Last update: 22 December 2013
Let be integrally dominant and let Define modules The following lemma is the main tool for studying the structure of these modules.
([Jan1980, Theorem 2.2], [Dix1996, Lemma 7.6.14]) Let be a finite dimensional module and let be a basis of consisting of weight vectors ordered so that if Suppose is a module generated by a highest weight vector of weight Set Then
|(a)||is a filtration of modules such that is or is a highest weight module of highest weight|
The braid group is the subgroup of generated by By restriction, both and are modules.
There is a unique contravariant form on the Verma module determined by where is the generating highest weight vector of As in (2.15), this form together with a nondegenerate contravariant form on gives contravariant forms and on and respectively.
With these notations at hand we use Lemma 4.1 to prove the fundamental facts about the modules and defined in (4.1).
Let be integrally dominant weights and
|(c)||Use the same notation for the contravariant form on and the contravariant form on obtained by restriction of to the subspace Then|
Assume is maximal length in the coset
|(e)||If is a dominant integral weight then|
(a) Let be the generating highest weight vector of and, for let be the image of in Then, since is integrally dominant, Lemma 4.1 shows that is a vector space isomorphism. This is a isomorphism since the action on commutes with and fixes
(b) By (a), and so It is sufficient to show that for all simple reflections of such that and Applying the exact functor to the Verma module inclusion an inclusion of Since there is a (vector space) isomorphism of weight spaces (This isomorphism can be realized by Lusztig’s braid group action [CPr1994, §8.1-8.2], Thus, by part (a), the inclusion is an isomorphism.
(c) Use the notations for the bilinear forms on and as given in the paragraph before the statement of the proposition. Let be an orthonormal basis of with respect to If then and so Conversely, if such that then So and thus By the contravariance of for integrally dominant weights with Thus where is the restriction of the form on to Thus where the isomorphism is a consequence of the fact that, because is an integrally dominant weight, the functor is exact (3.8).
(d) If is not a weight of then, by part (a), Since the functor is exact and is a quotient of is a quotient of Thus implies
Assume that is not the longest element of the double coset Then there is a positive root such that and Since there is an inclusion of Verma modules and is a quotient of On the other hand, by part (b), Thus
(e) When is a dominant integral weight see [Dix1996, 7.2.7]. Thus, by (c) and the vector space isomorphism (4.2) it follows that, as vector spaces, For any there is a nonzero constant such that and so, by induction, for some constant Thus is isomorphic to the vector space If then the contravariance of gives that Thus, by the nondegeneracy of
Proposition 4.2d gives a necessary condition on for the to be nonzero. The following will be useful for analyzing the combinatorics of the examples in Section 6. If denotes the weight lattice is an integrally dominant then the action of on by the dot action has fundamental domain and the following are equivalent:
|(b)||with integrally dominant and longest in the coset in|
|(c)||with longest in the double coset in|
In the classical case, when is type and is the dimensional fundamental representation the is a simple whenever it is nonzero (see [Suz1998]). As the following Proposition shows, this is a very special phenomenon.
Assume that for a dominant integral weight If the is irreducible (or for all all dominant integral weights and all integrally dominant weights then
|(a)||is type or and and|
|(b)||the action of the subgroup of generates|
(a) If is large dominant integral weight (for example, we may take then, as a where the sum is over a basis of consisting of weight vectors and is the weight of the vector The group is generated by the element which acts on a summand in by the constant Then is the component of and these are simple modules (for the various only if all the values as ranges over a weight basis of are distinct. It follows that all weight spaces of must be one dimensional. This means that
(b) Let As a where is an irreducible and the sum is over all dominant integral weights for which the irreducible appears in By restriction is an and this is the which, by assumption, is simple. Since is the trivial module acts on by the identity and so is simple as a Thus the simple in coincide exactly with the simple in and it follows that generates
Applying the functor to the Jantzen filtration of produces a filtration of An argument of Suzuki [Suz1998, Thm. 4.3.5], shows that this filtration can be obtained directly from the form on which is the restriction of the contravariant form on see (2.5) and (3.5)). To do this define and to obtain a filtration such that the quotients carry nondegenerate contravariant forms. Since, for different the subspaces are mutually orthogonal with respect to the contravariant form on On the other hand, if then write where and is an orthonormal basis of Then, for all and all and so So and the filtrations in (4.6) and (4.5) are identical.
Let and be integrally dominant weights and let be elements of maximal length in and respectively. Assume that the modules are simple. Then multiplicities of in the filtration (4.5) are given by where is the Kazhdan-Lusztig polynomial for the Weyl group
Since the functor is exact this result follows from the Beilinson-Bernstein theorem (2.6). The condition on is necessary for the module to be nonzero.
Let be such that is dominant and regular and let be a parabolic subgroup of the integral Weyl group Let be the longest element of and fix Following the method of [Che1987-2], applying the exact functor to the BGG resolution in (2.7) produces an exact sequence of where and the sum is over all of length (in Thus, in the Grothendieck group of the category of finite dimensional where and is the longest element of This identity is a generalization of the classical Jacobi-Trudi identity [Mac1995, I (5.4)] for expanding Schur functions in terms of homogeneous symmetric functions,
The braid group is the quotient of the affine braid group by the relation and so the modules are The following proposition determines the structure of as a module when is finite dimensional. This is a generalization of the Littlewood-Richardson rule.
Let be the set of dominant integral weights. Define the tensor product multiplicities by the decompositions Then
Let us abuse notation slightly and write sums instead of direct sums. Then, as a bimodule where As a bimodule Comparing coefficients of in these two identities yields the formula in the statement.
This is a typed version of the paper Affine Braids, Markov Traces and the Category by Rosa Orellana and Arun Ram*.
*Research supported in part by National Science Foundation grant DMS-9971099, the National Security Agency and EPSRC grant GR K99015.
This paper is a slightly revised version of a preprint of 2001. We thank F. Goodman, A. Henderson and an anonymous referee for their very helpful comments on the original preprint.