Last update: 13 November 2012
In view of the results in Section 1.18 we shall (for the remainder of this paper, except Sections 5–7 where we use the data in (1.1)) assume that in the definition of the group and see (1.2), (1.9), and (1.14). The Weyl group acts on
Let be a finite dimensional and let The space and the generalized t-weight space of are
is a decomposition of into Jordan blocks for the action of and we say that is a weight of if Note that if and only if A finite-dimensional
Remark. The term tame is sometimes used in place of the term calibrated particularly in the context of representations of Yangians, see [NTa1998]. The word calibrated is preferable since tame also has many other meanings in different parts of mathematics.
Let be a simple As an contains a simple submodule and this submodule must be one-dimensional since all irreducible representations of a commutative algebra are one dimensional. Thus, a simple module always has for some
The Pittie–Steinberg theorem, Theorem 1.17, shows that, as vector spaces,
and is the Iwahori–Hecke algebra defined in (1.8). Thus is a free module over of rank By Dixmier’s version of Schur’s lemma (see [44, Lemma 0.5.1]), acts on a simple by scalars and so it follows that every simple is finite dimensional of dimension Theorem 2.12(d) below will show that, in fact, the dimension of a simple module is
Let be a simple The central character of is an element such that
The element is only determined up to the action of since for all Because of this, any element of the orbit is referred to as the central character of
Because in the construction of a theorem of Steinberg [Ste1968-2, 3.15, 4.2, 5.3] tells us that the stabilizer of a point under the action of is the reflection group
Thus the orbit can be viewed in several different ways via the bijections
where the last bijection is the restriction of the map in (1.6). If the root system is generated by the simple roots that it contains then is a parabolic subgroup of and is the set of minimal length coset representatives of the cosets in
For let be the one-dimensional given by
The principal series representation is the defined by
The module has basis with acting by left multiplication.
If and then the defining relation (1.10) for implies that
where the sum is over in the Bruhat–Chevalley order and Let be the stabilizer of under the It follows from (2.7) that the eigenvalues of on are of the form and by counting the multiplicity of each eigenvalue we have
In particular, if is regular (i.e., when is trivial), there is a unique basis of determined by
Let The spherical vector in is
Up to multiplication by constants this is the unique vector in such that for all The following is due to Kato [Kat1981, Proposition 1.20 and Lemma 2.3],
Proposition 2.11. Let and let be the stabilizer of under the
The proof is accomplished in exactly the same way as done for the graded Hecke algebra in [KRa2002, Proposition 2.8]. The only changes which need to be made to [KRa2002] are
Part (b) of the following theorem is due to Rogawski [Rog1985, Proposition 2.3] and part (c) is due to Kato [Kat1981, Theorem 2.1]. Parts (a) and (d) are classical.
Theorem 2.12. Let and and define
(a) follows from (2.8) and the definition of calibrated. Part (b) accomplished exactly as in [KRa2002, Proposition 2.8] and (c) is a direct consequence of the definition of and Proposition 2.11.
(d) Let be a nonzero vector in If is as in the construction of in (2.6) then, as Thus, since induction is the adjoint functor to restriction there is a unique homomorphism given by
This map is surjective since is irreducible and so is a quotient of
The following proposition defines maps on generalized weight spaces of finite-dimensional These are “local operators” and are only defined on weight spaces such that In general, does not extend to an operator on all of
Proposition 2.14. Fix let be such that and let be a finite-dimensional Define
(a) The element acts on by times a unipotent transformation. As an operator on is invertible since it has determinant where Since this determinant is nonzero is a well defined operator on Thus the definition of makes sense.
Since is not an element of or it should be viewed only as an operator on in calculations. With this in mind it is straightforward to use the defining relation (1.10) to check that
for all and This proves (a)-(c).
(d) The operator acts on as times a unipotent transformation. Similarly for Thus, as an operator on if and only if Thus part (c) implies that and each factor in this composition, is invertible if and only if
(e) Let be regular. By part (a), the definition of the and the uniqueness in (2.9), the basis of in (2.9) is given by
where for a reduced word of Use the defining relation (1.10) for to expand the product of and compute
where and are rational functions in the By the uniqueness in (2.9), for all Since the values of and coincide on all generic points it follows that
whenever both sides are well defined operators on
Let and recall that
We say that the pair is a local region if Under the bijection (2.4) the set maps to the set of chambers whose union is the set of points which are
See the picture in Example 4.11(d). In this way the local region really does correspond to a region in This is a connected convex region in since it is cut out by half spaces in The elements index the chambers in the local region and, as runs over the subsets of the sets form a partition of the set (which, by (2.4), indexes the cosets in
Corollary 2.19. Let be a finite dimensional Let and let Then
Suppose We may assume that Then and Now, implies and implies Since and it follows from Proposition 2.14(d) that the map is well defined and invertible. It remains to note that if then where for all This follows from the fact that corresponds to a connected convex region in
This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.
Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).