Session S10 - Categorification, Higher Representation Theory, and Homological Knot Invariants

Thursday, July 15, 13:30 ~ 14:05 UTC-3

Restriction of square integrable representations

Jorge Vargas

Famaf-CIEM, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $G$ be a semisimple Lie group, and $(\pi,V)$ a irreducible square integrable representation for $G$. Thus, a model for $V$ is the $L^2$-kernel of a elliptic operator on a fiber bundle over the symmetric space $G/K$ attached to $G$. Let $H$ be a closed reductive subgroup for $G$. We say $\pi$ is $H$-discretely decomposable ( $H$-admissible) if the sum of the closed $H$-irreducible subspaces in $V$ is dense in $V$, ($H$-admissible if it is $H$-discretely decomposable and the multiplicity of each irreducible factor is finite). We give criteria for being $H$-$\cdots$ in language of spherical functions as well as in the language of differential intertwining operators. On a basic exposition we will present an overview of some aspects of branching problems and results in Orsted-Vargas, Branching problems in reproducing kernel spaces, Duke mathematical journal, Vol. 169, 3478-3537, 2020 and some consequences.

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