The Langlands classification for graded Hecke algebras

Abstract

We establish the Langlands classication for graded Hecke alge- bras. The proof is analogous to the proof of the classication of highest weight modules for semisimple Lie algebras.

Full text

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 124, Number 4, April 1996
THE LANGLANDS CLASSIFICATION
FOR GRADED HECKE ALGEBRAS
SAM EVENS
(Communicated by Roe Goodman)
Abstract. We establish the Langlands classification for graded Hecke alge-
bras. The proof is analogous to the proof of the classification of highest weight
modules for semisimple Lie algebras.
0. Introduction
In this paper, we prove that the classification of irreducible representations of
a graded Hecke algebra can be reduced to the classification of tempered represen-
tations. In particular, any irreducible representation V of a graded Hecke algebra
H can be realized as the unique irreducible quotient of H ⊗
H
P
U, where H
P
is the
graded Hecke algebra associated to a parabolic subgroup P and U is tempered on
the semisimple part of H
P
and has real part in a certain positive cone on the central
part. We call (P, U) Langlands data for V and write V = J(P, U). Moreover, we
show J(P, U)
∼
=
J(P
0
,U
0
) implies P = P
0
and U
∼
=
U
0
, so that irreducible represen-
tations of H are completely classified by Langlands data. Of course, for reductive
real groups, this result is due to Langlands [La], and for reductive p-adic groups,
this result is due independently to Silberger and Wallach [Si, BW]. For Hecke alge-
bras associated to p-adic groups with connected center and q not a root of unity,
and for graded Hecke algebras associated to p-adic groups, this result can be de-
duced from the classification of irreducible modules in [KL, Lu1, Lu2]. Thus, our
result is really new only for representations of graded Hecke algebras not associated
to p-adic groups with connected center (for example, with general unequal param-
eters). We also note that the case of unramified representations of p-adic groups is
equivalent to the case of representations of Hecke algebras when the Hecke algebra
parameter q is a prime power, and that the representation theory of graded Hecke
algebras is equivalent to the representation theory of Hecke algebras when q is not
a root of unity.
Our proof is much simpler than the proofs in the other cases, since the diffi-
culties that arise in the real and p-adic settings do not arise here. In particular,
the exponents are just weights, the Jacquet module is given by restriction, and
intertwining operators are unnecessary. As a consequence, the proof reduces to a
standard lemma of Langlands used in the other cases and some simple considera-
tions analogous to those used in the study of highest weight modules. Essentially
Received by the editors October 5, 1994.
1991 Mathematics Subject Classification. Primary 22E50.
Supported by NSF postgraduate fellowship.
c
1996 American Mathematical Society
1285
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1286 SAM EVENS
the same proof works for the case of Hecke algebras. We note that the classification
of tempered representations for graded Hecke algebras is likely to be nontrivial,
since it does not appear to reduce to the cases associated to p-adic groups, which
is known (see [BZ], [KL], [Lu1]).
1. The graded Hecke algebra
1.1. We give the definition of the graded Hecke algebra associated to a root system
and a system of parameters as in [Lu2]. Let (X, R, Y,
ˇ
R, Π) be a root system,
where as usual X and Y are free finitely generated abelian groups together with a
perfect pairing between them, R ⊂ X and
ˇ
R ⊂ Y are finite subsets with a bijection
between them, and Π is a subset of R. R is called the set of roots,
ˇ
R is called
the set of co-roots, and Π = {α
1
,...,α
l
} is called the set of simple roots. These
satisfy well-known properties (cf. [Lu2, 1.1]). There are induced systems of positive
roots R
+
and
ˇ
R
+
. We will assume that the system is reduced so that if α ∈ R,
then 2α 6∈ R. Let W be the associated Weyl group, which is generated by simple
reflections s
α
,α∈Π.A system of parameters c :Π→Nis a function such that
c(α
i
)=c(α
j
)ifthereisw∈W such that w(α
i
)=α
j
,together with a function
c
∗
: {α ∈ Π|ˇα ∈ 2Y }→N.
Let t = X ⊗ C, t
∗
= Y ⊗ C be the dual vector space, and t
R
and t
R
∗
be the real
spans of X and Y. For λ ∈ t
∗
,Re(λ) has the obvious meaning. Let A = S[t].
1.2. Definition. Given the root system (X, R, Y,
ˇ
R, Π) and the system of param-
eters c, the associated graded Hecke algebra is the tensor product of algebras
H
∼
=
C[W ] ⊗A⊗C[r]
subject to the cross relations
C[r] is in the center of H
and
x · s
α
− s
α
· (s
α
(x)) = (x − s
α
(x))(g(α) − 1),
where x ∈ t,α∈Πand
g(α)=1+
2c(α)r
α
,ˇα6∈ 2Y,
g(α)=1+
(c(α)+c
∗
(α))r
α
, ˇα ∈ 2Y.
We remark that when all c(α)=1andˇα6∈ 2Y for all α, then the second relation
can be written as x · s
α
− s
α
· (s
α
(x)) = 2rA
α
(x), where A
α
is the so-called BGG
operator.
1.3. Representation theory. Let a = {x ∈ t|ˇα(x)=0,α ∈ Π} and a
∗
=
{λ ∈ t
∗
|λ(α)=0,α ∈ Π}. Then S[a] is in the center of H and, as an algebra, H
decomposes into a tensor product H
∼
=
H
s
⊗ S[a]. Here H
s
is the graded Hecke
algebra associated with the root system (X
s
,R,Y
s
,
ˇ
R, Π), where X
s
and Y
s
are the
subsets of X and Y perpendicular to a
∗
and a respectively. If V is an irreducible
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THE LANGLANDS CLASSIFICATION FOR GRADED HECKE ALGEBRAS 1287
H–module, then V
∼
=
V
s
⊗ C
ν
, where V
s
is an irreducible representation of H
s
and
C
ν
is a one-dimensional character of S[a].
Let H be a graded Hecke algebra and let V be a finite dimensional A–module.
Then the abelian algebra A induces a generalized weight space decomposition V =
L
λ∈t
∗
V
λ
, where V
λ
is the largest subspace of V on which x−λ(x) acts nilpotently
for all x ∈ t. The set of λ ∈ t
∗
such that V
λ
6= 0 are called the weights of V .IfVis
any irreducible H–module, then the center of H acts by a character and it follows
that V is finite dimensional. In particular, V has a weight space decomposition.
1.4. Definition. Let V be an irreducible H–module. Then V is essentially tem-
pered if for every weight λ of V, Re(λ(x
i
)) ≤ 0, where x
i
is a fundamental coweight,
defined by the requirement that ˇα
j
(x
i
)=δ
ij
and ν(x
i
)=0,ν∈a
∗
.V is tempered
if V is essentially tempered and for every weight λ of V, Re(λ|a
R
)=0,where a
R
is the real span of the set of x ∈ X perpendicular to the coroots.
1.5. Parabolic subalgebras. Let Π
P
be a subset of Π and let R
P
be the set
of roots generated by Π
P
and
ˇ
R
P
the set of roots generated by ˇα, α ∈ Π
P
. Then
(X, R
P
,Y,
ˇ
R
P
,Π
P
) is a root system. Recall that H is the graded Hecke algebra
associated to a root system (X, R, Y,
ˇ
R, Π) and system of parameters c. Let H
P
be
the graded Hecke algebra associated to the root system (X, R
P
,Y,
ˇ
R
P
,Π
P
)andthe
restriction of the parameter set c. There is an obvious inclusion H
P
⊂ H given by
identifying the abelian algebras A for the two algebras and using the embedding
W
P
→ W given by identifying s
α
∈ W
P
with s
α
∈ W, for α ∈ Π
P
. We denote the
subalgebra H
s
described in 1.3 by H
M
s
.
For this ro ot system (X, R
P
,Y,
ˇ
R
P
,π
P
), we have the subspaces a = a
P
and
a
∗
= a
P
∗
as in 1.3. We denote
a
∗+
= {ν ∈ a
∗
|Re(ν(α)) > 0,α∈ Π−Π
P
}.
If U is a H
P
–module, then H ⊗
H
P
U is the induced H– module. If
˜
U is a H
M
s
-
module and C
ν
is a one dimensional representation of S(a) induced by ν ∈ a
∗
, then
we denote by
˜
U ⊗ C
ν
the corresponding H
P
module.
2. The Langlands classification
2.1. Theorem. (i) Let V be an irreducible H–module. Then V is a quotient of
H ⊗
H
P
U, where U =
˜
U ⊗C
ν
is such that
˜
U is a tempered H
M
s
–module and ν ∈ a
∗+
.
(ii) If U is as in (i),thenH⊗
H
P
Uhas a unique irreducible quotient, which we
denote by J(P, U).
(iii) If J(P,
˜
U ⊗ C
ν
)
∼
=
J(P
0
,
˜
U
0
⊗ C
ν
0
), then P = P
0
,
˜
U
∼
=
˜
U
0
as H
M
s
-modules,
and ν = ν
0
.
2.2. We first require two lemmas due to Langlands. Let Z be a real inner product
space of dimension n. Let { ˇα
1
, ··· , ˇα
n
} be a basis such that ( ˇα
i
, ˇα
j
) ≤ 0 whenever
i 6= j. Let {β
1
, ··· ,β
n
} be a dual basis so that ( ˇα
i
,β
j
)=δ
ij
. For a subset F of Π,
let
S
F
= {
X
j6∈F
c
j
β
j
−
X
i∈F
d
i
ˇα
i
|c
j
> 0,j 6∈ F, d
i
≥ 0,i∈F}.
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1288 SAM EVENS
2.3. Lemma [BW, IV, 6.11]. Let x ∈ Z. Then x ∈ S
F
for a unique subset F =
F (x).
If x ∈ Z, then let x
0
=
P
j6∈F
c
j
β
j
, where x ∈ S
F
and x =
P
j6∈F
c
j
β
j
−
P
i∈F
d
i
ˇα
i
. It is clear that if x
0
= y
0
,thenF(x)=F(y).Define a partial ordering
on Z by setting x ≥ y if x − y =
P
a
i
≥0
a
i
ˇα
i
.
2.4. Lemma [BW, IV, 6.13]. If x, y ∈ Z and x ≥ y, then x
0
≥ y
0
.
2.5. Proofof2.1.FirstwenotethatwemayassumethatRgenerates t and
hence
ˇ
R generates t
∗
, by an easy argument.
2.6. Proof of 2.1(i). Throughout the proof, denote the simple co-roots by ˇα
1
, ··· , ˇα
l
,
and let {β
1
, ··· ,β
l
}be a dual basis to the simple coroots relative to the Killing form
as in 2.2. Let V be an irreducible representation of H. Let λ be a weight of A on
V such that Re(λ) is maximal among real parts of weights of V. Let F = F (Re(λ))
and let Π
P
= F, a = a
P
, and a
∗
= a
P
∗
. Let a
s
be the elements of t perpendicular to
a
∗
. Similarly, let a
s
∗
be the elements of t
∗
perpendicular to a. We have a splitting
t
∗
= a
∗
⊕ a
s
∗
and we can identify restriction to a (resp. a
s
) with projection onto
a
∗
(resp. a
s
∗
) by using the Killing form.
Let ν = λ|
a
. By construction, ν ∈ a
∗+
. Let W be an irreducible representation of
H
P
appearing in V such that C[a] acts by ν on W. Let µ be a C[a
s
] weight appearing
in W. Then Re(µ)=−
P
i∈F
z
i
ˇα
i
.To show W is a tempered representation of H
M
s
we must show that all z
i
≥ 0.
Clearly, µ + ν is a weight of V and we have
Re(µ + ν)=
X
j6∈F
c
j
β
j
−
X
i∈F
z
i
ˇα
i
,c
j
>0,
while
Re(λ)=
X
j6∈F
c
j
β
j
−
X
i∈F
d
i
ˇα
i
,c
j
>0,d
i
≥0.
Let F
2
= {i ∈ F |z
i
< 0} and let F
1
= F − F
2
. Then Re(µ + ν) ≥
P
j6∈F
c
j
β
j
−
P
i∈F
1
z
i
ˇα
i
. Thus, Re(µ + ν)
0
≥
P
j6∈F
c
j
β
j
= Re(λ)
0
. But Re(λ) ≥ Re(µ + ν), so
Re(λ)
0
≥ Re(µ + ν)
0
by Lemma 2.4. Hence, Re(λ)
0
= Re(µ + ν)
0
, so F (Re(λ)
0
)=
F(Re(µ + ν)
0
). This implies that F
2
= ∅,soallz
i
≥0.
The inclusion of H
P
–modules, W ⊂ V, induces a nonzero map π : H ⊗
H
P
W → V
given by π(h ⊗ w)=h·w(by Frobenius reciprocity). Since V is irreducible, V is
a quotient of H ⊗
H
P
W. 
This argument is very similar to the argument given in the proof of the analogous
fact for real groups [BW, IV, 3.4]. Note that it follows from the argument that every
weight λ of W has F (Re(λ)) = F.
2.7. Proof of 2.1(ii). Note that U is naturally embedded in H ⊗
H
P
U. Let
W
P
= {w ∈ W |w(R
P
+
) ⊂ R
+
}.
Then all weights of (H ⊗
H
P
U)/U are of the form w
P
(λ), where w
P
∈ W
P
,w
P
6=1,
and λ is weight of U [BM, proof of 6.4].
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THE LANGLANDS CLASSIFICATION FOR GRADED HECKE ALGEBRAS 1289
Let λ be a weight of U ,andwriteRe(λ)=
P
j6∈F
c
j
β
j
−
P
i∈F
d
i
ˇα
i
with all
c
j
> 0andd
i
≥0.Then if w ∈ W
P
,
Re(wλ)=
X
j6∈F
c
j
wβ
j
−
X
i∈F
d
i
w ˇα
i
.
Define ρ ∈ t
∗
by ρ(ˇα)=1,ˇα∈
ˇ
Π.Since w :Π
P
→R
+
,ρ(w(ˇα
i
)) ≥ ρ(ˇα
i
),i∈F.
Since β
j
is a fundamental weight, w(β
j
) ≤ β
j
with equality if and only if in a
minimal expression for w, each simple reflection fixes β
j
. For w ∈ W
P
and j 6∈ F,
thiscanonlyhappenifw=1.It follows that if w ∈ W
P
and w 6=1,then
ρ(Re(w(λ))) <ρ(Re(λ)). Fix a weight λ of U such that ρ(Re(λ)) is maximal. It
follows that λ does not occur as a weight of the A–module (H ⊗
H
P
U)/U.
In particular, if a submodule Z of H ⊗
H
P
U contains the weight λ,thenZ=
H⊗
H
P
U. Let I
max
be the sum of all submodules of H ⊗
H
P
U not containing
the weight λ. It follows by a standard argument that I
max
is the maximal proper
submodule and H ⊗
H
P
U has a unique irreducible quotient. 
2.8. Proof of 2.1(iii). Assume that π : J(P, U)
∼
=
J(P
0
,U
0
). Let λ be a weight of U
such that ρ(Re(λ)) is maximal and let λ
0
be a weight of U
0
such that ρ(Re(λ
0
)) is
maximal. Suppose F (Re(λ)) 6= F (Re(λ
0
)). Then it follows that λ
0
is not a weight
of U (by the remark at the end of 2.6), so by the argument in 2.7, ρ(Re(λ)) <
ρ(Re(λ
0
)). Similarly, it would follow that ρ(Re(λ)) >ρ(Re(λ
0
)). Hence, F(Re(λ)) =
F (Re(λ
0
)), so P = P
0
. Moreover, π(U )=U
0
,since U (resp. U
0
) is the unique
H
P
submodule containing a weight λ
1
such that ρ(Re(λ)) = ρ(Re(λ
1
)) (resp.
ρ(Re(λ
0
)) = ρ(Re(λ
1
))). Hence U
∼
=
U
0
as H
P
modules. 
2.9. Kazhdan and Lusztig also implicitly establish a Langlands classification for
representations that are tempered with respect to an arbitrary group homomor-
phism V : C
∗
→ R
∗
. This can be done in our setting also using a linear map
C → R. We omit the details.
Acknowledgments
The author would like to thank Allen Moy for his encouragement and his sug-
gestions, Cathy Kriloff for correcting certain inaccuracies, and the University of
Michigan for its hospitality during the preparation of this paper.
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Department of Mathematics, University of Arizona, Tucson, Arizona 85721
E-mail address: evens@math.arizona.edu
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