→ ε(C)D + ε(D)C + ib+ aj.
By Corollary 2.6, the tangent space T[C,i,j](M) can then be identified with the quo-
tient ker dµ/ im dσ(C,i,j). Here, σ(C,i,j) : GV → Mst0 is the map given by g 7→
(gCg−1, gi, jg−1), and so
dσ(C,i,j) : LV,V → T(C,i,j)(Mst0 ) = ker dµ,
X 7→ (XC − CX,Xi,−jX).
Lemma 3.14. Let (C, i, j) ∈Mst0 . Then
1. dσ(C,i,j) is injective and dµ (at (C, i, j)) is surjective, and
2. the tangent space of M at [C, i, j] may be identified with the middle cohomology
of the following complex:
LV,Vdσ(C,i,j)
−−−−→ EV ⊕ LW,V ⊕ LV,Wdµ−−− LV,V . (3.4)
Proof: The proof of (1) is completely analogous to [15, Lemma 3.2]. Part (2), as
above, simply follows from Corollary 2.6.
It is worth noting that, up to isomorphism, the varieties Mst0 (V,W ) and M(V,W )
are parametrized by the graded dimensions of V and W . Indeed, let v = (vk),w =
(wk) ∈ NQ0 and, for i = 1, 2, let V i, W i be Q0-graded vector spaces such that
dimV ik = vk and dimW i
k = wk for all k ∈ Q0. Fix Q0-graded vector space isomor-
phisms f : V 1 → V 2 and h : W 1 → W 2. Then we define maps
Mst0 (V 1,W 1)→Mst
0 (V 2,W 2),
3. Quiver Varieties 52
(C, i, j) 7→ (fCf−1, fih−1, hif−1),
and
GV 1 → GV 2 ,
g 7→ fgf−1.
One can easily check that these maps are isomorphisms of varieties and groups, re-
spectively. Thus, Mst0 (V 1,W 1) ∼= Mst
0 (V 2,W 2) and, by Lemma 2.3, M(V 1,W 1) ∼=
M(V 2,W 2). Hence, we define one standard representative for each isomorphism class:
Mst0 (v,w) := Mst
0
⊕k∈Q0
Cvk ,⊕k∈Q0
Cwk
,
and
M(v,w) := M
⊕k∈Q0
Cvk ,⊕k∈Q0
Cwk
= Mst0 (v,w)/Gv,
where Gv =∏
k∈Q0GLvk(C).
From now on, we restrict ourselves to the case where Q is a quiver of type Am−1,
m ∈ N+ (with the case m = 1 corresponding to the quiver consisting of one vertex
and one loop). We label the vertices of Q by 0, 1, . . . ,m − 1, and hence we can
identify Q0 = Q0 with Zm. In this case, we have that Q is the quiver:
1 2 · · · m− 1
0
We denote by M(m; v,w) (resp. M(m; v,w)) the variety M(v,w) (resp. M(v,w))
corresponding to a quiver of type Am−1 (recall that these varieties do not depend on
the orientation ofQ, thus M(m; v,w) and M(m; v,w) are well-defined). Of particular
interest to us will be the case m = 1, for which we introduce the special notation:
M(s, n) := M(1; (n), (s)), M(s, n) := M(1; (n), (s)).
3. Quiver Varieties 53
Note that in this case
M(s, n) = Hom(Cn,Cn)⊕ Hom(Cn,Cn)⊕ Hom(Cs,Cn)⊕ Hom(Cn,Cs),
and so we will denote elements of M(s, n) by (A,B, i, j) and the elements ofM(s, n)
by [A,B, i, j], where, by convention, A represents the linear map associated to the
loop in Q1 and B the linear map associated to the loop in Q1. The moment map µ
then simplifies to
µ(A,B, i, j) = [A,B] + ij.
Thus, by [22, Theorem 2.1], M(s, n) is isomorphic to the moduli space of framed
torsion-free sheaves on P2 with rank s and second Chern class c2 = n.
Fix an (s+ 1)-dimensional torus
T = (C∗)s × C∗,
and denote elements of T by (e, t), where e = (e1, . . . , es) ∈ (C∗)s and t ∈ C∗. We
have a natural action of (C∗)s on Cs given by
e · (w1, . . . , ws) = (e1w1, . . . , esws),
for all e ∈ (C∗)s and (w1, . . . , ws) ∈ Cs. For each c ∈ Zs, we have an action of T on
M(s, n) given by
(e, t) · (A,B, i, j) = (tA, t−1B, ie−1t−c, etcj),
where
tc := (tc1 , . . . , tcs).
The torus action preserves the space M st0 (s, n) and commutes with the action of
GLn(C), thus we have a well-defined action on M(s, n):
(e, t) · [A,B, i, j] = [tA, t−1B, ie−1t−c, etcj]. (3.5)
3. Quiver Varieties 54
LetMc(s, n) denote the moduli spaceM(s, n) with the torus action given by Equa-
tion (3.5).
We restrict our focus for now to the case s = 1. It is known that (A,B, i, j) ∈
M st0 (1, n) =⇒ j = 0 (see [22, Proposition 2.7]), thus we will simply write (A,B, i)
for an element (A,B, i, 0) ∈ M st0 (1, n). Let ω := e2π
√−1/m. The finite cyclic group of
order m acts on Mc(1, n) via the embedding
Zm → T,
k 7→ (1, ωk).(3.6)
That is
k · [A,B, i] = [ωkA, ω−kB,ω−kci],
for all k ∈ Zm and [A,B, i] ∈Mc(1, n). Recall that the fixed point setMc(1, n)Zm is
a closed subvariety of Mc(1, n) (see the discussion preceding Lemma 2.9). The goal
for the remainder of this chapter will be to describe Mc(1, n)Zm in terms of quiver
varieties of type Am−1.
Remark 3.15. Let
Fc(n) :=
(A,B, i) ∈M st0 (1, n) | [A,B, i] ∈Mc(1, n)Zm
,
i.e. we let Fc(n) be the preimage of Mc(1, n)Zm under the projection M st0 (1, n)
Mc(1, n). Thus, Fc(n) is a closed subvariety of M st0 (1, n). We have that (A,B, i) ∈
Fc(n) if and only if there exists a g ∈ GLn(C) such that
ωA = g−1Ag,
ω−1B = g−1Bg,
ω−ci = g−1i.
(3.7)
Note that since GLn(C) acts freely on M st0 , such a g, if it exists, is unique. Assume
that (A,B, i) ∈ Fc(n). By Remark 3.3, every v ∈ V may be written as a finite sum
v =∑p,q
λpqApBqi,
3. Quiver Varieties 55
where λpq ∈ C (note that A and B commute since the moment map implies that
[A,B] = 0) and we identify the map i : C→ Cn with i(1). For any p and q,
g(λpqApBqi) = ω−pλpqA
pgBqi = ωq−pλpqApBqgi = ωq−p+cλpqA
pBqi.
Thus, we have a weight space decomposition
Cn =⊕k∈Zm
Vk(A,B, i), Vk(A,B, i) := v ∈ Cn | gv = ωkv.
Note that by the above calculation, it is easily seen that
Vk(A,B, i) = spanApBqi | (q − p) ≡ (k − c) mod m. (3.8)
We observe that i ∈ Vc (where c is the equivalency class of c in Zm) and the restrictions
of A and B to Vk yield maps
A|Vk(A,B,i) : Vk(A,B, i)→ Vk−1(A,B, i) and B|Vk(A,B,i) : Vk(A,B, i)→ Vk+1(A,B, i).
Conversely, suppose we have a weight space decomposition Cn =⊕
k∈Zm Vk such that
A|Vk : Vk → Vk−1 and B|Vk : Vk → Vk+1 and i ∈ Vc. Set
g =∏k∈Zm
ωk idVk .
Then g−1Ag = ωA, g−1Bg = ω−1B and g−1i = ω−ci, and so (A,B, i) ∈ Fc(n).
Moreover, Vk(A,B, i) = Vk for each k ∈ Zm.
Lemma 3.16. Let (A,B, i) ∈ Fc(n) and h ∈ GLn(C). Then
Vk(h · (A,B, i)) = hVk(A,B, i),
for all k ∈ Zm.
Proof: Write
(A′, B′, i′) = h · (A,B, i) = (hAh−1, hBh−1, hi).
3. Quiver Varieties 56
Let g ∈ GLn(C) be the (unique) element satisfying (3.7) for (A,B, i). Let g′ = hgh−1.
Then
(g′)−1A′g′ = (hg−1h−1)(hAh−1)(hg−1h−1) = h(g−1Ag)h−1 = ω(hAh−1) = ωA′.
Similarly, (g′)−1B′g′ = ω−1B′. Moreover,
(g′)−1i′ = (hg−1h−1)(hi) = h(g−1i) = ω−chi = ω−ci′.
Hence, g′ is the unique element in GLn(C) satisfying (3.7) for (A′, B′, i′). Thus,
Vk(A,B, i) = v ∈ Cn | gv = ωkv, and Vk(A′, B′, i′) = v ∈ Cn | g′v = ωkv.
For each v ∈ Vk(A,B, i),
g′(hv) = (hgh−1)(hv) = h(gv) = ωk(hv),
and so hv ∈ Vk(A′, B′, i′). Thus, h is a linear map Vk(A,B, i) → Vk(A′, B′, i′). Since
h is invertible, Vk(A′, B′, i′) = hVk(A,B, i).
Lemma 3.17. 1. The variety Mc(1, n)Zm is smooth.
2. The tangent space of Mc(1, n)Zm at [A,B, i] may be identified with the middle
cohomology of the following complex:
Ldσ(A,B,i)
−−−−−→ E− ⊕ E+ ⊕ Hom(C, Vc)⊕ Hom(Vc,C)dµ−−− L, (3.9)
where L =⊕
k∈Zm Hom(Vk, Vk) and E± =⊕
k∈Zm Hom(Vk, Vk±1).
Proof: For Part (1), note that Zm is a compact Lie group, and thus, the category
of finite-dimensional linear representations of Zm is semisimple. Therefore, by [9,
Proposition 1.3], Mc(1, n)Zm is smooth. For Part (2), we first note that the tangent
3. Quiver Varieties 57
space of Mc(1, n) is obtained by setting V = Cn and W = C in Complex (3.4), i.e.
the tangent space of Mc(1, n) is the middle cohomology of
Hom(Cn,Cn)dσ(A,B,i)
−−−−−→Hom(Cn,Cn)⊕ Hom(Cn,Cn)
⊕
Hom(C,Cn)⊕ Hom(Cn,C)
dµ−−− Hom(Cn,Cn). (3.10)
Therefore, by Lemma 2.9, the tangent space ofMc(1, n)Zm ∼= (ker dµ/ im dσ(A,B,i))Zm .
Let Zm act on Cn by
ω · v = gv,
for all v ∈ V , where g is the element in GLn(C) satisfying properties (3.7) for (A,B, i).
Likewise, let Zm act on C by
ω · u = ωcu,
for all u ∈ C. Then Hom(Cn,Cn), Hom(C,Cn) and Hom(Cn,C) naturally become
Zm-modules. In order for dσ(A,B,i) and dµ to be Zm-module maps, we introduce the
one-dimensional modules ω and ω−1, (on which ω acts by multiplication by ω and
ω−1, respectively), and modify Complex (3.10) to
Hom(Cn,Cn)dσ(A,B,i)
−−−−−→ωHom(Cn,Cn)⊕ ω−1 Hom(Cn,Cn)
⊕
Hom(C,Cn)⊕ Hom(Cn,C)
dµ−−− Hom(Cn,Cn),
where we use juxtaposition to indicate the tensor product. One checks that the
action of Zm on ker dµ/ im dσ(A,B,i) induced by the action on Cn and C matches
the action described in Remark 2.8. Thus, by Lemma 2.10, (ker dµ/ im dσ(A,B,i))Zm ∼=
ker dµZm/ im dσ(A,B,i)Zm , where dµZm and dσ
(A,B,i)Zm are the restrictions of dµ and dσ(A,B,i)
to the fixed points of their respective domains. Now, f ∈ Hom(Cn,Cn)Zm if and only
if f = gfg−1, which occurs if and only if f |Vk : Vk → Vk. Therefore,
Hom(Cn,Cn)Zm ∼=⊕k∈Zm
Hom(Vk, Vk).
3. Quiver Varieties 58
Similarly, one can show that
(ω±1 Hom(Cn,Cn))Zm ∼=⊕k∈Zm
Hom(Vk, Vk∓1), Hom(C,Cn)Zm ∼= Hom(C, Vc),
and Hom(Cn,C)Zm ∼= Hom(Vc,C),
which completes the proof.
Let (C, i, j) ∈ M(m; v,1c). We identify⊕
k∈Zm Cvk with C|v| by identifying 1k`
with 1v0+···+vk−1+`, where 1k`vk`=1 and 1`|v|`=1 denote the standard bases of Cvk and
C|v|, respectively. Let AC , BC ∈ Hom(C|v|,C|v|
)be the maps determined by
(AC)|Cvk = ε(ρ)Cρ and (BC)|Cvk = Cτ , (3.11)
for each k ∈ Zm, ρ : k → k− 1 in Q1 and τ : k → k+ 1 in Q1 (note that in the m = 2
case, we simply make a choice as to which arrow of Q1 corresponds to k → k+ 1 and
which one corresponds to k → k − 1). We thus have a mapping
M(m; v,1c)→M(1, |v|),
(C, i, j) 7→ (AC , BC , i, j).(3.12)
Suppose that µ(C, i, j) = 0. For every vertex k ∈ Q0, we have four arrows incident
to k as in the following diagram.
k − 1 k k + 1α
α
β
β(3.13)
Thus, the Hom(Cvk ,Cvk) component of ε(C)C is
(ε(C)C)k = ε(α)CαCα + ε(β)CβCβ = −Cα(ε(α)Cα) + (ε(β)Cβ)Cβ = [AC , BC ]|Cvk .
Hence, ε(C)C = [AC , BC ]. Therefore,
[AC , BC ] + ij = ε(C)C + ij = µ(C, i, j) = 0.
3. Quiver Varieties 59
Now, suppose (C, i, j) is stable. Furthermore, suppose S is a subspace of C|v| such
that AC(S) ⊆ S, BC(S) ⊆ S and im(i) ⊆ S. For each k ∈ Zm, let Sk be the vector
space spanned by elements of the form:
D`1D`2 · · ·D`pi(1),
where each `i ∈ 1, 2, p ∈ N, D1 = AC , D2 = BC and
#`i | `i = 2 −#`i | `i = 1 ≡ k − c mod m.
Then S =⊕
k∈Zm Sk and, by construction, S is thus a C-invariant subspace of C|v|.
Since (C, i, j) is stable, S = C|v|. Therefore, (AC , BC , i, j) is stable. In particular, this
means that if (C, i, j) ∈Mst0 (m; v,1c), then (AC , BC , i, j) ∈M st
0 (1, |v|). In particular,
by [22, Proposition 2.7], this implies j = 0. Since this fact turns out to be rather
important, we highlight it with the following lemma.
Lemma 3.18. If (C, i, j) ∈Mst0 (m; v,w) with |w| = 1, then j = 0.
By the discussion above, the mapping (3.12) maps Mst0 →M st
0 . Thus, by Equa-
tion (3.11), Lemma 3.18 and Remark 3.15, we may define a morphism of varieties
ϕv : Mst0 (m; v,1c)→ Fc(|v|),
(C, i, 0) 7→ (AC , BC , i).
Lemma 3.19. The map ϕv induces a morphism of varieties
ϕv : M(m; v,1c)→Mc(1, |v|)Zm ,
[C, i, 0] 7→ [AC , BC , i].
In particular, one has a morphism of varieties ϕ :∐|v|=nM(m; v,1c)→Mc(1, n)Zm,
given by ϕ =∐|v|=n ϕv.
3. Quiver Varieties 60
Proof: Let ψ : Gv → GL|v|(C) be the group homomorphism induced by our
identification of⊕
k∈Zm Cvk with C|v|, i.e. partitioning GL|v|(C) into m2 blocks of size
vi × vj, i, j = 0, 1, . . . ,m − 1, we embed GLvk(C) into the (k, k)-th diagonal block
of GL|v|(C), for each k = 0, 1, . . . ,m − 1. One then has the following commutative
diagram:
Gv ×Mst0 (m; v,1c) Mst
0 (m; v,1c)
GL|v|(C)× Fc(|v|) Fc(|v|),
ψ × ϕ ϕ
where the horizontal arrows represent the group action. The result then follows by
Lemma 2.3.
Theorem 3.20. The map ϕ from Lemma 3.19 is an isomorphism. Therefore,
Mc(1, n)Zm ∼=∐|v|=n
M(m; v,1c),
as varieties.
Proof: We first begin by showing that ϕ is bijective. Suppose that (C, i, 0) ∈
Mst0 (m; v,1c) and (D, a, 0) ∈Mst
0 (m; u,1c), with |v| = |u| = n, are such that
ϕ[C, i, 0] = ϕ[D, a, 0],
i.e. [AC , BC , i] = [AD, BD, a]. Then there exists g ∈ GLn(C) such that (AC , BC , i) =
g · (AD, BD, a). For each k ∈ Zm,
Cvk = Vk(AC , BC , i) = g(Vk(AD, BD, a)) = g(Cuk),
where the second equality follows from Lemma 3.16. In particular, v = u. Moreover,
since g|Cvk : Cvk → Cvk , we may view g as an element of Gv. One then easily verifies
that (C, i, 0) = g · (D, a, 0). Thus, [C, i, 0] = [D, a, 0], and so ϕ is injective.
3. Quiver Varieties 61
Now, let (A,B, i) ∈ Fc(n). Set vk = dimVk(A,B, i) and choose a graded linear
isomorphism
f :⊕k∈Zm
Vk(A,B, i)→⊕k∈Zm
Cvk .
Then [A,B, i] = [fAf−1, fBf−1, fi]. Define (C, a, 0) ∈M (m; v,1c) by
Cρ =
ε(ρ)(fAf−1)|Cvk , if ρ : k → k − 1,
(fBf−1)|Cvk , if ρ : k → k + 1,
and a = fi,
for all ρ ∈ Q1. We claim that (C, a, 0) ∈Mst0 (m; v,1c). To show that µ(C, a, 0) = 0,
recall that for each k ∈ Zm, one has 4 arrows incident to k as in Diagram (3.13).
Thus, the Hom(Cvk ,Cvk)-component of ε(C)C is
(ε(C)C)k = ε(α)CαCα + ε(β)CβCβ
= ε(α)ε(α)(fBf−1)|Cvk−1 (fAf−1)|Cvk + ε(β)2(fAf−1)|Cvk+1 (fBf−1)|Cvk
= [fAf−1, fBf−1]|Cvk .
Thus, applying the moment map to (C, a, 0),
µ(C, a, 0) = ε(C)C = [fAf−1, fBf−1] = f [A,B]f−1 = 0,
where the last equality follows from the fact that µ(A,B, i) = 0. Therefore, (C, a, 0) ∈
µ−1(0). To show that (C, a, 0) is stable, suppose that S =⊕
k∈Zm Sk is a C-invariant
subspace of⊕
k∈Zm Cvk with a ∈ S. Then by construction, A(f−1(S)), B(f−1(S)) ⊆
f−1(S) and i ∈ f−1(S). Since (A,B, i) is stable, f−1(S) =⊕
k∈Zm Vk, and thus
S =⊕
k∈Zm Cvk . Hence, (C, a, 0) is stable. Therefore, (C, a, 0) ∈Mst0 (m; v,1c) and
ϕ[C, a, 0] = [fAf−1, fBf−1, fi] = [A,B, i].
Hence, ϕ is surjective.
Next, we show that ϕ is an etale morphism. Recall that we identify the tangent
spaces of M andMZm with the middle cohomologies of Complex (3.4) and Complex
3. Quiver Varieties 62
(3.9), respectively. By construction of ϕ, the induced map dϕ on the tangent spaces
is
dϕ : T[C,i,0](M)→ T[AC ,BC ,i](MZm),
(D, a, b) + im dσ(C,i,j) 7→ (D−, D+, a, b) + im dσ(AC ,BC ,i),
where (D−)k = ε(ρ)Dρ:k→k−1 and (D+)k = Dρ:k→k+1 for all k ∈ Zm. Clearly, dϕ is
injective. Moreover, by Lemma 3.14 and Lemma 3.17, T[C,i,0](M) and T[AC ,BC ,i](MZm)
have the same dimension, and hence dϕ is an isomorphism. By Lemma 2.11, ϕ is an
isomorphism.
Chapter 4
Vector Bundles and Geometric
Operators
In this chapter, we describe how to obtain so-called “geometric operators” on the
(localized) equivariant cohomology of smooth algebraic varieties; this method was
first introduced in [3] . The methods outlined in this chapter will serve as our main
tool for constructing our geometric versions of the Clifford, Heisenberg and Chevalley
operators in the next chapter. We do not review equivarient cohomology theory here,
but instead refer the reader to such expository papers as [27] or [2]. Our constructions
rely heavily on the Localization Theorem (see Theorem 4.1). The reader may wish
to consult [1, Appendix to Chapter 6] for more information on localization.
Let G = (C∗)d be a d-dimensional torus and, for each j = 1, . . . , d, we denote
the 1-dimensional G-module
(g1, . . . , gd) 7→ gj,
by gj. Let pt denote the space consisting of a single point equipped with the trivial
action of the torus G. Let tj denote the first Chern class of
gj → pt,
63
4. Vector Bundles and Geometric Operators 64
for each j = 1, . . . , d. Note that the tj are elements of degree 2. Recall that the
equivariant cohomology of pt is
H∗G(pt) = C[t1, . . . , td].
Let X be a space with a G-action. Then H∗G(X) is an H∗G(pt)-module. We consider
the localized equivariant cohomology of X:
H∗G(X) := H∗G(X)⊗C[t1,...,td] C(t1, . . . , td).
Unless otherwise noted, “cohomology” will always mean “localized equivariant coho-
mology”. Let
i : XG → X,
be the inclusion of the G-fixed points and let
p : XG pt .
The advantage of localized equivariant cohomology over nonlocalized equivariant co-
homology is that its study can be reduced to the cohomology of the G-fixed points.
We will only be interested in the case where X is a smooth variety with finitely many
G-fixed points. In this situation, we have the following theorem.
Theorem 4.1 (Localization Theorem). The following map is an isomorphism of
algebras:
H∗G(X)→ H∗G(XG) =⊕x∈XG
H∗G(pt),
α 7→(i∗x(α)
eG(Tx)
)x∈XG
,
(4.1)
where ix : x → X, Tx is the tangent space of x in X, and eG(Tx) is the equivariant
Euler class of Tx. The inverse of (4.1) is given by the Gysin map i∗ : H∗G(XG) →
H∗G(X).
4. Vector Bundles and Geometric Operators 65
Proof: This is a restatement of [5, Proposition 9.1.2] in the case that X has
finitely many fixed points.
Suppose now that X has real dimension 4n, for some n ∈ N. We define a bilinear
form 〈-, -〉X on the middle degree localized equivariant cohomology H2nG (X) by
〈a, b〉X := (−1)np∗(i∗)−1(a ∪ b), (4.2)
where i∗ is invertible by the Localization Theorem. We can extend this idea to a
product of varieties. Indeed, suppose X1, X2 are varieties of real dimension 4n1 and
4n2, respectively. We define a bilinear form 〈-, -〉X1×X2 on H2(n1+n2)G (X1 ×X2) by
〈a, b〉 := (−1)n2p∗((i1 × i2)∗)−1(a ∪ b), (4.3)
where i1 and i2 are the inclusions of the G-fixed points into the first and second
factors, respectively. An element α ∈ H2(n1+n2)G (X1 ×X2) then defines an operator
α : H2n1G (X1)→ H2n2
G (X2), (4.4)
by using the bilinear form to define structure constants:
〈α(x), y〉X2 := 〈α, x⊗ y〉X1×X2 .
Thus, an element α ∈ H2(n1+n2)G (X1 ×X2) will be called a geometric operator.
Recall that the torus T = (C∗)s × C∗ acts on Mc(s, n) via
(e, t) · [A,B, i, j] = [tA, t−1B, ie−1t−c, etcj], (4.5)
for all (e, t) ∈ T and [A,B, i, j] ∈ Mc(s, n). Let T• = C∗ and define an action of T•
on Mc(s, n) via the embedding
T• → T
z 7→ (1, z, z2, . . . , zs−1, 1).
4. Vector Bundles and Geometric Operators 66
Similar to Remark 3.15, one sees that [A,B, i, j] ∈ Mc(s, n)T• if and only if there
exists a group homomorphism g : T• → GLn(C) such that
g(z)−1Ag(z) = A,
g(z)−1Bg(z) = B,
g(z)−1i = i(1, z−1, . . . , z1−s),
jg(z) = (1, z, . . . , zs−1)j.
(4.6)
Define
V k = V k(A,B, i, j) := v ∈ Cn | g(z)v = zk−1v. (4.7)
By the stability of (A,B, i, j), we have that Cn =⊕s
k=1 Vk. Moreover,
A(V k), B(V k), i(C1k) ⊆ V k, and j(Vk) ⊆ C1k,
for all k = 1, . . . , s. Conversely, if there exists a decomposition Cn =⊕s
k=1 Uk such
that A(Uk), B(Uk), i(1k) ⊆ Uk and j(Uk) ⊆ C1k, then we may define a group
homomorphism g : T• → GLn(C) by defining g(z)|Uk = zk−1 idUk . One easily checks
that g satisfies the conditions of (4.6) and Uk = V k(A,B, i, j). Thus, [A,B, i, j] ∈
Mc(s, n)T• .
Now suppose n = (n1, . . . ,ns) ∈ Ns such that |n| = n. Define
Mc(n) :=Mc1(1,n1)× · · · ×Mcs(1,ns).
Identify⊕
k Cnk with Cn by identifying 1k` with 1n1+···+nk−1+`, where 1k`nk`=1 is the
standard basis of Cnk . An element ([A1, B1, i1], . . . , [As, Bs, is]) ∈Mc(n) then deter-
mines an element [A,B, i, 0] ∈Mc(s, n)T• by defining
A|Cnk := Ak, B|Cnk := Bk, i = i1 + · · ·+ is. (4.8)
One can then check that we have a well-defined map∐|n|=n
Mc(n)→Mc(s, n)T• ,
([A1, B1, i1], . . . , [As, Bs, is]) 7→ [A,B, i, 0],
(4.9)
4. Vector Bundles and Geometric Operators 67
where [A,B, i, 0] is defined as in (4.8).
Lemma 4.2. The map (4.9) is an isomorphism of varieties. Thus, Mc(s, n)T• ∼=∐|n|=nMc(n).
Proof: This is a straight-forward generalization of [23, Lemma 3.2].
We fix once and for all an r ∈ N+ and a partition of r of length s,
r := (r1, . . . , rs).
Let R′ = lcmr1, . . . , rs and define
R :=
R′, if R′
(1rk
+ 1r`
)∈ 2Z for all k, `,
2R′, otherwise.
Consider the product variety Mc(n) =Mc1(1,n1)× · · · ×Mcs(1,ns). Each compo-
nent Mc`(1,n`), for ` = 1, . . . , s, carries with it the action of a 2-dimensional torus
T = C∗ × C∗ (given by setting s = 1 in Equation (4.5)). Let T? = (C∗)s × C∗ and
define a T?-action on Mc`(1,n`) via the map
T? → T,
(e, t) 7→ (e`, tR/r`).
That is, T? acts on Mc`(1,n`) by
(e, t) ? [A`, B`, i`] = (e`, tR/r`) · [A`, B`, i`] = [tR/r`A`, t
−R/r`B, i`e−1` t−c`R/r` ],
for all (e, t) ∈ T? and [A`, B`, i`] ∈ Mc`(1,n`). Then T? acts on the product Mc(n)
by acting on each of its components, i.e.
(e, t) ? ([A1, B1, i1], . . . , [As, Bs, is]) = ((e, t) ? [A1, B1, i1], . . . , (e, t) ? [As, Bs, is]).
4. Vector Bundles and Geometric Operators 68
Lemma 4.3. The set Mc(n)T? is in one-to-one correspondence with the set
(I1, . . . , Is) | I` is a semi-infinite monomial of charge c` and energy n`.
Proof: We first prove that
Mc(n)T? =Mc1(1,n1)T × · · · ×Mcs(1,ns)T .
Let ([A1, B1, i1], . . . , [As, Bs, is]) ∈ Mc(n)T? . Fix ` ∈ 1, . . . , s. For all (e, t) ∈ T =
C∗ × C∗, choose ξ ∈ C∗ such that ξR/r` = t. Then
(e, t) · [A`, B`, i`] = ((1, . . . , e, . . . , 1), ξ) ? [A`, B`, i`] = [A`, B`, i`].
Therefore,Mc(n)T? ⊆Mc1(1,n1)T ×· · ·×Mcs(1,ns)T . The reverse inclusion follows
by construction of the action of T? on Mc(n).
Now, by [21, Proposition 2.9], Mc`(1,n`)T is in one-to-one correspondence with
the set of Young diagrams of size n`, which is itself in one-to-one correspondence with
the set of semi-infinite monomials of charge c` and energy n` by Lemma 1.9.
In light of Lemma 4.3, we will henceforth identify points ofMc(n)T? with s-tuples
of semi-infinite monomials.
Define an action of ZR on Mc(n) via the embedding
ZR → T?,
k 7→ (1, ωk),
where ω = e2π√−1/R. Then ZR acts on the `-th component,Mc`(1,n`), ofMc(n) via
the embedding
ZR → T,
k 7→ (1, ωR/r`) = (1, e2π√−1/r`).
4. Vector Bundles and Geometric Operators 69
Thus,
Mc(n)ZR =Mc1(1,n1)Zr1 × · · · ×Mcs(1,ns)Zrs ,
where the action of Zr` on Mc`(1,n`) is defined as in Equation (3.6). By Theorem
3.20, we know that Mc`(1,n`)Zr` ∼=
∐|v`|=n`
M(r`; v`,1c`). Hence, we define
Mc(v1, . . . ,vs) := M(r1; v1,1c1)× · · · ×M(rs; v
s,1cs),
and obtain
Mc(n)ZR ∼=∐|v`|=n`
Mc(v1, . . . ,vs).
We summarize the various fixed point varieties with the following diagram of inclu-
sions:
Mc(s, n) ⊇Mc(s, n)T• ∼=∐
nMc(n) ⊇∐
nMc(n)ZR ⊇∐
nMc(n)T? .
∼=∐v` Mc(v
1, . . . ,vs)
(4.10)
We now consider the localized T?-equivariant cohomology ofMc(n). Denote the
one-dimensional T?-modules
(e, t) 7→ ek, and (e, t) 7→ t,
by ek and t, respectively, and denote the tensor product of such modules by juxtapo-
sition. Moreover, we denote the first Chern classes of
ek 7→ pt, and t 7→ pt,
by bk and ε, respectively. Thus,
H∗T?(Mc(n)) = HT?(Mc(n))⊗C[b1,...,bs,ε] C(b1, . . . , bs, ε).
4. Vector Bundles and Geometric Operators 70
SinceMc(n) has real dimension 4|n|, we define a bilinear form 〈-, -〉n,c onH2|n|T?
(Mc(n))
as in (4.2), induced by
i :Mc(s, n)T? →Mc(n) and p :Mc(n)T? pt .
We extend to a bilinear form 〈-, -〉 on⊕
n,cH2|n|T?
(Mc(n)) ∼=⊕
n,cH2nT?
(Mc(s, n)T•
)by
〈-, -〉 :=∑n,c
〈-, -〉n,c.
One also defines a bilinear form 〈-, -〉n,m,c,d on H2(|n|+|m|)T?
(Mc(n) ×Md(m)) as in
Equation (4.3), which we extend to a bilinear form 〈-, -〉 on⊕
n,m,c,dH2(|n|+|m|)T?
(Mc(n)×
Md(m)) by
〈-, -〉 :=∑
n,m,c,d
〈-, -〉n,m,c,d.
Our goal will be to construct geometric operators on⊕
n,cH2|n|T?
(Mc(n)) as in Equa-
tion (4.4). In order to simplify computations, it will be useful for us to introduce
an orthonormal C(b1, . . . , bs, ε)-basis for H∗T?(Mc(n)). For each I ∈ Mc(n)T? , the
T?-action on Mc(n) induces an action on the tangent space TI = TI(Mc(n)). The
decomposition of TI into one-dimensional T? modules is given in the following lemma.
Lemma 4.4. Let I = (I1, . . . , Is) ∈Mc(n)T?. Then, as a T?-module,
TI ∼=s⊕`=1
⊕(i,j)∈λ(I`)
(t−hλ(I`)(i,j)R/r` ⊕ thλ(I`)(i,j)R/r`)
,
where λ(I`) is the Young diagram associated to I` and hλ(I) is the relative hook length
(see equations (1.2) and (1.1)).
Proof: We have that
TI(Mc(n)) ∼=s⊕`=1
TI`(Mc`(1,n`)).
4. Vector Bundles and Geometric Operators 71
The tangent space of Mc`(1,n`) at I` may then be computed by replacing t by tR/r`
in [15, Proposition 2.2].
It will be convenient to use the decomposition TI = T +I ⊕ T
−I , where
T ±I :=s⊕`=1
⊕(i,j)∈λ(I`)
t±hλ(I`)(i,j)R/r`
.
Lemma 4.5. For I ∈Mc(n)T?, the equivariant Euler classes of T +I and T −I are given
by
eT?(T +I ) =
s∏`=1
∏(i,j)∈λ(I`)
hλ(I`)(i, j)R
r`ε
,
eT?(T −I ) =s∏`=1
∏(i,j)∈λ(I`)
−hλ(I`)(i, j)R
r`ε
= (−1)|n|eT?(T +I ).
Proof: This follows directly from the definitions of T +I and T −I .
For each I ∈Mc(n)T? , let
[I] :=i∗(1I)
eT?(T −I )∈ H2|n|
T?(Mc(n)),
where 1I is the unit in H∗T?(pt) and eT?(T −I ) is to be interpreted as an invertible ele-
ment in this ring. Since the elements 1I form a C(b1, . . . , bs, ε)-basis of the cohomlogy⊕n,cH∗T?
(Mc(n)T?
), by the Localization Theorem (Theorem 4.1), the elements [I]
form a C(b1, . . . , bs, ε)-basis of⊕
n,cH∗T?(Mc(n)).
Lemma 4.6. The [I] are orthonormal with respect to the bilinear form 〈-, -〉.
Proof: The proof is completely analogous to [15, Proposition 2.4].
4. Vector Bundles and Geometric Operators 72
Define
A = spanC
[I] | I ∈Mc(n)T? , n ∈ Ns, c ∈ Zs. (4.11)
Then A is a full C-lattice in⊕
n,cH∗T?(Mc(n)). It will be useful for us to consider
the following gradings on A:
A =⊕n,c
Ac(n), Ac(n) :=
[I] | I ∈Mc(n)T?,
and
A =⊕c∈Z
A(c), A(c) :=
[I] | I ∈Mc(n)T? s.t. |c| = c.
Corollary 4.7. The restriction of 〈-, -〉 to A is non-degenerate and C-valued.
Proof: This follows directly from Lemma 4.6.
Remark 4.8. Notice that via the Kunneth formula
H∗T?(Mc(n)) ∼= H∗T?(Mc1(1,n1))⊗ · · · ⊗ H∗T?(Mcs(1,ns)),
the element [I] ∈ H∗T?(Mc(n)), where I = (I1, . . . , Is), maps to
[I1]⊗ · · · ⊗ [Is] ∈ H∗T?(Mc1(1,n1))⊗ · · · ⊗ H∗T?(Mcs(1,ns)).
Therefore,
Ac(n) ∼= Ac1(n1)⊗ · · · ⊗Acs(ns).
Let X be an algebraic variety with a T?-action, and let E → X be an T?-
equivariant vector bundle. We will denote the k-th equivariant Chern class of E by
ck(E). Note that ck(E) ∈ H2kT?
(X). The following lemma will act as our main tool in
constructing geometric operators in the following chapter.
4. Vector Bundles and Geometric Operators 73
Lemma 4.9. Let I ∈ Mc(n)T? and J ∈ Md(m)T?, let E be an equivariant vector
bundle on Mc(n)×Md(m) and let β ∈ H2kT?
(Mc(n)×Md(m)). Then
〈β ∪ c|n|+|m|−k(E)[I], [J]〉 =βI,J ∪ c|n|+|m|−k(E(I,J))
eT?(T −I )eT?(T +J )
,
where c|n|+|m|−k(E(I,J)) ∈ H∗T?(pt) = C[b1, . . . , bs, ε] is the polynomial given by the
equivariant Chern class of the fiber E over the point (I,J) and βI,J = i∗I,J(β), where
iI,J : (I,J) →Mc(n)×Md(m) is the inclusion of the fixed point.
Proof: See [15, Lemma 2.6].
Remark 4.10. The precise statement of [15, Lemma 2.6] differs slightly from ours
(since we use the variety Mc(n) under the action of T? rather than Mc(s, n) under
the action of T ). However, every step in the proof of [15, Lemma 2.6] applies to
Lemma 4.9; thus the two lemmas are essentially the same.
The next step will be to construct T?-equivariant vector bundles over Mc(n) ×
Md(m) whose Chern classes will define the appropriate geometric operators (as our
ultimate goal is to construct geometric versions of the Heisenberg, Clifford and Cheval-
ley operators from Chapter 1). We begin by defining vector bundles
Cn ×GLn(C) Mst0 (s, n)→Mc(s, n), and Cs ×Mc(s, n)→Mc(s, n),
which we denote by V = V(c, s, n) and W = W(c, s, n), respectively. Note that V
and W are simply the associated bundles of the trivial GLn(C)-bundles
Cn ×M st0 (s, n)→M st
0 (s, n), and Cs ×M st0 (s, n)→M st
0 (s, n).
The bundles V andW are T -equivariant with respect to the trivial action of T on Cn
and the natural action of T on Cs, respectively. Consider the Hom-bundle Hom(V ,V)
4. Vector Bundles and Geometric Operators 74
on Mc(s, n). We can define a global section s : Mc(s, n) → Hom(V ,V) by defining
s[A,B, i, j] to be the (well-defined) linear map
Cn ×GLn(C) [A,B, i, j]→ Cn ×GLn(C) [A,B, i, j],
GLn(C) · (v, (A,B, i, j)) 7→ GLn(C) · (Av, (A,B, i, j)).
By a slight abuse of notation, we will denote the section s by A. We similarly define
sections B, i and j of Hom(V ,V), Hom(W ,V) and Hom(V ,W), respectively.
One can extend this construction to a product of moduli spaces. The bun-
dle V(c, s, n) → Mc(s, n) may be extended to a vector bundle over the product
Mc(s, n)×Md(s,m) by:
V(c, s, n)×Md(s,m)→Mc(s, n)×Md(s,m).
We denote this vector bundle by V1 = V1(c,d, s, n,m). Likewise, we extend the bun-
dles W(c, s, n) → Mc(s, n), V(d, s,m) → Md(s,m) and W(d, s,m) → Md(s,m)
to bundles W1 = W1(c,d, s, n,m), V2 = V2(c,d, s, n,m) and W2 = W2(c,d, s, n,m)
over Mc(s, n) ×Md(s,m). We then have bundles Hom(Vk,Vk), Hom(Wk,Vk) and
Hom(Vk,Wk) as vector bundles over Mc(s, n) ×Md(s,m) with sections Ak, Bk, ik
and jk, where k = 1, 2. We define a three-term, T -equivariant complex of vector
bundles on Mc(s, n)×Md(s,m) by
Hom(V1,V2)ζ−→tHom(V1,V2)⊕ t−1 Hom(V1,V2)
⊕
Hom(W1,V2)⊕ Hom(V1,W2)
τ−→ Hom(V1,V2), (4.12)
where
ζ(X) =
XA1 − A2X
XB1 −B2X
Xi1
−j2X
, and τ
C
D
a
b
= A2D −DA1 + CB1 −B2C + i2b+ aj1.
4. Vector Bundles and Geometric Operators 75
Note that here the modules t±1 are the one-dimensional T -modules (e, t) 7→ t±1. One
verifies that τ ζ = 0.
Remark 4.11. We explain the presence of the “t” and “t−1” terms in Complex
(4.12); they are the result of the action of T on the section Ak and Bk. Let ζ1 :
Hom(V1,V2)→ Hom(V1,V2) be the first component of ζ. Denote the section “A1” by
s. To simplify notation, for every (I1, I2) ∈M st0 (s, n)×M st
0 (s,m), we will denote the
corresponding pair of orbits in Mc(s, n) ×Md(s,m) by (I1, I2). Moreover, we will
write s(I1,I2) := s(I1, I2) and v(I1,I2) := (GLn(C) · (v, I1), I2) for elements in the fiber
of V1 over the point (I1, I2). We wish to compute the action of T on s. For every
(e, t) ∈ T and (I1, I2) ∈M st0 (s, n)×M st
0 (s,m), with I1 = (A1, B1, i1, j1), the map
(e, t) · s(I1,I2) : (Cn ×GLn(C) I1)× I2 → (Cn ×GLn(C) I1)× I2,
is given by
((e, t) · s(I1,I2))(v(I1,I2)) = (e, t) · s(e,t)−1·(I1,I2)((e, t)−1 · v(I1,I2))
= (e, t) · s(e,t)−1·(I1,I2)(v(e,t)−1·(I1,I2))
= (e, t) · t−1Av(e,t)−1·(I1,I2) = t−1Av(I1,I2),
for all v ∈ Cn. Thus, the action of T on A1 (viewed as a section) is given by
(e, t) · A1 = t−1A1. Similarly, (e, t) · A2 = t−1A2. Therefore,
ζ1((e, t) ·X) = ((e, t) ·X)A1 − A2((e, t) ·X)
= (e, t) · (X((e, t)−1 · A1)− ((e, t)−1 · A2)X)
= (e, t) · (t(XA1 − A2X)) = t(e, t) · ζ1(X),
for all X ∈ Hom(V1,V2) and (e, t) ∈ T . Hence, ζ1 : Hom(V1,V2) → tHom(V1,V2) is
T -equivariant, thus justifying the extra factor “t” in Complex (4.12). One can show,
by a similar argument, that ζ and τ in Complex (4.12) are both T -equivariant.
4. Vector Bundles and Geometric Operators 76
Lemma 4.12. The cohomology ker τ/ im ζ of Complex (4.12) is a vector bundle on
Mc(s, n)×Md(s,m).
Proof: See [15, Lemma 3.2].
For n,m ∈ N, we will denote the vector bundle
ker τ/ im ζ →Mc(s, n)×Md(s,m),
by Kc,d(s, n,m). Notice that, by construction, one has the following vector bundle
on Mc(s, n)T• ×Md(s, n)T• :
(ker τ/ im ζ)T• →Mc(s, n)T• ×Md(s, n)T• ,
which we denote by Kc,d(s, n,m)T• . By Lemma 4.2,
Mc(s, n)T• ×Md(s,m)T• ∼=∐
|n|=n, |m|=m
Mc(n)×Md(m),
and so we may identify these varieties and consider the restriction of Kc,d(s, n,m)T• :
Kc,d(n,m) := Kc,d(s, n,m)T•|Mc(n)×Md(m),
which is a vector bundle on Mc(n) ×Md(m). On Mc`(1,n`) ×Md`(1,m`), the
product of the `-th components of Mc(n) and Md(m), one has the vector bundle
Kc`,d`(1,n`,m`). Let
f` :Mc(n)×Md(m)→Mc`(1,n`)×Md`(1,m`),
denote the canonical projection. Then the vector bundle pullback, f ∗`Kc`,d`(1,n`,m`),
is a vector bundle on the full product Mc(n)×Md(m).
Lemma 4.13. Let X,X1, X2 be smooth algebraic varieties, G,G1, G2 reductive alge-
braic groups acting freely on X,X1, X2, respectively, such that X/G,X1/G1, X2/G2
4. Vector Bundles and Geometric Operators 77
are smooth, and let V, V1, V2 be representations of G,G1, G2, respectively. Moreover,
suppose that there exists an embedding G1×G2 → G such that V ∼= V1⊕V2 as G1×G2-
representations and that there exists a G1×G2-equivariant morphism ϕ : X1×X2 → X
such that the induced morphism ϕ : X1/G1 ×X2/G2 → X/G is an isomorphism. Let
fi : X1/G1 ×X2/G2 → Xi/Gi denote the projection. Then the vector bundles
f ∗1 (V1 ×G1 X1)⊕ f ∗2 (V2 ×G2 X2)→ X1/G1 ×X2/G2, and V ×G X → X/G,
are isomorphic.
Proof: For simplicity, we will view G1 × G2 as a subgroup of G and identify V
with V1 ⊕ V2, as G1 ×G2-modules. Define
ψ : (V1 ×X1)⊕ (V2 ×X2)→ V ×X,
(v1, x1)⊕ (v2, x2) 7→ (v1 ⊕ v2, ϕ(x1, x2)).
It is clear that ψ is a morphism of varieties. Moreover, one checks that the following
diagram commutes:
(G1 ×G2)× ((V1 ×X1)⊕ (V2 ×X2)) (V1 ×X1)⊕ (V2 ×X2)
G× (V ×X) V ×X
i× ψ ψ,
where i : G1 ×G2 → G is the inclusion map and the horizontal arrows represent the
group action. So, by Lemma 2.3, we have an induced morphism ψ : (V1 ×G1 X1) ⊕
(V2 ×G2 X2)→ V ×G X, which yields the following commutative diagram:
(V1 ×G1 X1)⊕ (V2 ×G2 X2) V ×G X
X1/G1 ×X2/G2 X/G.
ψ
ϕ
4. Vector Bundles and Geometric Operators 78
Since, by assumption, ϕ is an isomorphism, it remains only to show that ψ is an
isomorphism and that it induces linear maps on the corresponding fibers. The latter
is obvious, and so we focus on the former. By virtue of X/G, X1/G1 and X2/G2 being
smooth, we have that (V1 ×G1 X1)⊕ (V2 ×G2 X2) and V ×G X are smooth. Thus, by
Lemma 2.11, it suffices to show that ψ is bijective and etale. To show bijectivity, it
suffices to look at the fibers. For every (G1 · x1, G2 · x2) ∈ X1/G1×X2/G2, the linear
map induced by ψ on the fibers is given by
(V1 ×G1 G1 · x1)⊕ (V2 ×G2 G2 · x2)→ V ×G G · ϕ(x1, x2),
G1 · (v1, x1)⊕G2 · (v2, x2) 7→ G · (v1 ⊕ v2, ϕ(x1, x2)),
which is clearly bijective. To show that ψ is etale, by Corollary 2.6,
TG1·(v1,x1)⊕G2·(v2,x2)
(⊕k=1,2
(Vk ×Gk Xk)
)∼=⊕k=1,2
T(vk,xk)(Vk ×Xk)/ im dσ(vk,xk)
∼=⊕k=1,2
(Tvk(Vk)⊕ Txk(Xk)) / im dσ(vk,xk),
and
TG·(v1⊕v2,ϕ(x1,x2))(V ×G X) ∼= T(v1⊕v2,ϕ(x1,x2))(V ×X)/ im dσ(v1⊕v2,ϕ(x1,x2))
∼=(T(v1⊕v2)(V )⊕ Tϕ(x1,x2)(X)
)/ im dσ(v1⊕v2,ϕ(x1,x2)).
The differential dψ is given by
((y1 ⊕ z1) + im dσ(v1,x1))⊕ ((y2 ⊕ z2) + im dσ(v2,x2))
7→ ((y1 ⊕ y2)⊕ dϕ(z1, z2)) + im dσ(v1⊕v2,ϕ(x1,x2)).
We will first show that dψ is surjective. Let (y′⊕z′)+im dσ(v1⊕v2,ϕ(x1,x2)) ∈ (Tv1⊕v2(V )⊕
Tϕ(x1,x2)(X))/ im dσ(v1⊕v2,ϕ(x1,x2)). Since dϕ is an isomorphism, there exists a (z1 +
im dσx1)⊕ (z1 + im dσx2) ∈ Tx1(X1)/ im dσx1 ⊕ Tx2(X2)/ im dσx2 such that
dϕ((z1 + im dσx1)⊕ (z1 + im dσx2)) = dϕ(z1, z2) + im dσϕ(x1,x2) = z′ + im dσϕ(x1,x2).
4. Vector Bundles and Geometric Operators 79
Equivalently, z′ − dϕ(z1, z2) = dσϕ(x1,x2)(h), for some h ∈ T1(G). We also note that
σ(v1⊕v2,ϕ(x1,x2)) : G→ V ×X is the restriction of
σv1⊕v2 × σϕ(x1,x2) : G×G→ V ×X,
to the diagonal of G×G. Hence,
(y′, z′) + dσ(v1⊕v2,ϕ(x1,x2)) = ((y′ − dσv1⊕v2(h))⊕ ϕ(z1, z2)) + dσ(v1⊕v2,ϕ(x1,x2)).
Thus, (y′, z′) + dσ(v1⊕v2,ϕ(x1,x2)) ∈ im dψ. By dimension counting, dψ is an isomor-
phism, and so ψ is etale.
Lemma 4.14. There is an isomorphism of vector bundles
Kc,d(n,m) ∼=s⊕`=1
f ∗` (Kc`,d`(1,n`,m`)).
Proof: We identify each point
([A11, B
11 , i
11], . . . , [As1, B
s1, i
s1], [A1
2, B12 , i
12], . . . , [As2, B
s2, i
s2]) ∈Mc(n)×Md(m),
with its image, ([A1, B1, i1, 0], [A2, B2, i2, 0]) ∈Mc(s, |n|)T•×Md(s, |m|)T• , under the
isomorphism given in Equation (4.9). By Lemma 4.13,
Vk|Mc(n)×Md(m)∼=
s⊕`=1
f ∗` V`k,
where V`k = Vk(c`,d`, 1,n`,m`). Likewise,
Wk|Mc(n)×Md(m)∼=
s⊕`=1
f ∗`W`k,
where W`k = Wk(c`,d`, 1,n`,m`). Thus, Kc,d(s, |n|, |m|) restricted to Mc(n) ×
4. Vector Bundles and Geometric Operators 80
Md(m) is the middle cohomology of
s⊕k,`=1
Hom(f ∗kVk1 , f ∗` V`2)ζres−−→
s⊕k,`=1
Hom(f ∗kVk1 , f ∗` V`2)⊕ Hom(f ∗kVk1 , f ∗` V`2)
⊕
Hom(f ∗kWk1 , f
∗` V`2)⊕ Hom(f ∗kVk1 , f ∗`W`
2)
τres−−→
s⊕k,`=1
Hom(f ∗kVk1 , f ∗` V`2),
(4.13)
where ζres and τres are the restrictions of ζ and τ to the indicated domains (note
that since we only consider the action of T•, the “t” and “t−1” terms from Complex
(4.12) may be safely omitted). It is easy to see that we have the following equality of
sections:
Ak =s⊕`=1
f ∗`A`k, Bk =
s⊕`=1
f ∗`B`k, ik =
s⊕`=1
f ∗` i`k,
for k = 1, 2. Therefore, Complex (4.13) decomposes as a direct sum of complexes:
s⊕k,`=1
Hom(f ∗kVk1 , f ∗` V`2)ζk`−→
Hom(f ∗kVk1 , f ∗` V`2)⊕ Hom(f ∗kVk1 , f ∗` V`2)
⊕
Hom(f ∗kWk1 , f
∗` V`2)⊕ Hom(f ∗kVk1 , f ∗`W`
2)
τk`−→ Hom(f ∗kVk1 , f ∗` V`2)),
where
ζk`(X) =
Xf ∗k (Ak1)− f ∗` (A`2)X
Xf ∗k (Bk1 )− f ∗` (B2)X
Xf ∗k (ik1)
0
, and
τk`
C
D
a
b
= f ∗` (A`2)D −Df ∗k (Ak1) + Cf ∗k (Bk1 )− f ∗` (B`
2)C + f ∗` (i`2)b.
4. Vector Bundles and Geometric Operators 81
We therefore have that
Kc,d(n,m) =s⊕
k,`=1
(ker τk`/ im ζk`)T• .
The bundle ker τk`/ im ζk` is the vector bundle pullback of Kck,d`(1,nk,m`) by the
projection
Mc(n)×Md(m)→Mck(1,nk)×Md`(1,m`).
Thus the task of computing (ker τk`/ im ζk`)T• , reduces to computingKck,d`(1,nk,m`)
T• .
The T•-fixed points of Kck,d`(1,nk,m`) may be computed (as a set) by comput-
ing the T•-fixed points of the fibers. Over the point ([Ak1, Bk1 , i
k1], [A`2, B
`2, i
`2]) ∈
Mck(1,nk)×Md`(1,m`), we may identify the fiber of Vk1 with Cnk by
(Cnk ×GLnk(C) (Ak1, B
k1 , i
k1))× [A`2, B
`2, i
`2]→ Cnk ,(
GLnk(C) · (v, (Ak1, Bk1 , i
k1)), [A`2, B
`2, i
`2])7→ v.
Similarly, we identify the fiber of V`2 with Cm` , the fiber of Wk1 with C1k and the
fiber of W`2 with C1`. Via these identifications, the fiber of Kc,d(n,m) is the middle
cohomology of
Hom(Cnk ,Cm`)ζk`−→
Hom(Cnk ,Cm`)⊕ Hom(Cnk ,Cm`)
⊕
Hom(C1k,Cm`)⊕ Hom(Cnk ,C1`)
τk`−→ Hom(Cnk ,Cm`),
where here we view ζk` and τk` as linear maps. By Lemma 2.10, (ker τk`/ im ζk`)T• ∼=
ker τk`|T•/ im ζk`|T• , where ζk`|T• and τk`|T• are the restrictions of ζk` and τk` to the
T•-fixed points of their respective domains. Via our identification of Vk1 with Cnk , we
have that T• acts on Cnk via multiplication by zk−1 for all z ∈ T•. That is,
z · v = zk−1v,
for all z ∈ T• and v ∈ Cnk . Likewise, T• acts on Cm` by multiplication by z`−1, on
C1k by multiplication by zk−1, and on C1` by multiplication by z`−1, for all z ∈ T•.
4. Vector Bundles and Geometric Operators 82
Therefore, for all z ∈ T• and f ∈ Hom(Cnk ,Cm`),
z · f = zk−`f.
Thus,
Hom(Cnk ,Cm`)T• =
Hom(Cnk ,Cmk), if k = `,
0, if k 6= `.
Similarly, Hom(C1k,Cm`) and Hom(Cnk ,C1`) are zero when k 6= `. Since the fibers
of Kck,d`(1,nk,m`)T• are all zero when k 6= `, we conclude that Kck,d`(1,nk,m`)
T• = 0
for all k 6= `. Therefore,
Kc,d(n,m) =s⊕`=1
f ∗`Kc`,d`(1,n`,m`).
From our diagram of inclusions, (4.10), we have that
Mc(n)×Md(m) ⊇Mc(n)ZR ×Md(m)ZR
∼=∐|v`|=n`|u`|=m`
Mc(v1, . . . ,v`)×Md(u1, . . . ,us).
We may thus view Mc(v1, . . . ,vs) × Md(u1, . . . ,us) as a subvariety of Mc(n) ×
Md(m), and consider the restriction of Kc,d(n,m):
Kc,d(v1,u1, . . . ,vs,us) := Kc,d(n,m)|Mc(v1,...,vs)×Md(u1,...,us),
which is a vector bundle on Mc(v1, . . . ,vs)×Md(u1, . . . ,us). In the following section,
the vector bundles Kc,d(n,m) and Kc,d(v1,u1, . . . ,vs,us) will allows us to construct
geometric versions of the Heisenberg algebra, the Clifford algebra and glr.
Chapter 5
Geometric Realizations of the
Basic Representation
In this chapter, we present the main theorem of this paper, Theorem 5.21, which
describes our geometric realizations of the basic representation of glr. We will do so
by constructing the oscillator algebra, the Clifford algebra and glr as geometric oper-
ators (in the sense of (4.4)). Using the Localization Theorem, we will consider these
geometric operators as operators on the same cohomology. We will show that these
operators satisfy the same relations as those observed by their algebraic counterparts
in Lemma 1.21, and that the cohomology on which they act (or more specifically the
full C-lattice A) corresponds naturally to fermionic Fock space.
Throughout this chapter, for any T?-equivariant vector bundle E, we let ctnv(E)
denote the top nonvanishing T?-equivariant Chern class of E. We will also frequently
make use of the Kunneth formula
H∗T?(X × Y ) ∼= H∗T?(X)⊗H∗T?(Y ),
and simply identify the two rings.
We begin by constructing the geometric version of the oscillator/Heisenberg alge-
bra. The dimension ofMc(n) is 4|n|, thus elements of H2(|n|+|m|)T?
(Mc(n)×Md(m))
83
5. Geometric Realizations of the Basic Representation 84
will define operators H2|n|T?
(Mc(n)) → H2|m|T?
(Md(m)). Consider the vector bundle
Kc,d(n,m) overMc(n)×Md(m). The rank of Kc,d(n,m) is |n|+ |m|, which can be
seen from the following lemma.
Lemma 5.1. Let (I,J) ∈ Mc(n)T? ×Md(m)T?. The equivariant Euler class of the
restriction of Kc,d(n,m) to (I,J) is
eT?(Kc,d(n,m)(I,J)) =s∏`=1
∏x∈λ(I`)
(d` − c` − hλ(I`),λ(J`)(x))R
r`ε
×
∏y∈λ(J`)
(d` − c` + hλ(J`),λ(I`)(y))R
r`ε
.
Proof: By Lemma 4.14,
Kc,d(n,m) ∼=s⊕`=1
f ∗` (Kc`,d`(1,n`,m`)),
and so, by properties of the Euler class,
eT?(Kc,d(n,m)) =s∏`=1
f ∗` (eT?(Kc`,d`(1,n`,m`)))
=s∏`=1
(1⊗`−1 ⊗ eT?(Kc`,d`(1,n`,m`))⊗ 1⊗s−`
).
Now, by setting r = 1 and replacing ε by (R/r`)ε in [15, Lemma 3.3], we have that
eT?(Kc`,d`(1,n`,m`)(I`,J`)) =
∏x∈λ(I`)
(d` − c` − hλ(I`),λ(J`)(x))R
r`ε
×
∏y∈λ(J`)
(d` − c` + hλ(J`),λ(I`)(y))R
r`ε
.
The result follows.
5. Geometric Realizations of the Basic Representation 85
Recall from [15, Section 3.3], that the top nonvanishing Chern class ofKc,c(1, n,m)
is
ctnv(Kc,c(1, n,m)) =
c2n(Kc,c(1, n, n)), if n = m,
cn+m−1(Kc,c(1, n,m)) if n 6= m.
(5.1)
By Lemma 4.14, the top nonvanishing Chern class of Kc,c(n,m) may be computed
by
ctnv(Kc,c(n,m)) = ctnv
(s⊕`=1
f ∗` (Kc`,c`(1,n`,m`))
)=
s∏`=1
f ∗` (ctnv(Kc`,c`(1,n`,m`)))
=s∏`=1
(1⊗`−1 ⊗ ctnv(Kc`,c`(1,n`,m`))⊗ 1⊗s−`
).
Therefore, if n = m,
ctnv(Kc,c(n,n)) = c2|n|(Kc,c(n,n)),
whereas if n differs from m in exactly one component,
ctnv(Kc,c(n,m)) = c|n|+|m|−1(Kc,c(n,m)).
For each ` = 1, . . . , s, define α` ∈ H∗T?(Mc`(1,n`)T?×Md`(1,m`)
T?) =⊕
(I,J)H∗T?(pt)
to be the element with (I, J)-th component
α`(I,J) =
ε
eT? (T(I,J)), if n` = m`,
0, otherwise,
where T(I,J) is the tangent space of (I, J) inMc`(1,n`)×Md`(1,m`). Let α` := i∗(α`),
where i : Mc`(1,n`)T? × Md`(1,m`)
T? → Mc(1,n`) × Md(1,m`) is the natural
inclusion. Denote by iI,J the inclusion (I, J) →Mc(1,n`)×Md(1,m`). Then by
[5, Equation (9.3)],
i∗I,J(α`) = (i∗I,J i∗)(α`) = eT?(T(I,J)) ∪ α`(I,J) =
ε, if n` = m`,
0, otherwise.
Let γ` := f ∗` (α`) = 1⊗`−1 ⊗ α` ⊗ 1⊗s−` ∈ H∗T?(Mc(n)×Md(m)). Note that γ` is an
element of degree 2.
5. Geometric Realizations of the Basic Representation 86
Definition 5.2 (Geometric oscillator/Heisenberg operators). For ` = 1, . . . , s and
k ∈ Z, define operators
P`(k) :⊕n,c
H2|n|T?
(Mc(n))→⊕n,c
H2|n|T?
(Mc(n)),
by
P`(k)|H2|n|T?
(Mc(n))=
(R/r`)γ` ∪ ctnv(Kc,c(n,n− k1`)), if k < 0,
−(R/r`)γ` ∪ ctnv(Kc,c(n,n− k1`)), if k > 0,
∈ H2(2|n|−k)T (Mc(n)×Mc(n− k1`))
P`(0)|H2|n|T?
(Mc(n))= c` · ctnv(Kc,c(n,n)) = c` id .
The P`(k) will be called geometric oscillators (or, for k 6= 0, geometric Heisenberg
operators).
Theorem 5.3. The operators P`(k) preserve the space A and satisfy the commutation
relations
[P`(k),Pm(0)] = 0, and [P`(k),Pm(j)] =1
kδ`,mδk+j,0 id, k 6= 0.
In particular, the mapping
P`(k) 7→ P`(k),
defines a representation of the s-coloured oscillator algebra on A and the linear map
determined by
[I] 7→(sλ(I1) ⊗ qc(I1)
)⊗ · · · ⊗
(sλ(Is) ⊗ qc(Is)
),
is an isomorphism of s-coloured oscillator algebra representations A → B. This
isomorphism maps A(c)→ B(c), for all c ∈ Z.
Proof: Take m = n− k1`. We know that
ctnv(Kc,c(n,m)) =∏s
i=1 (1⊗i−1 ⊗ ctnv(Kci,ci(1,ni,mi))⊗ 1⊗s−i) .
5. Geometric Realizations of the Basic Representation 87
By [15, Lemma 3.10], for i 6= `, we have that ctnv(Kci,ci(1,ni,ni)) is the identity as an
operator H2niT?
(Mci(1,ni))→ H2niT?
(Mci(1,ni)), i.e. ctnv(Kci,ci(1,ni,mi)) = 1. Thus,
ctnv(Kc,c(n,m)) = 1⊗`−1 ⊗ ctnv(Kc`,c`(1,n`,n` − k))⊗ 1⊗s−`.
Therefore, via the Kunneth formula,
⊕n,cH
2|n|T?
(Mc(n)) ∼=⊕
n,cH2n1T?
(Mc1(1,n1))⊗ · · · ⊗ H2nsT?
(Mcs(1,ns)),
we have
P`(k)|H2|n|T?
(Mc(n))= 1⊗(`−1) ⊗ (R/r`)α
` ∪ ctnv(Kc`,c`(1,n`,n` − k))⊗ 1⊗(s−`).
By [15, Theorem 3.14], for a fixed ` ∈ 1, . . . , s, the operators P`(k) preserve the
space⊕
n`,c`Ac`(n`) (see Remark 4.8) and satisfy the 1-coloured oscillator algebra
relations. The general result then follows by extension to the whole tensor product.
For the geometric version of the Clifford algebra, we recall from [15, Section 3.2]
that
ctnv(Kc,c±1(1, n,m)) = cn+m(Kc,c±1(1, n,m)). (5.2)
Therefore, by equations (5.1) and (5.2), if ni = mi for all i 6= `, then
ctnv(Kc,c±1`(n,m)) = c|n|+|m|(Kc,c±1`(n,m)).
Definition 5.4 (Geometric Clifford operators). For ` = 1, . . . , s and k ∈ Z, define
operators
Ψ`(k),Ψ∗`(k) :⊕n,c
H2|n|T?
(Mc(n))→⊕n,c
H2|n|T?
(Mc(n)),
by
Ψ`(k)|H2|n|T?
(Mc(n)):= (−1)c1+···+c`−1ctnv(Kc,c+1`(n,n + (k − c` − 1)1`))
5. Geometric Realizations of the Basic Representation 88
∈ H2(2|n|+k−c`−1)T?
(Mc(n)×Mc+1`(n + (k − c` − 1)1`)),
Ψ∗`(k)|H2|n|T?
(Mc(n)):= (−1)c1+···+c`−1ctnv(Kc,c−1`(n,n− (k − c`)1`))
∈ H2(2|n|−k+c`)T?
(Mc(n)×Mc−1`(n− (k − c`)1`)).
The Ψ`(k) and Ψ∗`(k) will be called geometric Clifford operators.
Theorem 5.5. The operators Ψ`(k) and Ψ∗`(k) preserve the space A and satisfy the
relations
Ψ`(k),Ψ∗j(i) = δkiδ`j, Ψ`(k),Ψj(i) = Ψ∗`(k),Ψ∗j(i) = 0.
In particular, the mapping
ψ`(k) 7→ Ψ`(k), ψ∗` (k) 7→ Ψ∗`(k),
defines a representation of the s-coloured Clifford algebra on A and the linear map
determined by
[I] 7→ I,
is an isomorphism of Clifford algebra representations A → F. This isomorphism
maps A(c)→ F(c), for all c ∈ Z.
Proof: Using an argument analogous to the proof of Theorem 5.3, one can show
that
Ψ`(k)|HT? (Mc(n)) = (−1)c1+···+c`−1(1⊗`−1⊗ctnv(Kc`,c`+1(1,n`,n`+k−c`−1))⊗1⊗s−`),
Ψ∗`(k)|HT? (Mc(n)) = (−1)c1+···+c`−1(1⊗`−1 ⊗ ctnv(Kc`,c`−1(1,n`,n` − k + c`))⊗ 1⊗s−`).
By [15, Theorem 3.6], for fixed `, the operators (−1)c1+···+c`−1ctnv(Kc`,c`+1(1,n`,n` +
k − c` − 1)) and (−1)c1+···+c`−1ctnv(Kc`,c`−1(1,n`,n` − k + c`)) preserve the space⊕n`,c`
Ac`(n`) (see Remark 4.8) and satisfy the 1-coloured Clifford algebra relations.
5. Geometric Realizations of the Basic Representation 89
The general result then follows by extension to the whole tensor product.
Recall from Diagram 4.10 that, since
Mc(n)ZR ∼=∐|v`|=n`
Mc(v1, . . . ,vs),
we may view Mc(v1, . . . ,vs)×Md(u1, . . . ,us) as a subvariety of Mc(n)×Md(m),
where n` = |v`| and m` = |u`|. We therefore have an action of T? onMc(v1, . . . ,vs)
and ∐v`
Mc(v1, . . . ,vs)T? =
∐n
Mc(n)T? .
Let
Kc(n; `, k)± :=∐|vi|=ni
Kc,c(v1,v1, . . . ,v`,v` ± 1k, . . . ,v
s,vs),
which is a vector bundle on∐|vi|=ni
Mc(v1, . . . ,vs)×Mc(v
1, . . . ,v`±1k, . . . ,vs). By
construction, for any (I,J) ∈Mc(v1, . . . ,vs)T? ×Mc(v
1, . . . ,v` ± 1k, . . . ,v`)T? ,
ctnv(Kc(n; `, k)±(I,J)) = ctnv(Kc,c(n,n± 1`)(I,J)).
Thus,
ctnv(Kc(n; `, k)±) = c(2|n|±1)−1(Kc(n; `, k)±).
By the same reasoning, for (I,J) ∈Mc(v1, . . . ,vs)T? ×Mc(v
1, . . . ,vs)T? ,
ctnv(Kc,c(v1,v1, . . . ,vs,vs)(I,J)) = ctnv(Kc,c(n,n)(I,J)),
and so
ctnv(Kc,c(v1,v1, . . . ,vs,vs)) = c2|n|(Kc,c(v
1,v1, . . . ,vs,vs)).
Let h :∐
v`,u` Mc(v1, . . . ,vs) × Md(u1, . . . ,us) ∼= Mc(n)ZR × Md(m)ZR →
Mc(n)×Md(m) denote the natural inclusion and let β` := h∗(γ`) for all ` = 1, . . . , s.
For ` = 1, . . . , s and k = 0, . . . , r` − 1, we define
E`k(c,n) := −(R/r`)β` ∪ ctnv(Kc(n; `, k)−)
5. Geometric Realizations of the Basic Representation 90
∈ H4|n|−2T?
∐|vi|=ni
Mc(v1, . . . ,vs)×Mc(v
1, . . . ,v` − 1k, . . . ,vs)
,
F`k(c,n) := (R/r`)β` ∪ ctnv(Kc(n; `, k)+)
∈ H4|n|+2T?
∐|vi|=ni
Mc(v1, . . . ,vs)×Mc(v
1, . . . ,v` + 1k, . . . ,vs)
,
H`k(c; v1, . . . ,vs) :=
((1c)k −
r`−1∑j=0
a`kjv`j
)ctnv(Kc,c(v
1,v1, . . . ,vs,vs))
∈ H4(|v1|+···+|vs|)T?
(Mc(v1, . . . ,vs)×Mc(v
1, . . . ,vs)),
where a`kj is the (k, j)-th entry of the generalized Cartan matrix of type Ar`−1, i.e.
a`kj =
2, if k = j,
−1, if k = j ± 1,
0, otherwise,
if r` ≥ 2,
a`kj =
2, if k = j,
−2 if k = j ± 1,
if r` = 2,
a`00 = 0, if r` = 1.
Note that the elements E`k, F`k and H`k do not define operators since they do not
lie in the middle degree cohomology of their respective quiver varieties. However,
we can push these elements forward to the middle degree cohomology of the appro-
priate moduli spaces; we describe this process below. For what follows, it will be
convenient to view elements of H∗T?(Mc(v1, . . . ,vs)×Md(u1, . . . ,us)) as elements of
H∗T?(Mc(n)ZR ×Md(m)ZR). That is, by identifying
H∗T?(Mc(n)ZR ×Md(m)ZR) ∼= H∗T?
∐|vi|=ni|ui|=mi
Mc(v1, . . . ,vs)×Md(u1, . . . ,us)
5. Geometric Realizations of the Basic Representation 91
∼=⊕|vi|=ni|ui|=mi
H∗T?(Mc(v1, . . . ,vs)×Md(u1, . . . ,us)),
we will identify elements of H∗T?(Mc(v1, . . . ,vs)×Md(u1, . . . ,us)) with their images
under the inclusion
H∗T?(Mc(v
1, . . . ,vs)×Md(u1, . . . ,us))→ H∗T?(Mc(n)ZR ×Md(m)ZR).
In this way, we will consider
E`k(c,n) ∈ H∗T?(Mc(n)ZR ×Mc(n− 1`)ZR),
F`k(c,n) ∈ H∗T?(Mc(n)ZR ×Mc(n + 1`)ZR),
H`k(c; v1, . . . ,vs) ∈ H∗T?(Mc(n)ZR ×Mc(n)ZR).
Recall from the Localization Theorem (4.1) that we have isomorphisms f1 and f2 as
in the diagram below:
H∗T?(Mc(n)×Md(m))
H∗T?(Mc(n)T? ×Md(m)T?).
H∗T?(Mc(n)ZR ×Md(m)ZR)
f1 f2
Let
µ : H∗T?(Mc(n)T? ×Md(m)T?)→ H∗T?(Mc(n)T? ×Md(m)T?),
be the C(b1, . . . , bs, ε)-linear map determined by
1(I,J) 7→eT?(T ′I,J)
eT?(TI,J)1(I,J),
where TI,J and T ′I,J are the tangent spaces of (I,J) inMc(n)×Md(m) andMc(n)ZR×
Md(m)ZR , respectively. Let η : H∗T?(Mc(n)ZR×Mc(m)ZR)→ H∗T?(Mc(n)×Md(m))
be the composition
η := f−11 µ f2. (5.3)
5. Geometric Realizations of the Basic Representation 92
Lemma 5.6. The map η is degree-preserving. In particular, the images of E`k(c,n),
F`k(c,n) and H`k(c; v1, . . . ,vs) under η lie in the middle degree cohomology ofMc(n)×
Mc(n∓ 1`) and Mc(n)×Mc(n), respectively.
Proof: The map f2 decreases the degree by rk(eT?(T ′I,J)). The map µ increases
the degree by rk(eT?(T ′I,J))− rk(eT?(TI,J)). Finally, the map f−11 increases the degree
by rk(eT?(TI,J)). Thus, the composition of these maps is degree preserving.
The second statement in the lemma follows from the fact that E`k(c,n), F`k(c,n)
and H`k are elements of degree 4|n| − 2, 4|n|+ 2 and 4|n|, respectively.
Definition 5.7 (Geometric diagonal Chevalley operators). For ` = 1, . . . , s and k =
0, . . . , r` − 1, define operators
E`k,F
`k,H
`k :⊕n,c
H2|n|T?
(Mc(n))→⊕n,c
H2|n|T?
(Mc(n)),
by
E`k|H2|n|
T?(Mc(n))
:= η(E`k(c,n)) ∈ H4|n|−2T?
(Mc(n)×Mc(n− 1`)),
F`k|H2|n|
T?(Mc(n))
:= η(F`k(c,n)) ∈ H4|n|+2T?
(Mc(n)×Mc(n + 1`)),
H`k|H2|n|
T?(Mc(n))
:=∑|vi|=ni
η(H`k(c; v1, . . . ,vs)) ∈ H4|n|
T?(Mc(n)×Mc(n)).
The E`k, F`
k and H`k will be called geometric (diagonal) Chevalley operators.
Our next goal will be to describe the geometric Chevalley operators in terms of
geometric Clifford operators. To that end, we prove the following useful lemma.
Lemma 5.8. Let (I,J) ∈ Mc(n)T? × Md(m)T? and β ∈ H2kT?
(Mc(v1, . . . ,vs) ×
Md(u1, . . . ,us)). Let
Mc(n)×Md(m)i1×i2←−−−Mc(n)T? ×Md(m)T?
j1×j2−−−→Mc(n)ZR ×Md(m)ZR ,
5. Geometric Realizations of the Basic Representation 93
be the natural inclusions and let (i1 × i2)(I,J) and (j1 × j2)(I,J) denote the restrictions
of i1 × i2 and j1 × j2, respectively, to (I,J). If
(I,J) ∈Mc(v1, . . . ,vs)T? ×Md(u1, . . . ,us)T? ,
then
〈η(β ∪ c|n|+|m|−k(Kc,d(v1,u1, . . . ,vs,us))[I], [J]〉 = 〈β′ ∪ c|n|+|m|−k(Kc,d(n,m))[I], [J]〉,
where β′ is a preimage of (j1 × j2)∗(I,J)(β) under the map (i1 × i2)∗(I,J). Otherwise,
〈η(β ∪ c|n|+|m|−k(Kc,d(v1,u1, . . . ,vs,us)))[I], [J]〉 = 0.
Proof: To simplify notation, we write K = Kc,d(v1,u1, . . . ,vs,us), K = Kc,d(n,m)
and c = c|n|+|m|−k. We begin by explicitly computing η(β∪c(K)) (recall η = f−11 µf2
as in (5.3)). By definition of f2,
f2(β ∪ c(K)) =
((j1 × j2)∗(K,L)(β ∪ c(K))
eT?(T ′(K,L))
)(K,L)∈Mc(n)T?×Md(m)T?
=
(βK,L ∪ (j1 × j2)∗(K,L)(c(K))
eT?(T ′(K,L))
)(K,L)∈Mc(n)T?×Md(m)T?
,
where T ′(K,L) is the tangent space of (K,L) in Mc(n)ZR ×Mc(m)ZR , and βK,L =
(j1 × j2)∗(K,L)(β). By applying µ, we get
µ f2(β ∪ c(K)) =
(βK,L ∪ (j1 × j2)∗(K,L)(c(K))
eT?(T(K,L))
)(K,L)∈Mc(n)T?×Md(m)T?
,
where T(K,L) is the tangent space of (K,L) in Mc(n) ×Md(m). By definition, the
map f−11 = (i1 × i2)∗, and thus,
η(β ∪ c(K)) = (i1 × i2)∗
(βK,L ∪ (j1 × j2)∗(K,L)(c(K))
eT?(T(K,L))
)(K,L)∈Mc(n)T?×Md(m)T?
.
5. Geometric Realizations of the Basic Representation 94
By definition of the bilinear form,
〈η(β ∪ c(K))[I], [J]〉 = (−1)|m|p∗((i1 × i2)∗)−1(η(β ∪ c(K)) ∪ [I]⊗ [J])
= (−1)|m|p∗((i1 × i2)∗)−1
(η(β ∪ c(K))
eT?(T −I )eT?(T −J )∪ (i1 × i2)∗(1(I,J))
),
where the last equality comes from the definition of [I] and [J]. By the projection
formula,
〈η(β ∪ c(K))[I], [J]〉 = (−1)|m|p∗
((i1 × i2)∗(η(β ∪ c(K)))
eT?(T −I )eT?(T −J )∪ 1(I,J)
).
By [5, Equation 9.3], (i1 × i2)∗(i1 × i2)∗ is simply multiplication by the Euler class of
the tangent space. Hence,
(i1 × i2)∗η(β ∪ c(K)) =(βK,L ∪ (j1 × j2)∗(K,L)(c(K))
)(K,L)∈Mc(n)T?×Md(m)T?
.
Thus,
〈η(β ∪ c(K))[I], [J]〉 = (−1)|m|p∗
(βI,J ∪ (j1 × j2)∗(I,J)(c(K))
eT?(T −I )eT?(T −J )
).
By construction, if
(I,J) /∈Mc(v1, . . . ,vs)T? ×Md(u1, . . . ,us)T? ,
then
(j1 × j2)∗(I,J)(c(K)) = 0.
On the other hand, if
(I,J) ∈Mc(v1, . . . ,vs)T? ×Md(u1, . . . ,us)T? ,
then by functoriality of the Chern class and the construction of K,
(j1 × j2)∗(I,J)(c(K)) = c(K(I,J)) = c(K(I,J)).
Therefore,
〈η(β∪c(K))[I], [J]〉 = (−1)|m|βI,J ∪ c(K(I,J))
eT?(T −I )eT?(T −J )=
βI,J ∪ c(K(I,J))
eT?(T −I )eT?(T +J )
= 〈β′∪c(K)[I], [J]〉,
5. Geometric Realizations of the Basic Representation 95
where the last equality follows from Lemma 4.9.
Corollary 5.9. Let (I,J) ∈Mc(n)T? ×Mc(m)T?.
1. For m = n− 1`, if
(I,J) ∈Mc(v1, . . . ,vs)T? ×Mc(v
1, . . . ,v` − 1k, . . . ,vs)T? ,
then
〈E`k[I], [J]〉 = 〈P`(1)[I], [J]〉.
Otherwise
〈E`k[I], [J]〉 = 0.
2. For m = n + 1`, if
(I,J) ∈Mc(v1, . . . ,vs)T? ×Mc(v
1, . . . ,v` + 1k, . . . ,vs)T? ,
then
〈F`k[I], [J]〉 = 〈P`(−1)[I], [J]〉.
Otherwise
〈F`k[I], [J]〉 = 0.
3. For m = n,
〈H`k[I], [J]〉 =
((1c`)k −
∑j
a`kjv`j
)δI,J.
Proof: Statements (1) and (2) follow directly from Lemma 5.8 and the definitions
of P`(±1) (note that, in the notation of Lemma 5.8, (i1 × i2)∗(γ`) = (j1 × j2)∗(β`)).
The third statement follows from Lemma 5.8 and the fact that ctnv(Kc,c(n,n)) = id
as an operator on H2|n|T?
(Mc(n)) (see Definition 5.2).
5. Geometric Realizations of the Basic Representation 96
Proposition 5.10. Let (I,J) ∈ Mc(n)T? ×Mc(n + 1`)T?. If λ(Iα) = λ(Jα) for all
α 6= ` and λ(J`) can be obtained by adding one box to λ(I`), then
〈P`(−1)[I], [J]〉 = 1.
Otherwise, 〈P`(−1)[I], [J]〉 = 0.
Proof: This is a direct result of [15, Proposition 5.3 and the proof of Theorem
3.14]. Note that in our case, λ(J`)− λ(I`) consists of a single box.
Lemma 5.11. If I ∈ Mc(v1, . . . ,vs)T?, then v`k is equal to the number of boxes in
λ(I`) whose residue is congruent to k − c` modulo r`.
Proof: Let I` = [A,B, i] ∈ Mc`(1,n`)T? . By [24, Proposition 2.9], λ(I`) is
obtained from I` by drawing a box in the (p, q)-th position if Ap−1Bq−1i 6= 0 (note
that our Young diagrams are rotated 90 clockwise from those in [24]). From Equation
(3.8),
v`k = dim spanApBqi | q − p ≡ k − c` mod r`.
Since the nonzero ApBqi are linearly independent, the boxes (p, q) ∈ λ(I`) whose
residue is congruent to k−c` mod r` are in one-to-one correspondence with a basis of
spanApBqi | q−p ≡ k−c` mod r`. Thus, v`k is equal to the number of such boxes.
Lemma 5.12. Let (I,J) ∈Mc(n)T? ×Mc(n + 1`)T?. Then
〈F`k[I], [J]〉 = 1,
if λ(Iα) = λ(Jα) for all α 6= ` and λ(J`) can be obtained from λ(I`) by adding one box
whose residue is congruent to k − c` modulo r`. Otherwise, 〈F`k[I], [J]〉 = 0.
5. Geometric Realizations of the Basic Representation 97
Proof: This follows immediately from Corollary 5.9, Proposition 5.10 and Lemma
5.11.
Proposition 5.13. Let (I,J) ∈Mc(n)T? ×Mc(n + 1`)T?. Then for all i ∈ Z,
〈Ψ`(i)Ψ∗`(i− 1)[I], [J]〉 =
1, if J = ψ`(i)ψ∗` (i− 1)I,
0, otherwise.
Proof: This follows immediately from Theorem 5.5.
Theorem 5.14. For ` = 1, . . . , s and k = 0, 1, . . . , r` − 1,
E`k =
∑i∈Z
Ψ`(k + ir`)Ψ∗`(k + ir` + 1),
F`k =
∑i∈Z
Ψ`(k + ir` + 1)Ψ∗`(k + ir`),
as operators on⊕
n,cH2|n|T?
(Mc(n)).
Proof: Let I,J be two s-tuples of semi-infinite monomials of charge c. We first
prove that
〈F`k[I], [J]〉 =
⟨∑i∈Z
Ψ`(k + ir` + 1)Ψ∗`(k + ir`)[I], [J]
⟩. (5.4)
If there exists an α 6= ` such that Iα 6= Jα, then both sides of Equation (5.4) are zero
and we are done. Thus, we assume that Iα = Jα for all α 6= `. Write
I` = i1 ∧ i2 ∧ i3 ∧ · · · , and J` = j1 ∧ j2 ∧ j3 ∧ · · · ,
where im, jm ∈ Z. Recall that the number of boxes in the m-th row of λ(I`) is
im − c` + m− 1 (likewise for λ(J`)). Suppose that 〈F`k[I], [J]〉 = 1. Then by Lemma
5.12, λ(J`) is obtained by adding one box of the appropriate residue to λ(I`). This
5. Geometric Realizations of the Basic Representation 98
means that there exists an m ∈ N such that jn = in for all n 6= m and jm = im + 1.
The position of the added box is thus (m, jm− c` +m− 1). The residue of the added
box must be congruent to k − c` modulo r`, and so,
(jm − c` +m− 1)−m = jm − c` − 1 ≡ k − c` mod r`,
or equivalently
jm − k − 1 ≡ 0 mod r`.
Thus, jm = k + ir` + 1, for some i ∈ Z. Now, since jn = in for all n 6= m and
jm = im + 1, we have that
J = ψ`(jm)ψ∗` (jm − 1)I, and J 6= ψ`(j)ψ∗` (j − 1)I, for all j 6= jm.
Hence,⟨∑i∈Z
Ψ`(k + ir` + 1)Ψ∗`(k + ir`)[I], [J]
⟩= 〈Ψ`(jm)Ψ∗`(jm − 1)[I], [J]〉 = 1,
by Proposition 5.13. We have therefore proven that Equation (5.4) holds when
〈F`k[I], [J]〉 = 1.
Suppose now that 〈F`k[I], [J]〉 = 0. Then λ(J`) cannot be obtained by adding one
box to λ(I`). In particular, J 6= ψ`(i)ψ∗` (i− 1)I for all i ∈ Z. Thus,⟨∑
i∈Z
Ψ`(k + ir` + 1)Ψ∗`(k + ir`)[I], [J]
⟩= 0.
Therefore, Equation (5.4) holds in all cases, and so
F`k =
∑i∈Z
Ψ`(k + ir` + 1)Ψ∗`(k + ir`).
Now, since P`(−1) and P`(1) are adoint (see [15, Lemma 3.13]), it follows by
Corollary 5.9 that E`k and F`
k are adoint. Moreover, since Ψ`(i) and Ψ∗`(i) are adjoint
(see [15, Lemma 3.5]), it follows that∑i∈Z
Ψ`(k + ir` + 1)Ψ∗`(k + ir`), and∑i∈Z
Ψ`(k + ir`)Ψ∗`(k + ir` + 1),
5. Geometric Realizations of the Basic Representation 99
are adjoint. Therefore, for all s-tuples of semi-infinite monomials I, J,
〈E`k[I], [J]〉 = 〈[I],F`
k[J]〉 =
⟨[I],∑i∈Z
Ψ`(k + ir` + 1)Ψ∗`(k + ir`)[J]
⟩
=
⟨∑i∈Z
Ψ`(k + ir`)Ψ∗`(k + ir` + 1)[I], [J]
⟩.
Thus,
E`k =
∑i∈Z
Ψ`(k + ir`)Ψ∗`(k + ir` + 1).
Proposition 5.15. For all ` = 1, . . . , s and k = 0, 1, . . . , r` − 1,
[E`k,F
`k] = H`
k.
Proof: If r` = 1, then
E`0 = P`(1), F`
0 = P`(−1), H`0 = id,
and the result follows from Theorem 5.3.
Now, fix an ` ∈ 1, . . . , s such that r` ≥ 2. For any semi-infinite monomial I of
charge c ∈ Z, we will say that a box (p, q) ∈ λ(I) is a k-box if its residue is congruent
to k− cmod r`. We will say that a box (p, q) ∈ λ(I) is k-removable if (p, q) is a k-box
and (p, q) may be removed from λ(I) to create a new Young diagram. Denote the set
of k-removable boxes of λ(I) by R. We will say that a box (p, q) /∈ λ(I) is k-addable
if (p, q) is a k-box and (p, q) may be added to λ(I) to create a new Young diagram.
Denote the set of k-addable boxes of λ(I) by A.
By Theorem 5.14, we have that, for all I ∈Mc(v1, . . . ,vs)T?
E`k[I] =
∑J
[J], and F`k[I] =
∑K
[K],
5. Geometric Realizations of the Basic Representation 100
where the J run over all semi-infinite monomials such that Ji = Ii for all i 6= ` and
J` is obtained from I` by removing a k-removable box, and the K run over all semi-
infinite monomials such that Ki = Ii for all i 6= ` and K` is obtained from I` by
adding a k-addable box. Thus, it is easy to see that
[E`k,F
`k][I] = (|A| − |R|)[I],
where here A and R refer to the sets of k-addable and k-removable boxes of I`,
respectively. Therefore, it suffices to show that
|A|−|R| = (1c)k−r∑j=0
a`kjv`j = δc,k−2v`k+v`k+1+v`k−1 = δc,k+(v`k+1−v`k)+(v`k−1−v`k),
where, of course, the indices of v` are taken modulo r`.
For the remainder of the proof, we will identify I` with its Young diagram λ =
λ(I`). The case where λ is the empty Young diagram is trivial, so we assume λ
consists of at least one box. The k-border of λ is the set
B := k-boxes (p, q) ∈ λ | (p+ 1, q), (p, q + 1), or (p+ 1, q + 1) /∈ λ.
We partition B with respect to the main diagonal of λ as follows:
B = U ∪M ∪ L,
U = (p, q) ∈ B | q > p, M = (p, p) ∈ B, L = (p, q) ∈ B | p > q.
The sets B,U,M and L are illustrated in the following diagram:
5. Geometric Realizations of the Basic Representation 101
L
U
B
M
Here B is the set of k-boxes in the shaded region, U (resp. L) is the set of boxes in B
that lie above (resp. below) the dashed line, and M is the intersection of B with the
dashed line. Let ∆→ (resp. ∆↓) be the subset of B consisting of boxes which have a
right (resp. lower) neighbour. That is,
∆→ = (p, q) ∈ B | (p, q + 1) ∈ λ, and ∆↓ = (p, q) ∈ B | (p+ 1, q) ∈ λ.
Denote by ∆′→ and ∆′↓ the complements of ∆→ and ∆↓, respectively, in B.
We begin with the case where k 6≡ c` mod r` (so that M = ∅). In the portion
of λ above the main diagonal, every (k + 1)-box occurs as the right neighbour of a
k-box. Conversely, every right neighbour of a k-box (if it has one) is a (k + 1)-box.
If a k-box does not have a right neighbour, it must therefore lie in B. Hence, the
number of k-boxes less the number of (k+ 1)-boxes above the main diagonal is equal
to |U ∩ ∆′→|. In the portion of λ below and including the main diagonal, we can
dualize this argument and conclude that the number of (k+ 1)-boxes less the number
of k-boxes is equal to the number of (k + 1)-boxes (on the border) with no lower
neighbour. Any (k + 1)-box not in the first column and without a lower neighbour
has a left neighbour, which must be a k-box and lie in L and ∆→. Hence, the number
of (k + 1)-boxes less the number of k-boxes is equal to |L ∩∆→| + δ, where δ = 1 if
5. Geometric Realizations of the Basic Representation 102
the last box of the first column of λ is a (k + 1)-box and δ = 0 otherwise. Since v`k
(resp. v`k+1) is the number of k-boxes (resp. (k + 1)-boxes) in λ,
v`k+1 − v`k = |L ∩∆→|+ δ − |U ∩∆′→|. (5.5)
By a completely analogous argument,
v`k−1 − v`k = |U ∩∆↓|+ δ′ − |L ∩∆′↓|,
where δ′ = 1 if the last box of the first row of λ is a (k− 1)-box and δ′ = 0 otherwise.
Now, we note that a k-box is k-removable if and only if it has no right or lower
neighbours. Hence, R = ∆′→ ∩∆′↓ and so
|R| = |∆′→ ∩∆′↓| = |B| − |∆→ ∪∆↓|. (5.6)
We can add a k-box at the end of the first row (resp. column) if and only if the last
box of the first row (resp. column) is a (k−1)-box (resp. (k+1)-box). A k-box (p, q),
where p, q 6= 1, is k-addable if and only if both (p−1, q) and (p, q−1) are in λ, which
occurs if and only if (p− 1, q − 1) ∈ ∆→ ∩∆↓. Hence,
|A| = |∆→ ∩∆↓|+ δ + δ′ = |∆→|+ |∆↓| − |∆→ ∪∆↓|+ δ + δ′. (5.7)
Since M = ∅, we have that B = U ∪ L, and so
|∆→/↓| = |U ∩∆→/↓|+ |L ∩∆→/↓| (5.8)
Combining equations (5.6), (5.7) and (5.8), we get
|A| − |R| = |U ∩∆→|+ |L ∩∆→|+ |U ∩∆↓|+ |L ∩∆↓| − |B|+ δ + δ′
= |U | − |U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓|+ |L| − |L ∩∆′↓| − |B|+ δ + δ′
= −|U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓| − |L ∩∆′↓|+ δ + δ′
= (v`k+1 − v`k) + (v`k−1 − v`k).
5. Geometric Realizations of the Basic Representation 103
In the case where k ≡ c` mod r`, one only has to make a few modifications to the
above arguments owing to the fact that M now consists of a box. In the portion of
λ above and including the main diagonal, the number of k-boxes less the number of
(k+1)-boxes is |U∩∆′→|+|M∩∆′→|. In the portion of λ below the main diagonal, one
again has that the number of (k+1)-boxes less the number of k-boxes is |L∩∆→|+δ.
Hence,
v`k+1 − v`k = |L ∩∆→|+ δ − |U ∩∆′→| − |M ∩∆′→|.
And again by a completely analogous argument,
v`k−1 − v`k = |U ∩∆↓|+ δ′ − |L ∩∆′↓| − |M ∩∆′↓|. (5.9)
One can compute |A| and |R| exactly as in equations (5.7) and (5.6). The difference
now is that, since M 6= ∅, we have B = U ∪M ∪ L, and so
|∆→/↓| = |U ∩∆→/↓|+ |M ∩∆→/↓|+ |L ∩∆→/↓|.
Therefore, using similar calculations as before,
|A| − |R| = |U | − |U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓|+ |L| − |L ∩∆′↓| − |B|+ δ + δ′
+ |M ∩∆→|+ |M ∩∆↓|
= |U | − |U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓||L| − |L ∩∆′↓| − |B|+ δ + δ′
+ |M | − |M ∩∆′→|+ |M | − |M ∩∆′↓|
= −|U ∩∆′→|+ |L ∩∆→|+ |U ∩∆↓| − |L ∩∆′↓|+ δ + δ′ − |M ∩∆′→|
+ |M | − |M ∩∆′↓|
= 1 + (v`k+1 − v`k) + (v`k−1 − v`k),
which completes the proof.
With the operators E`k, F`
k and H`k, we are able to give the following partial
result towards our goal of giving a geometric realization of Vbasic.
5. Geometric Realizations of the Basic Representation 104
Proposition 5.16. The geometric diagonal Chevalley operators preserve the space
A. Moreover, when r` ≥ 2, the operators E`k, F`
k and H`k satisfy the Kac-Moody
relations for slr`. In particular, the mapping
Ek 7→ E`k, Fk 7→ F`
k, Hk 7→ H`k, (5.10)
for k = 0, 1, . . . , r` − 1, defines a representation of slr` on A and the linear isomor-
phism determined by [I] 7→ I, is an isomorphism of slr`-representations A→ F. The
mapping (5.10) together with
I ⊗ tn 7→ |n|r`P`(nr`), and I ⊗ 1 7→ P`(0),
for n ∈ Z − 0, defines a representation of glr` on A and the mapping [I] 7→ I is
then an isomorphism of glr`-representations A→ F.
When r` = 1, the mapping
I ⊗ tn 7→ |n|P`(n), I ⊗ 1 7→ P`(0), c 7→ idA,
for all n ∈ Z− 0, defines a representation of gl1 on A and the mapping [I] 7→ I is
an isomorphism of gl1-representations A→ F.
Proof: The fact that the geometric diagonal Chevalley operators preserve A
is clear from Corollary 5.9. By Theorem 5.5, we have the following commutative
diagram:
A F
A F
Ψ`(i),Ψ∗`(i) ψ`(i), ψ
∗` (i),
(5.11)
for each ` = 1, . . . , s and i ∈ Z. Fix a value of ` ∈ 1, . . . , s such that r` ≥ 2. For
each k = 0, 1, . . . , r` − 1, let
E`k = Er1+···+r`−1+k, F `
k = Fr1+···+r`−1+k, H`k = Hr1+···+r`−1+k.
Then by Lemma 1.21 and Theorem 5.14, we have the following commutative diagram:
5. Geometric Realizations of the Basic Representation 105
A F
A F
E`k,F
`k E`
k, F`k ,
for all k = 0, 1, . . . , r` − 1. By Proposition 5.15, we therefore also have the following
commutative diagram:
A F
A F
H`k H`
k.
Thus, when r` ≥ 2, we have a representation of slr` on A and the map A→ F is an
isomorphism of slr`-representations. The fact that the addition of the mapping
I ⊗ tn 7→ |n|r`P`(nr`), I ⊗ 1 7→ P`(0),
gives us a representation of glr` on A follows from Lemma 1.21 and Theorem 5.3.
The case where r` = 1 is obvious.
Proposition 5.16 shows that the operators E`k, F`
k, H`k and P`(nr`) on A corre-
spond to the operators associated to the (`, `)-th diagonal blocks of glr on F. Thus,
as was the case algebraically, to complete our realization of glr, it remains only to
construct the operators associated to the off-diagonal blocks.
Let c ∈ Zs and i 6= j ∈ 1, . . . , s. Suppose n,m ∈ Ns such that n` = m` for all
` 6= i, j. By equations (5.1) and (5.2), we have that
ctnv(Kc,c+1i−1j(n,m)) = c|n|+|m|(Kc,c+1i−1j(n,m)).
For i = 1, . . . , s− 1 and k ∈ Z, define
K+c (n; i, k) := Kc,c+1i−1i+1
(n,n + ((k + 1)ri − ci − 1)1i − (kri+1 − ci+1 + 1)1i+1),
5. Geometric Realizations of the Basic Representation 106
K−c (n; i, k) := Kc,c+1i+1−1i(n,n + ((k − 1)ri+1 − ci+1)1i+1 − (kri − ci)1i).
We also define
K+c (n; 0, k) := Kc,c+1s−11(n,n + (krs − cs − 1)1s − (kr1 − c1 + 1)11),
K−c (n; 0, k) := Kc,c+11−1s(n,n + (kr1 − c1)11 − (krs − cs)1s).
Definition 5.17 (Geometric off-diagonal Chevalley operators). For i = 0, 1, . . . , s−1,
define operators
E′i,F′i,H
′i :⊕n,c
H2|n|T?
(Mc(n))→⊕n,c
H2|n|T?
(Mc(n)),
by
E′i|H2|n|T?
(Mc(n))= (−1)ci
∑k∈Z
ctnv(K+c (n; i, k)),
F′i|H2|n|T?
(Mc(n))= (−1)ci
∑k∈Z
ctnv(K−c (n; i, k)),
for i = 1, . . . , s− 1, and
E′0|H2|n|T?
(Mc(n))= (−1)c1+···+cs−1
∑k∈Z
ctnv(K+c (n; 0, k)),
F′0|H2|n|T?
(Mc(n))= (−1)c1+···+cs−1
∑k∈Z
ctnv(K−c (n; 0, k)),
and finally, H′i := [E′i,F′i], for all i = 0, . . . , s− 1.
Remark 5.18. For the diagonal Chevalley operators, the space Ac(v1, . . . ,vs) is a
subspace of the the v`-weight space of glr` , thus allowing us to define the operators
H`k explicitly in terms of the v`. For the off-diagonal Chevalley operators, the v` no
longer encode information about the weights of glr. Hence, our defining H′i simply as
the commutator of E′i and F′i seems the best we can achieve.
Lemma 5.19. As operators on A,
E′i =∑k∈Z
Ψi((k + 1)ri)Ψ∗i+1(kri+1 + 1), F′i =
∑k∈Z
Ψi+1((k − 1)ri+1 + 1)Ψ∗i (kri),
5. Geometric Realizations of the Basic Representation 107
for i = 1, . . . , s− 1, and
E′0 =∑k∈Z
Ψs(krs)Ψ∗1(kr1 + 1), F′0 =
∑k∈Z
Ψ1(kr1 + 1)Ψ∗s(krs).
Proof: Fix i ∈ 1, . . . , s− 1. Consider the restriction of E′i to H2|n|T?
(Mc(n)). To
simplify notation, let d = c + 1i−1i+1 and mk = n + ((k+ 1)ri− ci− 1)1i− (kri+1−
ci+1 + 1)1i+1. Then
E′i = (−1)ci∑k∈Z
ctnv(Kc,d(n,mk)) = (−1)ci∑k∈Z
ctnv
(s⊕`=1
f ∗`Kc`,d`(1,n`, (mk)`)
)
= (−1)ci∑k∈Z
(s∏`=1
(1⊗`−1 ⊗ ctnv(Kc`,d`(1,n`, (mk)`))⊗ 1⊗s−`)
),
where the second equality follows from Lemma 4.14 and the third equality is the
application of the Kunneth formula. For ` 6= i, i+ 1,
ctnv(Kc`,d`(1,n`, (mk)`)) = ctnv(Kc`,c`(1,n`,n`)) = 1,
by [15, Lemma 3.10]. Thus,
E′i = (−1)ci∑k∈Z
(1⊗i−1 ⊗ ctnv(Kci,di(1,ni, (m
k)i))⊗ 1⊗s−i)×
(1⊗i ⊗ ctnv(Kci+1,di+1
(1,ni+1, (mk)i+1))⊗ 1⊗s−i−1
)=∑k∈Z
Ψi((k + 1)ri)Ψ∗i+1(kri+1 + 1).
The remaining equalities can be proved in an analogous manner.
For k = 0, 1, . . . , r, we can write
k = r1 + · · ·+ r`−1 + k′,
for unique 1 ≤ ` ≤ s and 0 ≤ k′ ≤ r` − 1. For all k such that k′ 6= 0, let
Ek := E`k′ , Fk := F`
k′ , Hk := H`k′ .
5. Geometric Realizations of the Basic Representation 108
For all k such that ` 6= 1 and k′ = 0, let
Ek := E′`−1, Fk := F′`−1, Hk := H′`−1.
For k = 0,
E0 := E′0, F0 := F′0, H0 = H′0.
Theorem 5.20. The operators Ek, Fk and Hk, k = 0, 1, . . . , s−1, preserve the space
A and satisfy the Kac-Moody relations for slr. In particular, the mapping
Ek 7→ Ek, Fk 7→ Fk, Hk 7→ Hk,
defines a representation of slr on A and the linear map determined by [I] → I is an
isomorphism of slr-representations A 7→ F.
Proof: This follows from the commutativity of Diagram (5.11) and comparison
of Theorem 5.14 and Lemma 5.19 with Lemma 1.21.
By Theorem 5.20, we have a geometric version of slr. As before, to get a similar
geometric version of glr, we need to add operators corresponding to the loops on the
identity. This leads us to our main theorem, from which we conclude that A(0) is a
geometric version of Vbasic.
Theorem 5.21. For k = 0, 1, . . . , s− 1, and n ∈ Z− 0, the mapping
Ek 7→ Ek, Fk 7→ Fk, Hk 7→ Hk, I⊗ tn 7→ |n|s∑`=1
r`P`(nr`), I⊗1 7→s∑`=1
P`(0),
defines a representation of glr on the space A and the linear map determined by
[I] 7→ I is an isomorphism of glr-representations A → F. This isomorphism maps
A(0)→ F(0), and thus A(0) ∼= Vbasic.
5. Geometric Realizations of the Basic Representation 109
Proof: From Theorem 5.20, the mapping
Ek 7→ Ek, Fk 7→ Fk, Hk 7→ Hk,
gives us a representation of slr on A. The additional mapping
I ⊗ tn 7→ |n|s∑`=1
r`P`(nr`), I ⊗ 1 7→s∑`=1
P`(0),
gives us a representation of glr on A by Lemma 1.21.
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