Last update: 31 March 2013
The Weyl group. Let be a reduced irreducible root system in fix a set of positive roots and let be the corresponding simple roots in Let be the Weyl group corresponding to Let denote the simple reflection in W corresponding to the simple root and recall that can be presented by generators and relations
The Iwahori-Hecke algebra. Fix such that is not a root of unity. The Iwahori-Hecke algebra is the associative algebra over defined by generators and relations
where are the same as in the presentation of For define where is a reduced expression for By [Bou1968, Ch. IV §2 Ex. 23], the element does not depend on the choice of the reduced expression. The algebra has dimension and the set is a basis of
The group The fundamental weights are the elements of given by The weight lattice is the lattice in given by
Let be the abelian group except written multiplicatively. In other words,
Let denote the group algebra of There is a on given by for which we extend linearly to a on
The affine Hecke algebra. The affine Hecke algebra associated to and is the algebra given by
where the multiplication of the is as in the Iwahori-Hecke algebra the multiplication of the is as in and we impose the relation
This formulation of the definition of is due to Lusztig [Lus1989] following work of Bernstein and Zelevinsky. The elements form a basis of
The torus is an abelian group with a given by For any element define the polar decomposition
for all Let There is a unique and a unique such that
In this way we identify the sets and with and respectively.
Theorem 1.4. (Bernstein, Zelevinsky, Lusztig [Lus1983, 8.1]) The center of is
Since has countable dimension, Dixmier’s version of Schur’s lemma implies that acts on an irreducible by scalars. Let be such that
Since it follows that for all The of is the central character of We shall often abuse notation and refer to any weight as “the central character” of
Weight spaces. Let be a finite dimensional For each the space of and the generalized space are the subspaces
respectively. If then In general but we do have
This is a decomposition of into Jordan blocks for the action of The set of weights of is the set
The calibration graph. Let Define a graph with
Proposition 1.6. ([Ram1998] Proposition 2.12) Let be a finite dimensional irreducible with central character Then
if and are in the same connected component of the calibration graph
For each subset define
where is the inversion set of Define a placed shape to be a pair such that and The elements of the set are called standard tableaux of shape
Proposition 1.9. ([Ram1998] Theorem 2.14) Let The connected components of the calibration graph are the sets
Calibrated representations. A finite dimensional is calibrated if for all
Proposition 1.10. ([Ram1998] Proposition 4.2)
Let and be simple roots in and let be the rank two root subsystem of which is generated by and Let be the Weyl group of the subgroup of generated by the simple reflections and A weight is calibratable for if one of the following two conditions holds:
A placed skew shape is a placed shape such that for all and all pairs of simple roots in the weight is calibratable for
Theorem 1.11. ([Ram1998] Theorem 3.1 and Proposition 4.1)
Remark 1.12. It follows from the results of Rodier [Rod1981] that if is an irreducible with regular central character (i.e. then satisfies the hypothesis of the statement of Theorem 1.11 (b).
Langlands classification. The following discussion follows the work of Evens [Eve1996] and [KLu0862716, §8]. For this subsection it is convenient to assume that and For the general case see [KLu0862716, §8]. Let and let be the polar decomposition of Define
A finite dimensional is tempered if for all weights of (as defined in (1.5)) we have
The module is square integrable if for all and all weights of
Let be a subset of the simple roots and let be the subalgebra of generated by and all We shall say that a finite dimensional is tempered if is the maximal set such that for all weights of
Theorem 1.13. (see [Eve1996]) Let and let be an irreducible tempered representation of
The Langlands parameters of an irreducible are given by the pair specified by Theorem 1.13 (b).
Classification by indexing triples. Kazhdan and Lusztig [KLu0862716] (see also the important work of Ginzburg [CGi1433132]) gave a refinement of the Langlands classification. Let be the simple complex algebraic group with root system and weight lattice An indexing triple consists of
and an irreducible representation of the component group where Let be the of the variety
By a theorem of Lusztig [Lus1985] is an The group also acts on and this action commutes with the action of The standard modules are the given by the decomposition
where the sum is over all irreducible representations of
Theorem 1.14. [KLu0862716]
In this way each irreducible corresponds to a unique (up to conjugation) indexing triple. One can replace by in the Lie algebra (see [CGi1433132, Ch. 8]) so that an indexing triple is
and an irreducible representation of the component group where and is taken with respect to the adjoint action of on We will use this form of the indexing triples in the examples in later sections.
Principal series modules. Let and let be the one dimensional corresponding to the character Specifically, is the one dimensional vector space with basis and given by
The principal series module corresponding to is the
Theorem 1.15. [Mat1977]
Theorem 1.16. Kato’s irreducibility criterion [Kat1981]) Let and let The principal series module is irreducible if and only if
Remark. Kato actually proves a more general result and thus needs a further condition for irreducibility. We have simplified matters by specifying the weight lattice in our construction of the affine Hecke algebra. One can use any lattice in and Kato works in this more general situation. When the one uses the weight lattice a result of Steinberg [Ste1968-2, 4.2, 5.3] says that the stabilizer of a point under the action of is always a reflection group. Because of this Kato’s criterion takes a simpler form.
Weights of induced modules. If define to be the subalgebra of generated by and all
Lemma 1.17. Let such that for all and let be the one dimensional with basis and given by
Let be the set of minimal length coset representatives of cosets of in Then the weights of the are and
The module has basis By writing for a reduced word and inductively using the defining relation (1.2) we get
where the sum is over which are less than in Bruhat order and This shows that the eigenvalues of on are The result follows by counting the multiplicity of each eigenvalue.
The operators. The maps defined below are local operators on in the sense that they act on each weight space of separately. The operator is only defined on weight spaces such that
Proposition 1.18. ([Ram1998] Proposition 2.7) Let such that and let be a finite dimensional Define
Lemma 1.19. Let such that and suppose that is an such that Let be the stabilizer of under the action of on Assume that is such that and are in the same connected component of Let be a minimal length coset representative of Then
Let be the two dimensional principal series module for the affine Hecke algebra of type (see §2 central character Then and has basis Let be a nonzero weight vector in There is a unique homomorphism
where we view as an by restriction to the parabolic subalgebra This homomorphism must be an injection since is irreducible. Thus the vectors span a two dimensional subspace of and acts on this subspace by the matrix
Let be a reduced expression of Since and are in the same connected component of we can use Proposition 1.18 (c) to show that the map
is well defined and bijective. Thus the vectors span a two dimensional subspace of and, by Proposition 1.18 (b) acts on this subspace by the matrix wt
This proves (a). Then
Since is the zero map and so
The relation is the same as This relation forces to have Jordan blocks of size 1 and eigenvalues It follows that and
This is an excerpt of a preprint entitled Representations of rank two affine Hecke Algebras, written by Arun Ram, Department of Mathematics, Princeton University, August 5, 1989.
Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.