Quantum Weyl Reciprocity for Cohomology

Classical Weyl reciprocity relates the representation theories of the general linear group GLnðkÞ and the symmetric group Sr by means of the ðGLnðkÞ;SrÞ-bimodule T :1⁄4 V , where V 1⁄4 k. For example, when k 1⁄4 C, the poset 0þðn; rÞ of partitions of r with at most n non-zero parts indexes the irreducible polynomial GLnðCÞ-modules Lð Þ, which are homogeneous of degree r, and a subset of the complex irreducible Sr-modules S . As a bimodule, T decomposes into a direct sum of bimodules Lð Þ S . In quantum Weyl reciprocity, GLnðkÞ is replaced by the quantum enveloping algebra Uq of type An 1 (over an arbitrary algebraically closed 6eld k), while Sr is replaced by the Hecke algebra H 1⁄4 HðSrÞ of type Ar 1. Further, q-tensor space Tq serves as a natural bimodule for the pair ðUq;HÞ, and the image Uq ! EndHðTqÞ de6nes the q-Schur algebra Sqðn; rÞ. If k 1⁄4 C and q is not a root of unity, results of Jimbo largely recover results analogous to the classical q 1⁄4 1 case. In general, Tq is not completely reducible for the action of Sqðn; rÞ or for that of H. However, in analogy with the classical results, there is a 6ltration of Tq with sections of the form 9ð Þ S , with 2 0þðn; rÞ [17]. Here, 9ð Þ denotes the standard module for Uq of highest weight , and S is the Specht module for H corresponding to . The category of left Sqðn; rÞ-modules and that of right H-modules are related by the contravariant functors