Local theta correspondence: the basic theory

TL;DR

This paper provides an accessible overview of local theta correspondence, discussing key conjectures, current status, and the unity of algebraic and smooth theories in classical invariant theory.

Contribution

It offers an elementary introduction to the theory, explains fundamental conjectures, and reports on progress in explicitly describing local theta correspondence.

Findings

Explanation of Howe duality conjecture and Kudla-Rallis relation

Status update on describing local theta correspondence via Langlands-Vogan parameters

Discussion on automatic continuity linking algebraic and smooth theories

Abstract

We give an elementary introduction to Classical Invariant Theory and its modern extension "Transcending Classical Invariant Theory", commonly known as the theory of local theta correspondence. We explain the two fundamental assertions of the theory: the Howe duality conjecture and the Kudla-Rallis conservation relation conjecture. We give a status report on the problem of explicitly describing local theta correspondence in terms of Langlands-Vogan parameters. We conclude with a discussion on a certain problem of automatic continuity, which manifests unity of the theory in algebraic and smooth settings.