Freeness of the Quantum Coordinate Algebras

The main purpose of this note is to prove that the quantum coordinate algebra A[U ] is free over the ring A = Z[v, v−1]. In [L1], Lusztig defined the quantum coordinate algebra over Q[v, v−1] and prove that it is free as Q[v, v−1]-module. In [APW], Andersen, Polo, and Wen defined the quantum coordinate algebra over the ring A(p,v−1) and proved that the coordinate algebra is free. The main idea was to specialize v to 1 from A(p,v−1) and then use the result from the representation theory of algebraic groups in characteristics p > 0. The two important consequences of specifying v to 1 from A(p,v−1) are that the set of weights of an integrable U -module is W -invariant and the universal highest weight modules D(λ) are free and finitely generated over A(p,v−1) with characters given by Weyl’s character formula using Kempf’s vanishing theorem in characteristic p. Using the ring A(p,v−1) gives restriction on ` when one wants to consider cases when q is an `th root of 1. In [AW], Andersen and Wen extended the freeness to the ring A1, which is still larger than A, by studying the representations at mixed case. In this paper we will use Lusztig’s canonical basis to replace Kempf’s vanishing theorem (or its analogy at mixed cases as in [AW]). But it is still not clear that whether the set of weights of an integrable U -module will be W -invariant (even in the rank 1 case). In Section 2, we introduce the quantum analogue of Joseph’s induction [Jo]. It is not clear that Joseph’s induction functor can be defined on all integrable U≥0-modules of type 1. In Section 3, the freeness of the universal highest weight module D(λ) defined by the Joseph induction over A is proved using the canonical basis constructed by Lusztig [L2]. We will treat the parabolic cases as well. As a consequence, we prove that integrable modules of type 1 are locally A-finite for the parabolic subalgebras. Then using the freeness of the universal highest weight modules, we follow more or less the argument as in [APW] to prove the freeness of the coordinate algebras of the quantum parabolic algebras over A in Section 4. In Section 5, the exactness of certain induction functors is proved using the exactness of the quantum coordinate algebras over A. As application,