Explicit Rieffel induction module for quantum groups

Abstract

For $\mathbb{G}$ an algebraic (or more generally, a bornological) quantum group and $\mathbb{B}$ a closed quantum subgroup of $\mathbb{G}$, we build in this paper an induction module by explicitly defining, on the convolution algebra of $\mathbb{G}$, an inner product which takes its value in the convolution algebra of $\mathbb{B}$, as in the original approach of Rieffel. In this context, we study the link with the induction functor defined by Vaes. In the last part, we illustrate our result with parabolic induction of complex semisimple quantum groups. We first show that our induction functor coincides with the one already defined in the case of parabolic induction. Then we use the tools developed in this paper to give a geometric interpretation to the parabolic induction functor, following the approach suggested by Clare in the classical case.