Automorphic representations - WikisIwasawa decomposition Assume that F = R. Gram-Schmidt: every...

Automorphic representations

Erez Lapid

Lausanne, 2011

Goals

To explain

I Langlands functoriality principle and applications

I spectral decomposition of L2(G (F )\G (A))

I Eisenstein series

I L-functions

I Local theory

I Structure theory of algebraic groups

Plan

Second week: Global theory

I Lecture 4: Langlands functoriality principle

I Lecture 3: spectral decomposition of L2(G (F )\G (A))

First week: Local theory

I Lecture 2: local representation theory

I Lecture 1: structure theory of algebraic groups

Structure theory of (affine) algebraic groups

Throughout the talk F is of characteristic zero.Affine algebraic group: can be realized as a subgroup of GLn

(invertible matrices of size n × n) given by the zero locus of(finitely many) polynomials (with coefficients in F ) in the entries.(Does not include: elliptic curves)(Henceforth we will omit the word ”affine”)It is sometimes convenient to distinguish between the group G as avariety and its F -points G = G(F ).

ExamplesI GLn, dim = n2; in particular, the (one-dimensional)

multiplicative group Gm = GL1

I SLn = {g ∈ GLn : det g = 1}, dim = n2 − 1

I additive group Ga = {(

1 x0 1

)}

I mixture of multiplicative and additive: {(a b0 1

): a 6= 0},

dim = 2I symplectic group: {g ∈ GLn : tgJg = J} where J is

anti-symmetric invertible (and n even), dim =(n+1

2

)I orthogonal group: same except that J is a symmetric

invertible matrix, dim =(n

2

)I spin groups – double covers of orthogonal groupsI exceptional groups: 1 G2 (14,7), F4 (52,26), E6 (78, 27), E7

(133, 56), E8 (248, 248) (the only simple, simply-connected,simply-laced adjoint group!)

1in parenthesis: the dimension of the group and the smallest dimension of arepresentation

Important difference between algebraic and Lie groups: the groupsGm(R) = R∗ and Ga(R) = R are (almost) isomorphic as Liegroups but Gm and Ga are not isomorphic as algebraic groups.Their algebraic representations are quite different. Every algebraicrepresentation of Gm is diagonalizable, while evidently this is notthe case for Ga.

Differences if F not algebraically closed

The group SO(2,R) = {(

cos θ sin θ− sin θ cos θ

)} is diagonalizable over C

but not over R. It is a form of Gm, i.e., the C-points areisomorphic.A form of Gr

m is called a torus of rank r . A group isomorphic toGr

m (over F ) is a split torus. A rational representation (over F ) ofa split torus is diagonalizable over F .In the list above:

I the multiplicative group of a division algebra is a form of GLn

I unitary groups are also forms of GLn

I different quadratic forms of a given rank give rise to forms oforthogonal groups (but there are others, even for symplecticgroups)

I Over R every simple group has a unique compact form

Classification

The basic idea is to linearize the problem, i.e. to break the groupinto one-dimensional subgroups with simple commutation relations.The closest to vector spaces: unipotent (can be realized as upperunitriangular matrices).They are hard to classify but their structure is fairly easy (nilpotentgroups).Opposite extreme: Reductive groups: groups with no normalunipotent subgroups (i.e., unipotent radical is trivial). Equivalently:reductive group are completely reducible: every (algebraic)representation decomposes as a direct sum of irreducible ones.Another equivalent definition: G can be embedded in GLn in sucha way that it is stable under transpose.Levi decomposition: every group is a semidirect productG = M nU of a reductive group M and the unipotent radical U ofG (the maximal normal unipotent subgroup of G ); M is uniquelydetermined up to conjugation.

Reductive groups are essentially (up to taking quotients by a finitecentral group) the direct product of a torus with almost simplegroups (finite center). However, we need to consider reductivegroups even if we are only interested in simple groups.For simplicity we consider only split reductive groups, such as GLn,SLn, Sp2n, SO(n, n + 1), SO(n, n)They have a maximal torus (i.e., CG (T ) = T ) which is split over F(along with many others which are not split if F is not algebraicallyclosed). Such a torus is unique up to conjugation in G (F ).The rank of G is by definition the rank of T .The Weyl group W = NG (T )/CG (T ) = NG (T )/T , a finite group.

Examples of split maximal tori: the diagonal subgroup in GLn

(rank n) or of SLn (rank n − 1), W = SnSymplectic group Sp2n with respect to

J =

1

. ..

1−1

. ..

−1

T = {

t1

. . .

tnt−1n

. . .

t−11

: t1, . . . , tn ∈ F ∗}

(rank n) W = Sn n (Z/2Z)n

same for SO(n, n) with respect to J =

1

. ..

1

(W is of

index two in Sn n (Z/2Z)n). For SO(n, n + 1)

T = {

t1

. . .

tn1

t−1n

. . .

t−11

: t1, . . . , tn ∈ F ∗}

(again, rank n), W = Sn n (Z/2Z)n

Roots

G acts on g = Lie(G ) by the adjoint representation Ad. (For GLn

this is conjugation on Matn = gln = Lie(GLn); if G ⊂ GLn this isjust conjugation on g ⊂ gln.)The action of T on g is diagonalizable.

g = t⊕⊕α∈Φ

gα

where Φ ⊂ X ∗(T ) := Hom(T ,Gm) is the (finite) set of roots and

gα := {X ∈ g : Ad(t)X = α(t)X ∀t ∈ T}

are the T -eigenspaces which happen to be one-dimensional.

Example: GLn: the roots are the characters ti/tj , i 6= j ; rootspaces are ei ,j ; |Φ| = n(n − 1) same for SLn.roots of Sp2n: ti/tj , (ti tj)

±1, i 6= j , t±2i ; |Φ| = 2n2

roots of SO(n, n + 1): ti/tj , (ti tj)±1, i 6= j , t±1

i ; |Φ| = 2n2

roots of SO(n, n): ti/tj , (ti tj)±1, i 6= j ; |Φ| = 2n(n − 1)

We usually view the roots additively in the real vector spaceX ∗(T )⊗Z R. They form a root system, which I won’t define here.To a large extent G can be recovered from its root system.

Parabolic subgroups

They measure the various “infinities” of GThey are the “largest” subgroups of G in the sense that P\G isprojective (equivalently P(C)\G (C) is compact).One way to define them (for local fields): they are the groups ofthe form Px = {g ∈ G (F ) : {xngx−n, n = 1, 2, . . . } is bounded}where x ∈ G (F ).Fixing a minimal parabolic subgroup P0 (= a maximal solvablesubgroup = a Borel subgroup), there are precisely 2r (parabolic)subgroups containing P0. They are called standard. Any parabolicsubgroup is conjugate to a unique standard one.By definition, two parabolic subgroups are associate if their Leviparts are conjugate.

Example: GL(V )The rank is dimV (semisimple rank dimV − 1), parabolicsubgroups are the stabilizers of flags 0 ( V1 ( · · · ( Vk = V of V .The standard parabolic subgroups correspond to compositionsn = n1 + · · ·+ nk , n1, . . . , nk ≥ 1. Pn1,...,nk consists of the blockupper-triangular

A1,1 A1,2 . . . A1,k−1 A1,k

0 A2,2 . . . A2,k−1 A2,k

0 0. . .

. . ....

0 . . . 0 Ak−1,k−1 Ak−1,k

0 0 0 0 Ak,k

where Ai ,j ∈ Matni×nj , Ai ,i ∈ GLni . The standard Levidecomposition is Pn1,...,nk = Mn1,...,nkUn1,...,nk whereMn1,...,nk ' GLn1 × · · · × GLnk consists of the block diagonalmatrices diag(g1, . . . , gk), gi ∈ GLni , i = 1, . . . , k and Un1,...,nk

consists of the block upper unitriangular matrices (whereAi ,i = Idni in the notation above).

Two standard parabolic subgroups Pn1,...,nk and Pm1,...,mlare

associate if and only if the underlying partitions are the same, i.e.,if m1, . . . ,ml is a permutation of n1, . . . , nk (in particular k = l).Thus, associate classes of parabolic subgroups correspond topartitions n = n1 + · · ·+ nk , n1 ≥ · · · ≥ nk of n.Let h be a symplectic, symmetric or hermitian form on V . Considerthe isometry group G of (V , h). The parabolic subgroups of G arethe stabilizers of isotropic flags (i.e., h vanishes on Vi × Vi ).

Bruhat decomposition

Gauss elimination (for invertible matrices): Any invertible matrixcan be written as n1wan2 where w is a permutation matrix, a isdiagonal (uniquely determined), n1, n2 are upper unitriangular, (if

w is

1

. ..

1

then n1, n2 are unique).

For general groups: let P0 = T0 n N0 be a Borel subgroup. Bruhatdecomposition

G =∐w∈W

P0wN0.

There is a unique open double coset (the big cell) Bw0 = P0w0N0

and the map P0 × N0 → Bw0 , (p, n) 7→ pw0n is an isomorphism ofalgebraic varieties. Thus G is birationally equivalent to an affinespace. (Everything is defined over F .)

Iwasawa decomposition

Assume that F = R.Gram-Schmidt: every matrix can be written uniquely as a productof an upper triangular matrix with positive diagonal entries and anorthogonal matrix, i.e. GLn(R) = T (R)0N(R)K where N – upperunitriangular matrices, K = O(n) – maximal compact subgroup ofGLn(R).More generally for any algebraic group G over R, G (R) contains amaximal compact subgroup K which is unique up to conjugation.It can be realized as K = G (R) ∩ O(n) in some representationG ⊂ GLn. If P0 = T0 n N0 is a Borel subgroup then we have

G (R) = T0(R)0N0(R)K

Moreover, (t, n, k) 7→ tnk defines a diffeomorphismT (R)0 × N0(R)× K → G (R). (Iwasawa decomposition). Inparticular, G (R) is homotopic to K since T (R)0 and N0(R) arediffeomorphic to Euclidean spaces.

p-adic: maximal compact subgroups are not unique up toconjugation: They are unique for GLn(Qp) but for SLn(Qp) thereare n of them, all conjugate in GLn(Qp); for PGL2(Qp) there aretwo and they are quite different.There is a deep and elaborate theory of the maximal compactsubgroups of G due to Bruhat-Tits.Iwasawa decomposition: G (F ) = T0(F )N0(F )K (not uniquely)where K is a maximal compact. For GLn – elementaryCartan decomposition: F either R or p-adic G (F ) = KA+K(uniquely) where

A+ = {t ∈ T0(F ) : |α(t)| ≤ 1 for all roots α of T0 in Lie(N0)}

e.g., for GLn(F ), F p-adic

GLn(F ) = GLn(O){

$k1

. . .

$kn

k1 ≥ · · · ≥ kn}GLn(O).

Thus, ”from far away” G (F ) looks like a cone in a lattice!

Automorphic representations - lecture #2(4)

Lausanne, 2011

In this talk we will review some elements of representation theoryof reductive groups over local fields.The founding fathers: Bernstein, Casselman, Harish-Chandra,Gelfand, Jacquet, Kazhdan, Langlands, Zelevinsky

From now on G will be a reductive group (often split) defined overa local field of characteristic 0 (i.e. F = R,C or p-adic) andG = G(F ) with the topology coming from embedding it in someGLn.Which representations (π,V ) of G do we consider?First, V is a vector space over C (not finite-dimensional in general).Certainly the action map G × V → V better be continuous; butwhat topology on V ?p-adic case: discrete topology on V i.e. the stabilizer of everyv ∈ V is an open subgroup of G . (We call these smoothrepresentations.) This is reasonable since the topology of G has abasis of compact open subgroups.There are no irreducible smooth finite-dimensional representations(except characters) unless G has anisotropic quotients

Let S(G ) be the algebra of compactly supported locally constantfunctions on G (under convolution), i.e. S(G ) =

⋃K Cc(K\G/K )

where K ranges over the compact open subgroups of G .smooth representations of G ←→ non-degenerate modules ofS(G ) (i.e., S(G )M = M)Admissibility condition: for any open subgroup K ⊂ G , VK isfinite-dimensional.It turns out: π is finitely generated and admissible iff π has finitelengthWe denote by R(G ) the category of representations of G of finitelength. Π(g) ⊂ R(G ) the irreducible representations.The contragredient (π, V ) of (π,V ) ∈ R(G ) is the smooth part ofV ∗, i.e., V = {` ∈ V ∗ :there exists an open K ⊂ G such that ` ◦ π(g) = ` for all g ∈ K}with π(g)` = ` ◦ π(g)−1.

What about F = R? (We treat groups over C as real groups byrestriction of scalars, so we don’t need to treat F = C separately.)Representations of real groups are both simpler and morecomplicated than representations of p-adic groups!More complicated: analysisSimpler: arithmetic of F(If we considered representations of p-adic groups over Qp thenthe analysis becomes very difficult.)

One option is to consider unitary representation on Hilbert spaces.There are two problems:

1. This class is not rich enough for some purposes.

2. The representation is too big. (For instance, not enoughcontinuous functionals.)

To address the first problem we can consider Banachrepresentations, i.e., (continuous) representations on Banachspaces.Remark: by the uniform boundedness principle π : G × V → V iscontinuous iff π is separately continuous in G and V .Even more: embed G in SLn and define ‖g‖ as the maximum ofthe entries. Then there exist C and N such that‖π(g)v‖ ≤ C‖g‖N‖v‖ for all v ∈ V .

To address the second problem (too few continuous functionals)we look at smooth vectors V∞: the vectors v such that the mapg 7→ π(g)v from G to V is smooth. We can get smooth vectorsby taking π(f )v =

∫G f (g)π(g)v dg for any v and f ∈ C∞c (G ). In

fact, any smooth vector is of this form (Dixmier-Malliavin).On V∞ acts not only G but also the Lie algebra g (compatiblywith G ) and hence, the universal enveloping algebra U(g). (Wedenote the resulting representation by dπ.) We topologize V∞

with the Sobolev norms

pk(v)2 =∑

n≤k,X1,...,Xn∈B‖dπ(X1 . . .Xn)v‖2

where B is a basis of g. We get a smooth Frechet representation ofmoderate growth, (SFMG) i.e., every vector is smooth and for anyk there exist C , l and N such that pk(π(g)v) ≤ C‖g‖Npl(v) forall v .

Define: S(G ) = {f ∈ C∞(G ) :(X ∗ f )(g)‖g‖N is bounded for all N ∈ N,X ∈ U(g)}.SFMG representations ←→ non-degenerate S(G )-modulesFinally, we can impose the admissibility condition which is nowthat for any τ ∈ K , HomK (τ,V ) is finite-dimensional.If we assume that (π,H) is a representation on a Hilbert space and(S(G ),H∞) is finite length then it is admissible. However, thereare many different H’s with the same H∞.Denote by R(G ) the category of finite length (with respect toS(G )) admissible SFMG representations (any such can be realizedas H∞ for some Hilbert representation)Casselman-Wallach Theorem: V 7→ K -finite part of V defines anequivalence of categories between R(G ) and U(g)-modules offinite length which are also K -modules in a compatible way.contragredient: (π, V ) is defined by V = span of{` ◦ π(f ) : ` ∈ V ∗ (continuous dual) } in V ∗.

Infinitesimal character

The center of U(g) is a polynomial algebra in r variables and itsspectrum corresponds to Hom(T0(R)0,C∗)/W = X ∗(T0)⊗ C/WThe map Π(G )→ X ∗(T0)⊗ C/W

π 7→ character by which z(U(g)) acts on π

is finite-to-one.

Characters

How to define the character of a representation?In the finite-dimensional case χπ(g) = tr π(g); but this won’t workotherwise..For any f ∈ S(G ) π(f ) is of trace class; f 7→ tr π(f ) is adistribution on S(G )It is represented by a locally L1-function χπ which is G -invariant,i.e. tr π(f ) =

∫G χπ(g)f (g) dg (!)

Quotient measures

Let G be a locally compact group with right Haar measure dg .Denote the modulus function by δ : G → R+.If H ⊂ G is a closed subgroup we can define a right-invariantfunctional on continuous functions on G compactly supportedmodulo H satisfying f (hg) = δH

δG(h)f (g) for all h ∈ H, g ∈ G . We

denote it by ∫H\G

f (g)dg

dh.

It is characterized by∫H\G

(∫Hϕ(hg)

δGδH

(h) dh

)dg

dh=

∫Gϕ(g) dg

for every ϕ ∈ Cc(G ).

Induced representations

For any subgroup H of G and a representation (σ,Vσ) of H we canconstruct a representation IndG

H σ of G on the space

{ϕ : G → Vσ smooth| ϕ(hg) = (δHδG

)12 (h)σ(h)[ϕ(g)] ∀h ∈ H, g ∈ G}

with G acting by right translation.If σ is unitary then we can consider

L2- IndGH σ = {ϕ as above | ‖ϕ‖2 :=

∫H\G‖ϕ(g)‖2 dg

dh<∞}.

It is a unitary representation of G .If P = M n U is a parabolic subgroup and σ is a representation of

M then we can pullback σδ12P to P and induce it to G . We write

this as IP(σ) (parabolic induction).We get a functor IP : R(M)→ R(G ). (!)

Matrix coefficients

We would like to realize representations on spaces of functions.The simplest possibility: let ` ∈ π and considerv 7→ `(π(·)v) ∈ C (G ) (matrix coefficients). This is an intertwiningoperator from π to C (G ), with G acting by right translation. Ifπ ∈ Π(G ), this procedure will not be too sensitive on `. (The storyis different if ` is only assumed continuous.)

Hierarchy of representations

Definitions An irreducible representation is called

1. supercuspidal if it has a non-zero matrix coefficient which iscompactly supported in G (and then all its matrix coefficientsare compactly supported

2. square-integrable: unitary central character and matrixcoefficients are square-integrable modulo the center

3. tempered if its matrix coefficients times (1 + log‖g‖)−N are inL2(G ) for some N.

supercuspidal =⇒ square-integrable =⇒ tempered

supercuspidal can only exist if Z (G ) is compactIf G itself is compact then every irreducible representation issupercuspidalIn the Archimedean case a supercuspidal representation exists onlyif G is compactIn the p-adic case any G with compact Z admits manysupercuspidal representationsBasic example: let ρ be a cuspidal representation of G(Fq) andpull it back to a representation of K = G(O) via the canonicalprojection O → Fq. Then IndG

K ρ has finite length and itsconstituents are supercuspidal.essentially supercuspidal: matrix coefficients compactly supportedmodulo the centeressentially square-integrable/tempered: tempered after twist by anunramified character (i.e.

∏ki=1|χi |si where χ1, . . . , χk is a basis

for X ∗(G )).

Every π ∈ Π(G ) is a submodule of IP(σ) where σ ∈ Πess.cusp.(M).(In the real case M is the maximal torus.) Moreover in the p-adiccase, if IP′(σ′) contains π as a subquotient and σ′ ∈ Πess.cusp.(M

′)then (P, σ) and (P ′, σ′) are associate.The data (M, σ)/ ∼ is the p-adic analogue of the infinitesimalcharacter in the Archimedean case.Harish-Chandra classified the square-integrable representations forreal groups parameterizing them in a way very similar to theCartan-Weyl theory for compact Lie groups.For GLn there exists an elegant description of square-integrablerepresentations in terms of supercuspidal representations.(Bernstein-Zelevinsky). However, for other groups the class ofsquare-integrable representations is more stable than the class ofsupercuspidal representations.The tempered representations are the irreducible constituents ofIndG

P σ where σ is square-integrable of M.

Langlands classification

Let P = MU be a parabolic subgroup and τ an essentiallytempered representation of M such that |ωτ | is positive withrespect to P. Then IndG

P τ admits a unique irreducible quotientcalled Langlands quotient or equivalently a unique maximal propersubmodule (which consists of the vectors whose matrix coefficientsare of smaller growth).We write the Langlands data as (P, σ, λ) where σ ∈ Πtemp(M) andτ = σ[λ].Every π ∈ Π(G ) can be obtained as a Langlands quotient; the data(P, σ, λ) is unique up to conjugation.λ measures the (non-)temperedness of π, i.e. the growth of matrixcoefficients.The data for π is (P, σ,−λ) where P is the opposite parabolic toP (i.e., same Levi part M and P ∩ P = M)

In particular, if π = LQ(P, σ, λ) is unitary then necessarily π ' π(where π is the representation π(g) = π(g) on Vπ – the conjugatevector space (same additive group as Vπ but with conjugate scalarmultiplication)). Thus, (P, σ, λ) ∼ (P, σ,−λ).Also, the matrix coefficients are (at the very least) bounded – thisgives a bound on λ.spherical (unramified) representations: having fixed vector underK = G (O)For any unramified character χ of T0 the representation IP0(χ)admits a unique irreducible spherical representation J(χ)Every spherical representation is of this type. Moreover,J(χ) = J(χ′) if and only if χ′ = χw for some w ∈W . Thus,

spherical representations←→ X ∗(T0)⊗ (C/2πi log q)/W

Example: GLn the possible (P, σ) are (n1, . . . , nk ,m, nk , . . . , n1)(possibly m = 0) and (σ1, σ2, . . . , σk , σ, σk , . . . , σ1),(λ1, . . . , λk , 0,−λk , . . . ,−λ1). λ1 > · · · > λk > 0.There is a precise description of the unitary dual of GLn due toVogan in the Archimedean case and Tadic in the p-adic case Thenon-degenerate part is 0 < λ1, . . . , λk <

12 .

Spectral decomposition of L2(G (F ))Recall the Hilbert-Schmidt class E(H) of bounded linear operatorson H with

‖A‖2HS =

∑ei orthonormal basis

‖Aei‖2 = trA∗A = trAA∗ <∞

(does not depend on the basis). We have

E(H) ≡ H⊗H∨ as Hilbert spaces

Peter-Weyl For any compact group G we have an equivalence ofunitary representations of G × G

L2(G ) ' ⊕π∈Π(G)π ⊗ π ' ⊕π∈ΠE(Hπ)

where on the left M(g1, g2)f (x) = f (g−11 xg2), given by

f 7→ (√dππ(f ))π∈Π(G) where dπ = dimπ.

Equivalently, for dense subspace of continuous functions(containing the smooth functions for compact Lie groups) we have

f (1) =∑

π∈Π(G)

dπ tr π(f ).

Plancherel formula in general

L2(G ) ' ⊕([P])

∫Π2(M)

E(Hπ)µpl(π)

(sum over associate classes of parabolic subgroups) In particularthe support of the Plancherel measure is the tempered spectrum.For G semisimple µpl

∣∣Π2(G)

=∑

π∈Π2(G) dπδπ where dπ is the

formal dimension of π.Equivalent statement

f (1) =∑[P]

∫Π2(M)

tr π(f )µpl(π)

for f ∈ S(G ).


Erez Lapid

Lausanne, 2011

L-functions

Roughly speaking, L-functions provide means to glue local data (ofa global object) into a global analytic object. They are Eulerproducts of the form

LS(s,O) =∏p/∈S

L(s,Op)

where L(s,Op) = Pp(p−s)−1, Pp is a monic polynomial (dependingon p) of degree n (the degree of the L-function – independent ofp). Often, but not always, Pp has integer coefficients.As a Dirichlet series

L(s,O) =∞∑n=1

anns

where an is multiplicative and for all p the sequence apk , k ∈ Nsatisfies a linear recurrence relation of order n.Analytic information about L(s,O) yields (at least potentially)valuable arithmetic information.

Functional equation

LS(s,O) = γS(s,O)LS(1− s, O)

where γS is again a local object (handling bad primes, including∞).Another way to write it

L(s,O) = ε0(O)Q12−sL(1− s, O)

where Q ∈ N is the conductor and ε0(O)ε0(O) = 1.The basic and most important L-function is the Riemann zetafunction

ζ(s) =∏p

(1− p−s)−1 =∞∑n=1

1

ns.

Basic problem: there is no technique to obtain meromorphiccontinuation of an Euler product as such. We need an alternativerealization.

Estermann phenomenon

Consider the simplest type of Euler products E (s) =∏

p f (p−s)−1

where f ∈ Z[x ] is a fixed monic polynomial (independent of p).There is a dichotomy: if f is a product of cyclotomic polynomials(equivalently, if we can write

f =

∏ki=1(1− xni )∏lj=1(1− xmj )

for some n1, . . . , nk ,m1, . . . ,ml ∈ N) then

E (s) =

∏ki=1 ζ(ni s)∏lj=1 ζ(mjs)

and hence, E (s) admits meromorphic continuation. Otherwise,E (s) has a natural boundary (i.e., a line Re s = c consists ofaccumulation points of the poles).

Cusp formsLet G = GLn. Consider cuspidal automorphic forms: smooth ϕ onG (Q)\G (A) of uniform moderate growth such that∫

Mm×n−m(Q)\Mm×n−m(A)ϕ(

(Im X0 In−m

)g) dX = 0

for all g ∈ G (A) and 1 ≤ m < n. Equivalently, for any properparabolic subgroup P = MU (defined over Q)∫

U(Q)\U(A)ϕ(ug) du = 0

for all g ∈ G (A). Cusp forms are rapidly decreasing.Let G (A)1 = Ker|det ·| : G (A)→ R>0. Note thatG (A) ' G (A)1 × R>0, G (Q) ⊂ G (A)1 andvol(G (Q)\G (A)1) <∞.L2

cusp(G (Q)\G (A)1) (the closure of the ϕ’s above inL2(G (Q)\G (A)1)) decomposes discretely ⊕π as a representationof G (A) under right translation.

Standard L-functions

Let π be a cuspidal automorphic representation of G (A). We canwrite (abstractly) π = ⊗pπp.

For p /∈ S we can write πp = IndG(Qp)B(Qp) χp where

χp(diag(t1, . . . , tn)) = |t1|s1,p . . . |tn|sn,p and(s1,p, . . . , sn,p) ∈ Cn/ 2πi

log pZn uniquely determined up to

permutation. We call A(πp) = diag(p−s1,p , . . . , p−sn,p) (or rather,its conjugacy class) the Frobenius-Hecke parameters of πp.Define

LS(s, π) =∏p/∈S

L(s, πp) =∏p/∈S

det(1− p−sA(πp))−1

=∏p/∈S

n∏i=1

(1− p−(s+si,p))−1

Integral representationWe want to use Riemann’s method (or Tate’s thesis). This wascarried out by Godement-Jacquet. Let S(An) = ⊗pS(Qn

p)(restricted tensor product with respect to 1Zn

p) – the space of

Schwartz-Bruhat functions. Introduce the zeta integral

Z (f ,Φ, s) =

∫G(A)

Φ(g)f (g)|det g |s+(n−1)/2 dg , Φ ∈ S(Mn(A))

for Re s � 0 where f is a matrix coefficient, i.e.

f (g) =

∫G(Q)\G(A)1

ϕ(xg)ϕ′(x) dx .

This is factorizable: if ϕ, ϕ′ are factorizable vectors in π and πrespectively then f (g) =

∏p fp(gp) where fp is local matrix

coefficients. If also Φ(x) =∏

p Φp(xp) then

Z (f ,Φ, s) =∏p

∫G(Qp)

Φp(gp)fp(gp)|det gp|s+(n−1)/2p dgp.

How to compute these local factors?

Unramified representations

Recall B = T0 n U0 is the Borel subgroup of upper triangularmatrices, T0 – diagonal matrices, U0 – upper unitriangular

matrices. Suppose that Φp = 1Mn(Zp), πp = IndG(Qp)B(Qp) χp,

χp(diag(t1, . . . , tn)) = |t1|λ1 . . . |tn|λn , fp(g) = 〈πp(g)ϕ0, ϕ∨0 〉

where ϕ0, ϕ∨0 are the standard sections{ϕ0(utk) = δB(Qp)(t)

12χp(t)

ϕ∨0 (utk) = δB(Qp)(t)12χ−1

p (t), u ∈ U0(Qp), t ∈ T0(Qp), k ∈ K .

(This is valid for almost all p.) Then

fp(g) =

∫B(Qp)\G(Qp)

ϕ0(xg)ϕ∨0 (x) dx =

∫Kϕ0(kg) dk

zonal spherical function. There exists an explicit formula for these(Macdonald, Casselman), but we will not use it.

∫T0(Qp)

(∫U0(Qp)

Φp(ut) du

)χp(t)δ

− 12

B(Qp)(t)|det t|s+(n−1)/2p dt.

What is∫U0(Qp) Φp(ut) du ? We need to know when does ut have

integral entries. If t = diag(t1, . . . , tn) ∈ T0(Qp) andu = (ui ,j)1≤i ,j≤n ∈ U0(Qp) then ut has t1, . . . , tn on the diagonaland ui ,j tj above the diagonal. So the condition is|t1|p, . . . , |tn|p ≤ 1 and |ui ,j |p ≤ |tj |−1

p for all i < j . Hence,∫U0(Qp)

Φp(ut) du =

{∏nj=1|tj |

−(j−1)p if |t1|p, . . . , |tn|p ≤ 1,

0 otherwise.

On the other hand, δ− 1

2B (t) = |det t|−(n−1)/2

p∏n

j=1|tj |j−1p . We get∫

t1∈Q∗p :|t1|p≤1

. . .

∫tn∈Q∗

p :|tn|p≤1|t1|s+λ1

p . . . |tn|s+λnp dt1 . . . dtn

=[(1− p−(s+λ1)) . . . (1− p−(s+λn))

]−1

provided that Re(s + λi ) > 0 for all i .

Analytic continuationFor a cusp form ϕ on G (Q)\G (A) consider

I (ϕ,Φ, s) =

∫G(A)

Φ(g)ϕ(g)|det g |s+(n−1)/2 dg .

This is not factorizable because ϕ is not a factorizable function.However, by Fubini

Z (f ,Φ, s) =

∫G(A)

Φ(g)

∫G(Q)\G(A)1

ϕ(xg)ϕ′(x)|det g |s+(n−1)/2 dx dg

=

∫G(Q)\G(A)1

∫G(A)

Φ(g)ϕ(xg)ϕ′(x)|det g |s+(n−1)/2 dg dx .

By changing the variable g 7→ x−1g∫G(Q)\G(A)1

∫G(A)

Φx(g)ϕ(g)ϕ′(x)|det g |s+(n−1)/2 dg dx

=

∫G(Q)\G(A)1

I (ϕ,Φx , s)ϕ′(x) dx

where Φx(g) = Φ(x−1g).

It remains to analytically continue I (ϕ,Φ, s). LetG (A)≷1 = {g ∈ G (A) : |det g | ≷ 1} ' G (A)1 × R≷1. FollowingRiemann we write

I (ϕ,Φ, s) =

∫G(Q)\G(A)

∑γ∈G(Q)

Φ(γg)ϕ(6 γg)|det 6 γg |s+(n−1)/2 dg

=

∫G(Q)\G(A)>1

Θ∗Φ(g)ϕ(g)|det g |s+(n−1)/2 dg

+

∫G(Q)\G(A)<1


where Θ∗Φ(g) =∑

γ∈G(Q) Φ(γg). Let

ΘΦ(g) =∑

δ∈Mn(Q)

Φ(δg) = Θ∗Φ(g) + boundary terms.

By Poisson summation formula we have the functional equation

ΘΦ(g) = |det g |−nΘΦ(g ι)

where g ι = tg−1 is the Cartan involution.

So up to boundary terms

I (ϕ,Φ, s) =

∫G(Q)\G(A)>1


+

∫G(Q)\G(A)<1


=

∫G(Q)\G(A)>1


+

∫G(Q)\G(A)<1

Θ∗Φ

(g ι)ϕ(g)|det g |s−(n+1)/2 dg

=

∫G(Q)\G(A)>1


+

∫G(Q)\G(A)>1

Θ∗Φ

(g)ϕ(g ι)|det g |1−s+(n−1)/2 dg

= I (ϕι, Φ, 1− s)

where ϕι(g ι) = ϕ(g).

Boundary termsTo show that the boundary terms (usually) don’t contribute, itsuffices to show the vanishing of∫

G(Q)\G(A)1

∑γ∈Mn(Q) singular

Φ(γg)ϕ(g) dg .

We write this as∫G(Q)\G(A)1

∑η

∑δ∈Gη(Q)\G(Q)

Φ(ηδg)ϕ(g) dg

=

∫G(Q)\G(A)1

∑η

∑δ∈Gη(Q)\G(Q)

Φ(ηδg)ϕ(δg) dg

=∑η

∫Gη(Q)\G(A)1

Φ(ηg)ϕ(g) dg

where η is a set of representatives of the singular orbits of Mn(Q)under G (Q) (acting by right-multiplication) and Gη(Q) is thestabilizer {x ∈ G (Q) : ηx = η}.

Clearly, Gη(Q) is conjugate in G (Q) to G(0 00 Ir

)(Q) which is

{(∗ ∗0 Ir

)} = L(Q)V (Q) where L = {

(∗ 00 Ir

)} and

V = {(

In−r ∗0 Ir

)}. Now,

∫Gη(Q)\G(A)1

Φ(ηg)ϕ(g) dg

=

∫L(Q)V (A)\G(A)1

∫V (Q)\V (A)

Φ(ηvg)ϕ(vg) dv dg

=

∫L(Q)V (A)\G(A)1

Φ(ηg)

(∫V (Q)\V (A)

ϕ(vg) dv

)dg = 0

by cuspidality for r > 0. OTOH r = 0 gives

Φ(0)

∫G(Q)\G(A)1

ϕ(g) dg

which is non-zero only if n = 1 and π = 1.

Langlands’s philosophy

Simply put, any L-function is a product of standard L-functions!In particular, they have meromorphic continuation with finitelymany poles and a functional equation.This can be thought of as a (higher dimensional) reciprocity law.It is extremely deep. For most L-functions we don’t have a cluehow to meromorphically continue them (let alone, the finiteness ofthe poles).The most striking success in this direction so far is the work ofWiles and Taylor-Wiles who showed that the Hasse-Weil zetafunction of an elliptic curve over Q (with some assumptions,subsequently removed) is the L-function of a modular form. (TheTaniyama-Shimura-Weil modularity conjecture.) By earlier workthis implies (but is more fundamental than) Fermat’s LastTheorem.

Another more recent result follows from the work ofKhare-Wintenberger: the Artin L-function of an oddtwo-dimensional Artin representation

ρ : Gal(Q/Q)→ GL2(C), tr ρ(complex conjugation) = 0

(factors through a finite Galois extension of Q) is modular.Even more recently, Taylor and his collaborators have spectacularresults on potential modularity (implying meromorphic continuationand functional equation but without finiteness of poles) for manyhigher degree L-functions (cf. Serre’s minicourse last week).


Erez Lapid

Lausanne, 2011

Whittaker coefficients for GLn

Fix a non-trivial character ψ of A which is trivial on Q, i.e.ψ : Q\A→ C. (All other such characters are of the formψa = ψ(a·) for some a ∈ Q∗, i.e. the Pontryagin dual of Q\A isQ.) Let ψ0 : U0(A)→ C be the characterψ0(u) = ψ(u1,2 + · · ·+ un−1,n). Then ψ0 is trivial on U0(Q). Forany cusp form φ define the Fourier coefficient by

W ψφ (g) =

∫U0(Q)\U0(A)

φ(ug)ψ−10 (u) du, g ∈ G (A).

It satisfiesW ψφ (ug) = ψ0(u)W ψ

φ (g).

Moreover, Wφ(t) is rapidly decreasing for |t1| ≥ · · · ≥ |tn|. Finally,

W ψa

φ (g) =

∫U0(Q)\U0(A)

φ(ug)ψa0(u)−1 du =

∫U0(Q)\U0(A)

φ(ug)

ψ−10 (taut

−1a ) du =

∫U0(Q)\U0(A)

φ( 6 t 6−16a utag)ψ−1

0 (u) du = W ψφ (tag)

where ta = diag(an−1, an−2, . . . , 1).

Recovering φ from Wφ

Let Pn be the subgroup of G consisting of matrices whose last rowis v0 = (0, 0, . . . , 0, 1). Pn is the stabilizer of v0 for the right actionof G on row vectors.We have the following remarkable formula due toPiatetskii-Shapiro and Shalika

φ(g) =∑

γ∈U0(Q)\Pn(Q)

Wφ(γg)

Note that it is not a priori clear that the right-hand side isG (Q)-invariant!In particular, Wφ 6≡ 0 if φ 6≡ 0. (Not true for other split groups.)

Proof of φ(g) =∑

γ∈U0(Q)\Pn(Q) Wφ(γg)

We write Fourier expansion of φ(·g) along the last column

Un−1(R) = {un(ξ) =

(In−1 ξ

0 1

)ξ ∈ Rn−1} ' Rn−1

Doing Fourier analysis on Qn−1\An−1

φ(g) =∑

ξ∈Qn−1

∫Qn−1\An−1

φ(un(t)g)ψ−1(〈ξ, t〉) dt

=∑

ξ∈Qn−1\{0}

∫Qn−1\An−1

φ(un(t)g)ψ−1(〈ξ, t〉) dt

by cuspidality.

Note that Pn(Q) acts transitively (by conjugation) on thenon-trivial characters of Un−1(Q)\Un−1(A) (i.e., on Qn−1 \ {0}).

The stabilizer of ψ0 is Pn−1 = {

∗ ∗ ∗1 ∗1

}. We can therefore

rewrite the above as∑γ∈Pn−1(Q)\Pn(Q)

∫Un−1(Q)\Un−1(A)

φ(ug)ψ−10 (γuγ−1) du

=∑

γ∈Pn−1(Q)\Pn(Q)

∫Un−1(Q)\Un−1(A)

φ(γ−1uγg)ψ−10 (u) du

=∑


∫Un−1(Q)\Un−1(A)

φ(uγg)ψ−10 (u) du.

We repeat this procedure for∫Un−1(Q)\Un−1(A) φ(ug)ψ−1

0 (u) du. We

expand using Fourier on Qn−2\An−2 in the n − 1-column. Again,the constant term vanishes by cuspidality and we replace the sumover Qn−2 \ {0} by Pn−2(Q)\Pn−1(Q). We get∫

Un−1(Q)\Un−1(A)φ(ug)ψ−1

0 (u) du

=∑

δ∈Pn−2(Q)\Pn−1(Q)

∫Un−2(Q)\Un−2(A)

φ(uδg)ψ−10 (u) du

where Un−2 = {

In−2 ∗ ∗0 1 ∗0 0 1

}.Summing over γ ∈ Pn−1(Q)\Pn(Q) we get φ(g) =∑

δ∈Pn−2(Q)\Pn−1(Q)

∑γ∈Pn−1(Q)\Pn(Q)

∫Un−2(Q)\Un−2(A)

φ(uδγg)ψ−10 (u) du

=∑


∫Un−2(Q)\Un−2(A)

φ(uγg)ψ−10 (u) du.

Continuing this way we get by descending induction on i

φ(g) =∑

γ∈Pi (Q)\Pn(Q)

∫Ui (Q)\Ui (A)

φ(uγg)ψ−10 (u) du

where Pi = {(∗ ∗0 u

): u upper unitriangular in GLn+1−i} and Ui

is the group of upper unitriangular matrices of the form

(Ii ∗0 ∗

),

i.e., Ui is the unipotent radical of the parabolic of type(i , 1, . . . , 1). For i = 1, U1 = P1 = U0 and we get

φ(g) =∑

γ∈U0(Q)\Pn(Q)

∫U0(Q)\U0(A)

φ(ug)ψ−10 (u) du

as required.

Local Whittaker modelsFor a local field F we can consider the space of Whittaker functions

W = {W : G (F )→ C | W smooth of moderate growth and

W (ug) = ψ0(u)W (g) for all u ∈ U0(F ), g ∈ G (F )}.In the p-adic case smooth means locally constant (=invariantunder right translation by a small open subgroup), moderategrowth is redundant. G (F ) acts on W by right-translation. Forany irreducible π there exists at most one intertwining operator upto a scalar π →W. Alternatively, there exists at most onecontinuous functional Λ ∈ π∗ such that Λ(π(u)v) = ψ0(u)Λ(v) forall u ∈ U0(F ), v ∈ Vπ. (Gelfand-Kazhdan – p-adic case, Shalika -Archimedean case). For this we used Frobenius reciprocity

HomG(F )(π,W) ' HomU0(F )(π, ψ0)

A 7→ (v 7→ Av(e))Λ 7→ (v 7→ (g 7→ Λ(π(g)v)))It may be that π does not have a Whittaker model (for instance, ifπ is the trivial representation). If it has, we denote it by W(π).

The map φ 7→Wφ gives a non-zero intertwining map from π to theglobal Whittaker space

W = {W : G (A)→ C |W (ug) = ψ0(u)W (g)

∀u ∈ U0(A), g ∈ G (A)} = ⊗Wp

By multiplicity one we get an intertwining mapπ →W(π) := ⊗W(πp) (restriced tensor product with respect tothe spherical Wp normalized by Wp(e) = 1.This is very useful because although cusp forms in π are notfactorizable as functions, their Whittaker functions are.In other words for a factorizable vector φ in π = ⊗pπp we canwrite Wφ(g) =

∏p Wp(gp), g = (gp)p ∈ G (A).

An immediate consequence of φ(g) =∑

γ∈U0(Q)\Pn(Q) Wφ(γg) anduniqueness of Whittaker model: multiplicity one for cuspidalrepresentations on GLn.

Rankin-Selberg integral

Let π, π′ be cuspidal automorphic representations of GLn(A) andGLm(A) respectively. We want to study the degree nm L-function

LS(s, π × π′) =∏p/∈S

det(1− p−sA(πp)⊗ A(π′p))−1

=∏p/∈S

n∏i=1

m∏j=1

(1− p−(s+si,p+s′j,p))−1

where we recall that A(πp) are the Frobenius-Hecke parameters.Sometime this is called Rankin-Selberg convolution.n = 2,m = 1 : Hecke (1920’s)m = n = 2: Rankin and Selberg (independently) - 1940general case: Jacquet–Piatetskii-Shapiro–Shalika in the 1980’s.Final touches by Jacquet 2009.We will discuss the case m = n (most interesting).

Special Eisenstein series for GLn

Let V be a vector space over Q of dimension n.A = {tIn : t ∈ R>0} ' R>0 – the subgroup of central matriceswhich are positive at ∞ and 1 at the p-adic places. We haveG (A) ' G (A)1 × A where G (A)1 = Ker|det ·| : G (A)→ R>0.For Φ ∈ S(V (A)) define for Re s > 1

EΦ(g , s) =

∫A

∑v∈V (Q)\{0}

Φ(vag)|det ag |s da =

∫Aθ∗Φ(ag)ans da.

It is the Mellin transform of the truncated theta function

θ∗Φ(g) =∑

v∈V (Q)\{0}

Φ(vg) = θΦ(g)− Φ(0)

Alternatively, we can write

EΦ(g , s) =

∫A

∑γ∈Pn(Q)\G(Q)

Φ(v0γag)|det ag |s da.

Functional equation: θΦ(g) = |det g |−1θΦ(g ι) where g ι = tg−1.

Tate’s thesis revisited

EΦ(g , s) = |det g |s∫A>1

θ∗Φ(ag)ans da

+ |det g |s−1

∫A>1

θ∗Φ

(ag ι)a−n(s−1) da

+ vol(Q∗\I1)

(|det g |s−1 Φ(0)

s − 1− |det g |s Φ(0)

s

)= EΦ(g ι, 1− s).

The first two terms in the middle expression are entire. Theresidue at s = 1 is the constant function Φ(0).Let F be a number field of degree n over Q. We can embedT := F ∗ ↪→ GLn(Q), i.e. T (Q) = F ∗, T (A) = IF , T (A)1 = I1

F .For a Hecke character χ : F ∗\IF → C∗ of F , Φ ∈ S(AF )∫

T (Q)\T (A)1

EΦ(t, s)χ(t) dt =

∫T (Q)\T (A)

∑x∈F∗

Φ(xt)χ(t)|t|s dt

=

∫IF

Φ(t)χ(t)|t|s dt = local factors × LS(s, χ)

Case m = nLet φ, φ′ be cusp forms in the space of π, π′ resp.. The integral

I (φ, φ′,Φ, s) =

∫G(Q)\G(A)1

φ(g)φ′(g)EΦ(g , s) dg

admits meromorphic continuation. The only possible pole is ats0 = 0, 1 and it occurs if and only if π′ = π. The residue isproportional to

∫G(Q)\G(A)1 φ(g)φ′(g) dg . We may write the

integral as ∫Pn(Q)\G(A)

φ(g)φ′(g)Φ(v0g)|det g |s dg

where v0 = (0, . . . , 0, 1). Using φ(g) =∑

γ∈U0(Q)\Pn(Q) Wφ(γg)we get∫

Pn(Q)\G(A)

∑γ∈U0(Q)\Pn(Q)

Wφ(γg)φ′(g)Φ(v0g)|det g |s dg

=

∫Pn(Q)\G(A)

∑γ∈U0(Q)\Pn(Q)

Wφ(γg)φ′(γg)Φ(v0γg)|det γg |s dg

Casselman-Shalika formula

For λ = (λ1, . . . , λn) ∈ Zn set pλ = diag(pλ1 , . . . , pλn) ∈ T0(Qp).The normalized spherical Whittaker function of πp is given by

Wp(u$λk) = ψ0(u)

{δB(Qp)($λ)

12χλ(A(πp)) λ dominant,

0 otherwise

where χλ denotes the character of the irreducible representation ofGL(n,C) with highest weight λ. Incidently, by the Weyl characterformula

χλ(diag(x1, . . . , xn)) =det(x

λj+n−ji )i ,j=1,...,n

det(xn−ji )i ,j=1,...,n

(The Schur polynomial sλ – they form a basis for the symmetricpolynomials in x1, . . . , xn for λ1 ≥ · · · ≥ λn ≥ 0)

Unramified computationBy Iwasawa decomposition∫

U0(Qp)\G(Qp)Wp(gp)W ′

p(gp)Φp(v0gp)|det gp|s dgp

=∑

λ1≥···≥λn≥0

χλ(A(πp))χλ(A(π′p))p−|λ|s , |λ| = λ1 + · · ·+ λn.

We claim that for Re s � 0 this is equal to

det(1− p−sA(πp)⊗ A(π′p))−1 =∞∑k=0

p−sk tr Symk(A(πp)⊗ A(π′p)).

In other words, for any A,A′ ∈ GLn(C) such that |λ| < |λ′| for alleigenvalues λ of A and λ′ of A′ we have∑

λ1≥···≥λn≥0

χλ(A)χλ(A′−1)

= tr((A,A′)|Sym(Cn ⊗ (Cn)∨)) = tr((A,A′)| Sym(Mn(C))).

The unramified identity

∑λ1≥···≥λn≥0

χλ(A)χλ(A′−1) = tr((A,A′)| Sym(Mn(C))).

Thus, we need to show that under the right and left action ofGLn(C)⊗ GLn(C) on Mn(C) we have

Sym(Mn(C)) ' ⊕λ1≥···≥λn≥0Vλ ⊗ Vλ.

Decomposition of the C[G ] (regular functions on GLn) asG × G -module is

∑Vλ ⊗ Vλ (algebraic Peter-Weyl – formally

equivalent to it). As a G × G -submodule of C[G ], Sym(Mn(C))exactly corresponds to λn ≥ 0 (non-negative power of thedeterminant).

Inner product in GLn

Taking residue at s = 1 in the identity

I (φ, φ′,Φ, s) =

∫G(Q)\G(A)1

φ(g)φ′(g)EΦ(g , s) dg =

LS(s, π × π′)∏p∈S

Ip(Wp,W′p,Φp, s)

we get

Φ(0)(φ, φ′) = Φ(0)

∫G(Q)\G(A)1

φ(g)φ′(g) dg

= ress=1 LS(s, π × π′)

∏p∈S

Ip(Wp,W′p,Φp, 1)

However,

Ip(Wp,W′p,Φp, 1) =

∫U0(Qp)\G(Qp)

Wp(g)W ′p(g)Φp(v0g)|det g | dg

=

∫Pn(Qp)\G(Qp)

∫U0(Qp)\Pn(Qp)

Wp(pg)W ′p(pg)Φp(v0 6 pg)|det pg |

δPn(p)−1 dp dg .

Since δPn = |det| we get∫Pn(Qp)\G(Qp)

(∫U0(Qp)\Pn(Qp)

Wp(pg)W ′p(pg) dp

)Φp(v0g)|det g | dg .

For this to be proportional to Φp(0),∫U0(Qp)\Pn(Qp) Wp(pg)W ′

p(pg) dp must be independent of g , i.e.,

[Wp,W′p]p :=

∫U0(Qp)\Pn(Qp)

Wp(p)W ′p(p) dp

is an invariant pairing!

Finally, since ∫Pn(Qp)\G(Qp)

Φp(v0g)|det g | dg = Φ(0)

(polar coordinates) we get∫G(Q)\G(A)1

φ(g)φ′(g) dg = ress=1 LS(s, π × π′)

∏p∈S

[Wp,W′p]p.

Langlands’s functoriality conjecture (in the context of GLn)

For any algebraic (finite-dimensional) representationρ : GLn(C)→ GLN(C) and any cuspidal representation π ofGLn(A) there exists an automorphic representations σ of GLN(A)(not necessarily cuspidal) such that A(σp) = ρ(A(πp)) for p /∈ S .In particular,

LS(s, σ) = LS(s, π, ρ) :=∏p/∈S

det(1− p−sρ(A(πp)))−1.

so that LS(s, π, ρ) is “nice”.For GL2 we know the principle of functoriality for Sym2

(Gelbart-Jacquet), Sym3, Sym4 (Kim-Shahidi).The principle of functoriality makes sense (and is extremelyprofound) for other groups as well. Recently, Arthur establishedthe functorial transfer from classical groups to GLn (fromSO(2n + 1) to GL(2n), Sp(2n) to GL(2n + 1), SO(2n) to GL(2n)).A key ingredient is the Fundamental Lemma proved by Ngo.

Automorphic representations - WikisIwasawa decomposition Assume that F = R. Gram-Schmidt: every...

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