Analytic continuation of equivariant distributions

TL;DR

This paper develops a method to extend equivariant distributions from open subsets to entire algebraic varieties, enabling new constructions of functionals and models in representation theory for real and p-adic groups.

Contribution

It introduces a novel approach based on Bernstein's theory for constructing equivariant distributions, leading to new proofs and extensions in representation theory.

Findings

Constructed $H$-equivariant functionals on principal series representations.

Proved the existence of generalized Whittaker models for degenerate principal series.

Extended the method to p-adic groups using recent results.

Abstract

We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein's theory of analytic continuation of holonomic distributions. We use this to construct $H$-equivariant functionals on principal series representations of $G$, where $G$ is a real reductive group and $H$ is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations, and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the $p$-adic case using a recent result of Hong and Sun.