arXiv subjects
Publications and source records attributed to Yvann Gaudillot-Estrada.
If $G$ is an algebraic affine group acting on an affine variety $X$, there is a natural notion of covariant representation for the pair $(G,X)$. In this paper, we classify the irreducible covariant representations for any such pair by adapting the Mackey machine to this algebraic setting. Next, we give applications for continuous representations of motion groups on Banach spaces and other related examples.
When $G$ is a real semisimple group, there is a surprising interplay between its representation theory and that of its motion group $G_0$, known as the Mackey analogy. The present paper extends this analogy to the framework of $q$-deformations, for $G = \mathrm{SL}(2,\mathbb{R})$. In fact, we construct a deformation of $\mathrm{SL}(2,\mathbb{R})$ parametrized by $(q,t) \in \mathbb{R}_+^* \times \mathbb{R}$, where $q$ is the quantization parameter and $t$ is the Mackey parameter. We show how the representation theory varies along this deformation and we prove an analogue of the Connes-Kasparov isomorphism for the $q$-deformed reduced group C*-algebra.
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 $\mathfrak{sl}(2,\mathbb{R})$.
We consider Gromov-Hausdorff convergence of state spaces for spectral truncations of a compact metric group $G$. We work in the context of order-unit spaces and consider orthogonal projections $P_Λ$ in $L^2(G)$ corresponding to finite subsets of irreducible representations $Λ\subseteq \widehat G$. We then prove that the sequence of truncated state spaces $\{ S(P_ΛC(G) P_Λ)\}_Λ$ Gromov-Hausdorff converges to the original state space $S(C(G))$, when these are equipped with a metric associated to a Lip-norm which in turn is induced by the action of $G$.