Lectures on local theta correspondence

TL;DR

This paper provides an overview of local theta correspondence focusing on Archimedean theory, including Howe's Duality, conservation relations, invariants of representations, and applications to unitary representation theory.

Contribution

It offers a comprehensive lecture-based exposition emphasizing the invariant-theoretic aspects and applications of local theta correspondence for classical Lie groups.

Findings

Proof of conservation relations illustrating invariant-theoretic nature

Behavior of fundamental invariants under local theta correspondence

Applications to the theory of unitary representations

Abstract

This set of lecture notes on local theta correspondence is the written version of a mini-course the author gave in March of 2025 for the program ``Representation Theory and Noncommutative Geometry" at the Institut Henri Poincar\'e, Paris. The emphasis is on the Archimedean theory, which concerns representations of classical Lie groups. Section 1 is about the basic theory, including Howe's Duality Theorem, and the conservation relations. Section 2 highlights the invariant-theoretic nature of local theta correspondence via the proof of the conservation relations. Sections 3 and 4 explain how two fundamental invariants of representations behave under local theta correspondence. The final section discusses applications to unitary representation theory.