Vanishing of quasi-invariant generalized functions

TL;DR

This paper reviews new techniques for proving the vanishing of quasi-invariant generalized functions, which are crucial in representation theory, by combining geometric and analytic methods to simplify existing proofs.

Contribution

It introduces and explains three innovative techniques—metrical properness, unipotent χ-incompatibility, and first occurrence in Howe correspondence—for establishing vanishing results in representation theory.

Findings

Techniques simplify proofs of uniqueness in representation theory.

Geometric notions like metrical properness aid in vanishing proofs.

Analytic methods from Howe correspondence are effective in this context.

Abstract

Determination of quasi-invariant generalized functions is important for a variety of problems in representation theory, notably character theory and restriction problems. In this note, we review some new and easy-to-use techniques to show vanishing of quasi-invariant generalized functions, developed in the recent work of the authors (Uniqueness of Ginzburg-Rallis models: the Archimedean case, Trans. Amer. Math. Soc. 363, (2011), 2763-2802). The first two techniques involve geometric notions attached to submanifolds, which we call metrical properness and unipotent $\chi$-incompatibility. The third one is analytic in nature, and it arises from the first occurrence phenomenon in Howe correspondence. We also highlight how these techniques quickly lead to two well-known uniqueness results, on trilinear forms and Whittaker models.