A non-unitary approach to the $q$-deformation of $\mathrm{SL}(2,\mathbb{R})$
Yvann Gaudillot-Estrada
TL;DR
This work develops a non-unitary framework for the $q$-deformation of SL(2,$\mathbb{R}$), constructing a quantum enveloping algebra $U_q(\frak{g})$ and a quantized Hecke algebra $R_q(\frak{g},K)$ to study $(\frak{g},K)_q$-modules. It introduces an $\mathbb{A}$-form to connect the quantum and classical theories, defines a quantum analogue of parabolic induction, and classifies irreducible representations in terms of induced modules with detailed subquotient structure. A parallel classical limit is established via an algebraic $\mathbb{A}$-form, showing that the $q$-deformed non-unitary dual converges to the classical admissible dual as $q \to 1$. The results extend Harish-Chandra-type structures to the quantum setting and provide a robust bridge between quantum and classical representation theories for SL(2,$\mathbb{R}$).
Abstract
We study the representation theory of various convolution algebras attached to the $q$-deformation of $\mathrm{SL}(2,\mathbb{R})$ from an algebraic perspective and beyond the unitary case. We show that many aspects of the classical representation theory of real semisimple groups can be transposed to this context. In particular, we prove an analogue of the Harish-Chandra isomorphism and we introduce an analogue of parabolic induction. We use these tools to classify the non-unitary irreducible representations of $q$-deformed $\mathrm{SL}(2,\mathbb{R})$. Moreover, we explicitly show how they converge to the classical admissible dual of $\mathrm{SL}(2,\mathbb{R})$. For that purpose, we define a version of the quantized universal enveloping algebra defined over the ring of analytic functions on $\mathbb{R}_+^*$, which specializes at $q = 1$ to the enveloping $\ast$-algebra of $\mathfrak{sl}(2,\mathbb{R})$.