Induction for representations of coideal doubles, with an application to quantum $SL(2,\mathbb{R})$
Kenny De Commer
TL;DR
The paper develops a general induction framework for representations arising from coideal subalgebras of compact quantum group Hopf *-algebras via their Drinfeld doubles. It combines algebraic and analytic induction within the Doi-Koppinen (DK) data setting, and introduces two canonical unitary DK representations that model regular and intrinsic actions. The framework is then specialized to quantum SL(2,R)_t, where the Podleś sphere O_q(S_t^2) and the quantum circle O_q(SO(2)_t) yield a concrete infinitesimal double U_q(sl(2,R)_t); irreducible unitary representations are classified and their branching rules are established, with principal-series representations arising from induction. Finally, the regular representation of quantum SL(2,R)_t is decomposed into principal and discrete series (including exceptional discrete series) mirroring the classical SL(2,R) decomposition, supported by a localization technique and a Heisenberg-double factorization. The results illuminate how coideal doubles encode representation-theoretic data for quantum groups and provide explicit decompositions analogous to the classical theory, with potential for Plancherel-type formulas and higher-rank generalizations.
Abstract
We investigate the theory of induction in the setting of doubles of coideal $*$-subalgebras of compact quantum group Hopf $*$-algebras. We then exemplify parts of this theory in the particular case of quantum $SL(2,\mathbb{R})$, and compute the decomposition of the regular representation for quantum $SL(2,\mathbb{R})$ into irreducibles.