Bernstein components via Bernstein center

TL;DR

This paper characterizes when invariant distributions in the Bernstein center are supported on compact elements of a reductive p-adic group, linking support properties to constancy on irreducible representations.

Contribution

It establishes an if-and-only-if condition connecting support on compact elements with constant functions on Bernstein components, and shows this forms a subalgebra of the Bernstein center.

Findings

Invariant distributions supported on compact elements are constant on Bernstein components.

The set of Bernstein center elements supported on compact elements forms a subalgebra.

The proof modifies existing arguments to establish the characterization.

Abstract

Let G be a reductive p-adic group. Let $\Phi$ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that $\Phi$ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modiification of the arguments of J.F.Dat who proved our result in one direction.