A combinatorial approach to representations of quantum linear groups

In this paper we deal with Manin's quantization of GL,. Using the bideterminant bases, we prove that for a particular kind of elements w epsilon G(n), the canonical morphism from Ind G(q)/B-q lambda to IteInd(w) lambda (see Section 1) is surjective, and, H-0(lambda) is isomorphic to IteInd(w) lambda if w = w(0), the longest element in G(n). Our approach is to construct a basis consisting of bideterminants for the above iterated induced modules. We believe that such a basis is also interesting and useful. Moreover, most of the well-known homological properties of GL(n,q), such as Grothendieck vanishing, Kempf vanishing: Demazure character formula and Bott-Borel-Weil Theorem, are reobtained as consequences of the above surjectivity.