Branching integrals and Casselman phenomenon

TL;DR

This paper explores the geometric reasons behind the holomorphic continuation of matrix elements in real semisimple Lie groups and surveys related infinite-dimensional representation continuations.

Contribution

It provides a geometric explanation for the Casselman phenomenon and offers a comprehensive survey of holomorphic continuations of infinite-dimensional representations.

Findings

Holomorphic continuation of matrix elements is explained geometrically.

Singularities occur at a prescribed divisor in the complexification.

The paper surveys various methods of continuation in representation theory.

Abstract

Let $G$ be a real semisimple Lie group, $K$ its maximal complex subgroup, and $G_C$ its complexification. It is known that all the $K$-finite matrix elements on $G$ admit holomorphic continuation to branching functions on $G_C$ having singularities at the a prescribed divisor. We propose a geometric explanation of this phenomenon. The note also contsins a general survey of holomorphic continuations of infinite-dimensional representations.