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 the Howe Duality Theorem, and the conservation relations. Second 2 highlights the invariant and distributional 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.