Lectures on local theta correspondence

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 Poincaré, 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.

To appear in the proceedings of the IHP trimester on Representation Theory and Noncommutative Geometry