A non-unitary approach to the $q$-deformation of $\mathrm{SL}(2,\mathbb{R})$

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 $\mathfrak{sl}(2,\mathbb{R})$.