A spectral decomposition of orbital integrals for $PGL(2,F)$

0.26 MB·

by  David Kazhdan

A SPECTRAL DECOMPOSITION OF ORBITAL INTEGRALS FOR
PGL(2,F) (WITH AN APPENDIX BY S. DEBACKER)
DAVID KAZHDAN
Abstract. Let F be a local non-archimedian field, G a semisimple F-group, dg a Haar
measure on G and S(G) be the space of locally constant complex valued functions f on G
7 with compact support. For any regular elliptic congugacy class Ω = hG ⊂ G we denote by
1 IΩ the G-invariant functional on S(G) given by
0
2 IΩ(f)= f(g−1hg)dg
n ZG
a ThispaperprovidesthespectraldecompositionoffunctionalsIΩ inthecaseG=PGL(2,F)
J and in the last section first steps of such an analysis for the general case.

4
2

Dedicated to A. Beilinson on the occasion of his 60th birthday.
]
T
R Acknowledgments. Many thanks for J.Bernstein, S. Debacker and Y. Flicker who corrected
. a number of imprecisions in the original draft and S. Debacker for writing an Appendix.
h
t I am partially supported by the ERC grant 669655-HAS.
a
m
[
1. Introduction

2
v Let F be a local non-Archimedean field, O be the ring of integers of F, P ⊂ O the
9 maximal ideal, k = O/P the residue field, ̟ a generator of P, q = |k|, val : F× → Z the
9
9 valuation such that val(̟) = 1 and kak = q−val(a), a ∈ F×, the normalized absolute value.
4
ForanyanalyticF-varietyY we denotebyS(Y)thespaceoflocallyconstant complex-valued
0
. functions on Y with compact support and by S∨(Y) the space of distributions on Y.
1
Let G be a group of F-points of a reductive group over F,Z be the center of G, dz a Haar
0
7 measure on Z and dg a Haar measure on G. We denote by H(G) the space of compactly
1
supported measures on G invariant under shifts by some open subgroup. The map 7→ fdg
:
v defines an isomorphisms between the spaces S(G) and H(G).
i
X We denote by Gˆ ⊂ Gˆ ⊂ Gˆ ⊂ Gˆ the subsets of cuspidal, square-integrable and
cusp 2 t
r tempered representations. For any π ∈ Gˆ there exists a notion of the formal degree d(π,dg)
a 2
of π which depends on a choice of a Haar measure dg. We chose this measure in such a way
that the formal degree of the Steinberg representation is equal to 1 and write d(π) instead of
d(π,dg). (See Section 3 for definitions in the case G = PGL(2,F)).
Given a regular elliptic conjugacy class Ω ⊂ G we denote by I the functional on the
Ω
space H(G) given by fdg 7→ f(ghg−1)dg/dz,h ∈ Ω where dg/dz the invariant measure
G/Z
on G/Z corresponding to ourRchoice of measures dg and dz.
Date: January 25, 2017.
1
2 DAVIDKAZHDAN
Remark 1.1. The functional I does not depend on a choice of a Haar measure dg. In
Ω
particular these functionals are canonically defined in the case when G is semisimple.
ˆ
WedenotebyGthesetofequivalent classesofsmoothirreduciblecomplexrepresentations.
ˆ
For any π ∈ G we denote by χ the character of π which the functional on H(G) given by
π
χ (µ) = tr(π(µ)) where π(µ) = π(g)µ.
π G
R
Conjecture 1.1. For any regular elliptic conjugacy class Ω ⊂ G there exists unique measure
µ on the subset Gˆ ⊂ Gˆ of tempered representations such that
Ω t
I = χ µ .
Ω π Ω
Zπ∈Gˆ
We say that the measure µ gives the spectral description of the functional I . If t ∈ G is
Ω Ω
a regular ellipltic element we will write I := I ,Ω = tG.
t Ω
The main goal of this paper is to find the spectral description of the functionals I in the
Ω
case G = PGL(2,F). When the residual characteristic of F is odd such a description (based
on the knowledge of formuals for characters χ ) was given in [SS84].
π
We discuss the general of case of a general reductive group in the last Section 10, but until
Section 10 we assume that G = PGL(2,F).
Let B ⊂ G = PGL(2,F) be the subgroup of upper triangular matricies. Then B = AU
where A ⊂ B is the subgroup of diagonal and U ⊂ B of unipotent matrices. We denote by
A(O) the maximal compact subgroup of A and by X be the group of characters of A(O).
For any x ∈ X we define in Section 2 the notion of depth d(x) ∈ Z of x.
+
We denote by U ⊂ G the subset of non-trivial unipotent elements and by ν a G-invariant
measure on U.
For any x ∈ X we denote by ρ the representation of G induced from the character x
x
of A(O)U ⊂ B and define Gˆ ⊂ Gˆ as the subset of irreducible representations of G which
x
appear as subquotients of ρ . Then Gˆ = Gˆ ,i(x) = x−1 and we have a decomposition of
x x i(x)
Gˆ in the disjoint union
Gˆ = ∪ Gˆ ∪Gˆ
x∈X/i x cusp
where Gˆ ⊂ Gˆ is the subset of cuspidal representations. This decomposition induces a
cusp
direct sum decomposition
(⋆)S(G) = ⊕ S(G) ⊕S(G)
x∈X/i x cusp
and the analogous direct sum decomposition of the space S∨(G) of distributions.
Let δ,ω be the distributions on G given by
δ(f) = f(e),ω(f) = f(u)ν
Z
U
We denote by δ ,ω ∈ S∨(G) the components of δ and ω in the decomposition (⋆) and for
x x
r ≥ 0 define
δ = δ
r x
X
x∈X/i|d(x)≤r
A SPECTRAL DECOMPOSITION OF ORBITAL INTEGRALS 3
and
ω = ω .
r x
X
x∈X/i|d(x)≤r
In Section 2 we define the discriminant d(Ω) ∈ Z of a regular elliptic conjugacy class
+
Ω ⊂ G.
Theorem 1.2. For any elliptic torus T ⊂ G there exist functions c (t),c (t) on T such that
e U
any regular elliptic conjugacy class Ω = tG ⊂ G,t ∈ T we have an equality
I = d(π)χ (t)+c (t)δ +c (t)ω
t π e d(Ω) U d(Ω)
X
π∈Gˆcusp
of distributions.
The Plancerel formula 8.1 and Claim 8.2 provide spectral descriptions of functionals δ
r
and ω and there the spectral descriptions of I .
r r
2. The structure of groups A and PGL(2,F)
For any r ∈ Z we define U ⊂ O× by
≥0 r
U = O×; U = 1+Pr, r > 0.
0 r
We denote by da the Haar measure on F with da = 1 and by d×a the Haar measure on
O
F× with d×a = 1. Then d×a = (1−q−1)−1dRa/kak.
O×
We denRote by Θ the group of characters of F×, by i the involution of Θ given by θ 7→ θ−1
and by Θ ⊂ Θ the subgroup of characters θ such that θ2 = Id. We can consider θ ∈ Θ as
2 2
a character of F×/(F×)2.
We denote by Θ ⊂ Θ the subgroup of unramified characters and write Θ = Θ ∩Θ .
un 2,un 2 un
It is clear that |Θ | = 2.
2,un
We denote by X the group of characters of O× and by X ⊂ X the subgroup of characters
2
x such that x2 = Id. For any x ∈ X we denote by Θ ⊂ Θ the subset of characters θ such
x
that θ|O× = x. For any x ∈ X we define
2
Θ = Θ ∩Θ .
2,x x 2
It is clear that |Θ | = 2 and that the group Θ acts simply transitively on Θ for all
2,un 2,un 2,x
x ∈ X. So |Θ | = 2 for all x ∈ X.
2,x
It is clear that the map
Θ → C×, θ 7→ θ(̟),
defines a bijection Θ → C× for any x ∈ X. This isomorphism induces a structure of an
x
algebraic variety on Θ , for each x ∈ X. We denote by C[Θ ] the algebra of regular functions
x x
on Θ which is isomorphic to C[z,z−1].
x
In the case when x2 = Id the involution i acts on Θ . We denote by C[Θ /i] ⊂ C[Θ ] the
x x x
subring of invariant functions. It is clear that C[Θ /i] is isomorphic to C[z′],z′ = z +z−1.
x
Definition 2.1. For any x ∈ X we denote by d(x) the minimal integer r ≥ 0 such that the
restriction of x2 to U is trivial. Thus x2|U = 1 and x2|U 6= 1 where subgroups U
r d(x) d(x)−1 r
are defined in the beginning of this Section.
4 DAVIDKAZHDAN
G˜ = GL(2,F),G′ = {g ∈ GL(2,F)|max kg k = 1}
i,j∈(1,2) ij
p˜ : G˜ → G = PGL(2,F) be the natural projection, and p the restriction of p˜ on G′. The
map p : G′ → G is surjective and the group O× acts simply transitively on fibers of p.
Claim 2.1. For any µ ∈ H = H(G) there exists unique O×-invariant measure µ˜ ∈ H(G˜)
supported on G′ such that p µ˜ = µ.
⋆
We denote by p⋆ : H(G) → H(G˜) the map µ → µ˜. It is clear that the map p⋆ is G-
equivariant.‘
We often describe elements g ∈ G = PGL(2,F) in terms of a preimage g in GL(2,F)
under the map p : GL(2,F) → G and matrix coefficients of g . For any g ∈ G the ratio
ij
eg2 g2 e
11 does not depend on a choice of a representative g. We denote it by 11 .
det(eg) e det(g)
We denote by K ⊂ G the image of GL(2,O) and by A the image of the group of diagonal
e
matrices. We use the map
a11 0 7→ a /a
0 a22 11 22
to identify A with F× and A(O) wit(cid:0)h O×, w(cid:1)here A(O) = A∩K. We denote by ∈˜ GL(2,F)
the matrix
(1 0 ) 7→ a /a
0 ̟ 11 22
and by t ∈ A the image of t˜in A.
The following is well known.
Claim 2.2. The subsets KtnK ⊂ G, n ≥ 0, are disjoint, and G = ∪ KtnK.
n≥0
We write G≤m := ∪ KtnK.
0≤n≤m
We define Γ = K and for any d ≥ 1 we denote by Γ ⊂ K the image of the subgroup Γ˜
0 d d
of matrices g in GL(2,O) with g ∈ Pd. So I := Γ is an Iwahori subgroup of G. We denote
21 1
˜ ˜
by p : Γ → Γ the restriction of p on Γ .
1 1 1 1
We denote by U ⊂ G the image of the subgroup of matrices of the form
g = (1 u),
u 0 1
write B = AU, and denote by b 7→ ¯b the projection B → B/U ≃ A ≃ F×. For any θ ∈ Θ
¯
we denote by the same letter θ the character of B given by b 7→ θ(b).
We denote by det the map
2
det : G → F×/(F×)2, g 7→ det(g)(F×)2
2
and by G0 the kernel of det . If char(F) 6= 2 then G0 is an open subgroup of G.
2 e
For any character x ∈ X such that d = d(x) > 0 we denote by x : Γ → C× the map
d
g2
g 7→ x 11 . e
(cid:18)det(g)(cid:19)
If d(x) = 0, that is x2 = Id, we define a character x of Γ = K by x(g) = x(det (g)).
0 2
Claim 2.3. For any x ∈ X the map x is a character of Γ .
e d(x) e
Definition 2.2. For any regular elliptic conjugacy class Ω ⊂ G we let d(Ω) be the biggest
e
number d such that Ω intersects Γ .
d
A SPECTRAL DECOMPOSITION OF ORBITAL INTEGRALS 5
3. Basic structure of representations of G
We say that a measure µ on G is smooth if it is R-invariant for some open subgroup
R ⊂ G. Let H be the space of complex-valued compactly supported smooth measures µ on
G. For any open compact subgroup R ⊂ G we denote by ch ⊂ H the normalized Haar
R
measure on R.
Convolution, denoted by ∗, defines an algebra structure on H. The algebra H acts on
S(G)G by convolution from the right, (f,µ) 7→ f ∗µ, and also from the left.
The group G acts on H by conjugation. We denote by H the space of coinvariants which
G
is equal to the quotient H/[H,H].
We denote by C the category of smooth complex representations of G and by G the set of
equivalence classes of smooth irreducible representations of G.
b
The group G acts on P1(F) and therefore on the spaces S(P1(F)). It is clear that the
subspace C of constant functions invariant and we obtain the Steinberg representation St
of G on the space S(P1(F))/C. It is well known (see [GGP] ) the representation St of
G is irreducible. For any θ ∈ Θ we denote by C the one-dimensional representation
2 θ
g 7→ θ(det (g)), and define St = St⊗C .
2 θ θ
For any (π,V) ∈ C, µ ∈ H, we define
π(µ) = π(g)µ ∈ End(V).
Z
G
For irreducible representations π of G the operator π(µ) is of finite rank for any µ ∈ H and
we define the character χ on G, as a generalized function (a functional on H) by
π
χ (µ) = trπ(µ), µ ∈ H.
π
By [JL70], there exist a locally L1-function on G, (that we denote by χ ) such that
π
χ (µ) = χ µ
π π
Z
G
We define a map κ : µ 7→ µ from H to functions on G by
µ(π) = trπ(µ).
b b
It is clear that κ descends to a map from H to functions on G.
b G
Wesaythatanirreduciblerepresentation(π,V)ofGissquare-integrableifitisunitarizable
b
(that is, there exists a nonzero G-invariant Hermitian form (,) on V), and for every v ∈ V
the function g 7→ (π(g)v,v) on G belongs to L2(G). We denote by G ⊂ G the subset of
2
square-integrable representations. Let dg be a Haar measure on G.
b b
The following Claim follows from [HC70].
Claim 3.1. a) For every (π,V) ∈ G there exists a number deg(π) = deg(π,dg) > 0, called
2
the formal degree of π, such that
b
1
|m (g)|2dg = , m (g) = (π(g)v,v),
v v
Z deg(π)
G
for any v ∈ V,(v,v) = 1, where dg is a Haar measure on G.
b) There exists a unique choice of dg with deg(St,dg) = 1.
6 DAVIDKAZHDAN
c) For any irreducible square-integrable representation (π,V) and any v ∈ V,(v,v) = 1,
the sequence of locally constant functions
(I (v))(g) := m (hgh−1)dh
n v
Z
h∈G≤n
on G converges as a generalized function to the the character χ /deg(π,dg). In other words,
π
for any µ ∈ H the sequence { I (v)µ} converges to µˆ(π)/deg(π).
n
R
For any smooth representation (π,V) of G we denote by J(V) the normalized Jacquet)
functor which is a representation of A acting on the space V/V(U) where V(U) is the span
of {π(u)v −v,u ∈ U,v ∈ V}. We define the action of A on JV) by a 7→ kak1/2π(a), a ∈ A,
of A on V . Here kak = kt /t k for a represented by t1 0 .
U 1 2 0 t2
We say that V is cuspidal if J(V) = {0}. (cid:0) (cid:1)
We denote by C the subcategory of cuspidal representations and by G ⊂ G the
cusp cusp
subset of equivalence classes of irreducible cuspidal representations. Since matrix coefficients
b b
of cuspidal representation of G have compact support (see [JL70]) we have an inclusion
G ⊂ G .
cusp 2
b b 4. Induced representations
For any θ ∈ Θ we denote by (π ,R ) the representation of G unitarily induced from the
θ θ
¯
character b 7→ θ(b) of B. So R is the space of locally constant complex valued functions f
θ
on G such that
f(gb) = θ(¯b)k¯bk1/2f(g), g ∈ G, b ∈ B,
and G acts on R by left shifts: (π (x)f)(g) = f(x−1g).
θ θ
Since G = KB, the restriction to K identifies the space R with the space R , x = θ|O×,
θ x
where R is the space of locally constant functions f on K such that
x
¯
f(kb) = θ(b)f(k), k ∈ K, b ∈ B ∩K.
It is clear that in this realization the operator π (µ) ∈ End(R ) is a regular function on
θ x
θ ∈ Θ for any µ ∈ H and so the function
x
µ : θ 7→ µ(π ), θ ∈ Θ ,
x θ x
belongs to C[Θ ].
x b b
The following result is well known, see [JL70].
Proposition 4.1. a) For any θ ∈ Θ we have End (R ) = C.
G θ
b) A representation π is reducible if and only if θ(a) = θ (a)kak1/2 or θ(a) = θ (a)kak−1/2
θ 2 2
where θ ∈ Θ . In the second case π has a one-dimensional subrepresentation C , and the
2 2 θ θ2
quotient is isomorphic to St . In the first case π has St as a subrepresentation and the
θ2 θ θ2
quotient is isomorphic to C .
θ2
c) Let θ,θ′ ∈ Θ be such that π ,π are irreducible. Then the representations π ,π are
θ θ′ θ θ′
isomorphic iff θ′ = θ±1.
d) We have a disjoint union decomposition
G = G ∪(∪ C )∪ ∪ π .
2 θ2∈Θ2 θ2 θ∈(Θ−Θ2)/i θ
(cid:0) (cid:1)
b b
A SPECTRAL DECOMPOSITION OF ORBITAL INTEGRALS 7
e) We have a disjoint union decomposition
G = G ∪(∪ St ).
2 cusp θ2∈Θ2 θ2
f) For any θ ∈ Θ the character bχ :=bχ is given by a locally L1-function on G supported
θ πθ
on split elements such that
θ(a/b)+θ(b/a)
χ ((a 0)) = .
θ 0 b k(a−b)2/abk1/2
Definition 4.1. (1) For any x ∈ X, we denote by (τ ,V ) the representation of G by left
x x
shifts on the space of locally constant compactly supported functions f on G such that
f(gtu) = x(t)f(g), g ∈ G, t ∈ A(O), u ∈ U.
(2) We denote by f ∈ V the function supported on Γ U, d = d(x), and such that
0 x d
f (γu) = x(γ), γ ∈ Γ , u ∈ U.
0 d
(3) We denote by pθ, θ ∈ Θx, the mape
p : V → R , (p (f))(g) = f(gtn)qnθ(̟n),
θ x θ θ
Xn∈Z
where as before t is the image in G of
(1 0 ) ∈ GL(2,F).
0 ̟
We also have
Proposition 4.2. If M ⊂ V is a G-invariant subspace such that p (M) = R for all θ ∈ Θ
x θ θ x
then M = V .
x
Proof. As follows [Be92] it suffices to show that there is no nonzero morphism from V /M to
x
an irreducible representation of G. But as follows from [BZ76] all morphisms from V to an
x
irreducible representation of G are factorizable through a projection p for some θ ∈ Θ . (cid:3)
θ x
Corollary 4.3. If x2 6= Id then the function f generates V as an H-module.
0 x
Proof. It is clear that f := p (f ) ∈ R is not equal to 0. Moreover, it follows from
θ,0 θ 0 θ
Proposition 4.1 a),b) that it generates R as an H-module. But then Proposition 4.2 implies
θ
that f generates V as an H-module. (cid:3)
0 x
The following result follows from Corollary 4.3. We assume that x2 6= Id and use the
identification of the ring C[Θ] with C[z,z−1] as in the Introduction. Let
α : C[Θ ] ≃ C[z,z−1] → End (V )
x G x
be the algebra morphism defined by ((α(z))(f))(g) = q−1f(gt−1), f ∈ V .
x
8 DAVIDKAZHDAN
Corollary 4.4. a) For any S ∈ End (V ), and θ ∈ Θ , the map S preserves the subspace
G x x
ker(p ) ⊂ V and so defines S(θ) ∈ End (R ) = C.
θ x G θ
b) For any S ∈ End (V ), the function S on Θ belongs to C[Θ ] ≃ C[z,z−1].
G x b x x
c) The maps
b
End (V ) → C[Θ ],S 7→ S
G x x
and
b
C[Θ ] → End (V ),s 7→ α(s)
x G x
are mutually inverse.
5. Structure of the representation (τ ,V ) when d(x) > 0
x x
In this section we fix a character x ∈ X such that d(x) > 0 (so x2 6= Id).
Definition 5.1. (1) We denote by ρ the representation indG x of G on the space W of
x Γd x
locally constant functions φ on G such that
e f
φ(gγ) = x(γ)φ(g), g ∈ G, γ ∈ Γ ,
d
and by W ⊂ W the subspace of functions with compact support.
x x e
(2) Denote by φ ∈ W the function supported on Γ and equal to x there, and define
0 x d
f
µ := φ ch ∈ H.
0 0 Γd e
(3) Let A : V → W , B : W → V be the G-morphisms defined by
x x x x
f
A(f) = f ∗µ , B(φ) = φ , φ (g) = φ(gu)du,
0 U U
Z
U
where du is the Haar measure on U which is normalized by du = 1.
U∩K
R
Lemma 5.1. a) B(φ ) = f .
0 0
b) A(f ) = φ .
0 0
c) A defines an isomorphism A : V → W .
x x
d) End (V ) ≃ End (W ).
G x G x
Proof. Part a) is clear. It is also clear that the restriction of A(f ) to Γ is equal to x and
0 d
that supp(A(f )) ⊂ Γ UΓ . So to prove (b) it suffices to check that for any u ∈ F,kuk > 1,
0 d d
we have e
f (g γ)x(γ)−1dγ = 0,
0 u
Z
Γd
where dγ is the normalized Haar measure oneΓ and g = (1 u). To see this, write γ as γ γ ,
d u 0 1 0 1
γ = (1 0), γ = (a b). Note that x(γ) = x(γ ) = x(a/d). The integral equals
0 c 1 1 0 d 1
Z f0((10 u1)(1c 10)γ1)x(γ1)−1dcdγe1 = Z fe0(cid:16)(cid:16)1+cuc (1+u0c)−1(cid:17)γ1(cid:16)01 ad11+uuc (cid:17)(cid:17)x(a/d)−1dcdγ1
e
= x(1+uc)2dc, {c ∈ Pd; 1+uc ∈ O×},
Z
A SPECTRAL DECOMPOSITION OF ORBITAL INTEGRALS 9
since f is supported on Γ U (so it vanishes unless |1 + uc| = 1)and we are integrating a
0 d
nontrivial character the integral is 0.
The part c) follows from the parts a),b) since by Corollary 4.3‘ the function f generates
0
V as an H-module and ,as easy to check, the function φ generates W as an H-module.
x 0 x
The part d) follows from c).
(cid:3)
6. Algebras of endomorphisms
As before we fix (in this section) a character x ∈ X such that x2 6= 1.
Lemma 6.1. Let H.′ ⊂ H be the subalgebra of measures µ with
x
l (µ) = r (µ) = x(γ)µ, γ ∈ Γ ,
γ γ d
where l , r are left and right shifts by γ. Then
γ γ e
(1) µ = φ ch is the unit of H′.
0 0 Γd x
(2) Convolution on the right defines an isomorphism β : H′ → End (W ).
x G x
Proof. (1) is clear.
For (2), note that the map S 7→ S(φ )ch defines a morphism β˜ : End (W ) → H′.
0 Γd G x x
˜ ˜
One checks that the compositions β ◦β and β ◦β are the identity maps. So we can identify
End (W ) with H′.
G x x
(cid:3)
As follows from from Lemma 5.1 we identify the ring End (W ) with End (V ) and
G x G x
therefore (by Corollary 4.4 ) with the ring C[z,z−1].
For any n ∈ Z we denote by φ ∈ W the function supported on Γ tnΓ with
n x d d
φ (γ′tnγ′′) = x(γ′γ′′), γ′, γ′′ ∈ Γ ,
n d
and write µ = φ dg ∈ H′.
n n x e
The following result follows from the commutativity of the algebra H′ and Frobenius
x
reciprocity.
Claim 6.2. Let p : (ρ ,W ) → (π,W) be an irreducible quotient of W . Then
x x x
a) The action of End (W ) = H on W preserves ker(p) and therefore induces a homomor-
G x x x
phism p : H′ → End (W) = C.
x G e
b) For any µ ∈ H′ we have
x
e
p(µ) = µ(π).
Lemma 6.3. a) The map µ 7→ µ(πθ), θe∈ Θx,bdefines an isomorphism Hx′ → C[Θx].
b) µ (θ) = czn, c 6= 0 where we identify the space Θ with C×, θ 7→ z = θ(̟).
n x
c) The set {µ ; n ∈ Z} is a babsis of the space H′.
n x
b
Proof. a) The first part follows immediately from Lemma 5.1 and Proposition 6.2.
b) Let
R0 = {f ∈ R |π (γ)f = x(γ−1)f, γ ∈ Γ }.
θ θ θ d
e
10 DAVIDKAZHDAN
It follows from Lemma 5.1 and Claim 6.2 that dim(R0) = 1 and that this space is equal
θ
to C·f , where f was defined to be p (f ) in the proof of Corollary 4.3. Since f (e) = 1,
θ,0 θ,0 θ 0 θ,0
it is sufficient to show that
((π (µ ))(f ))(e) = czn,
θ n θ,0
but this is immediate. Since the map µ → µˆ is not a zero map we see that c 6= 0.
The part c) follows from b). (cid:3)
7. Categories of representations
Fix x ∈ X. Let G ⊂ G be the set of equivalence classes of irreducible representations
x
of G which appear as subquotients of (τ ,V ). It can be described as the set of equivalence
x x
b b
classes of irreducible subquotients of the representations {R } for θ ∈ Θ . It follows from
θ x
Proposition 4.1 that the set G depends only on the image of x in X/i and that for distinct
x
x′,x ∈ X/i the sets G and G are disjoint.
x bx′
We denote by C ⊂ C the subcategory of representations the equivalence classes of whose
x
b b
irreducible subquotients belong to G .
x
The following result is well known (see [BZ76] for parts a) and b) and [BDK86] for part
b
c)).
Proposition 7.1. a) We have a decomposition
(1)C = C ⊕ ⊕ C .
cusp x∈X/i x
(cid:0) (cid:1)
This decomposition defines the direct sum decompositions
(2)S(G) = S(G) ⊕ ⊕ S(G)
cusp x∈X/i x
(cid:0) (cid:1)
and
(3)S∨(G) = S∨(G) ⊕ ⊕ S∨(G)
cusp x∈X/i x
and (cid:0) (cid:1)
(4)H = H ⊕ ⊕ H .
G G,cusp x∈X/i G,x
b) For any x ∈ X, x2 6= Id, and µ ∈ H , the f(cid:0)unction µ is(cid:1)supported on Θ , and the map
G,x x
κ : µ 7→ µ defines an isomorphism from H to C[Θ ].
G,x x
c) For any x ∈ X , µ ∈ H , the function µ is supportedbon
2 G,x
b
(Θ /i)∪ ∪ St ,
x b θ∈Θ2,x θ
(cid:0) (cid:1)
and the map κ : µ 7→ µ defines an isomorphism from H to
G,x
C[Θ /i]⊕ ⊕ C .
b x θ∈Θ2,x θ
(cid:0) (cid:1)
Lemma 7.2. a) For x ∈ X −X we have H′ ⊂ H .
2 x x
b) The map H′x → H is an isomorphism.
G,x
Proof. a) Let (π,V) be an irreducible representation such that π(µ) 6= 0 for some µ ∈ H .
x
We want to show that π ∈ G .
x e
Since π(µ) 6= 0 we see that V0 6= {0}, where
b
V0 = {v ∈ V|π(γ)v = x(γ−1)v, γ ∈ Γ }.
d
Since V|Γ = V0 6= {0},
d e