The character correspondence in the stable range over a p-adic field

TL;DR

This paper verifies a weaker version of a conjecture relating character correspondences in the stable range for dual pairs over p-adic fields, extending known results from the real case.

Contribution

It confirms a weaker form of the character correspondence conjecture for p-adic dual pairs in the stable range, building on prior real case results.

Findings

Verified a weaker version of the conjecture for p-adic dual pairs

Extended the stable range character correspondence to p-adic fields

Provided evidence supporting the conjecture's validity in the p-adic setting

Abstract

Given a real irreducible dual pair there is an integral kernel operator which maps the distribution character of an irreducible admissible representation of the group with the smaller or equal rank to an invariant eigendistribution on the group with the larger or equal rank. If the pair is in the stable range and if the representation is unitary, then the resulting distribution is the character of the representation obtained via Howe's correspondence. This construction was transferred to the p-adic case and a conjecture was formulated. In this note we verify a weaker version of this conjecture for dual pairs in the stable range over a p-adic field.