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Lectures on local theta correspondence

Chen-Bo Zhu

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.

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.

Paper Structure

This paper contains 24 sections, 39 theorems, 310 equations, 1 figure.

Table of Contents

  1. The basic theory
  2. Reductive dual pairs and classification
  3. Oscillator representations and Howe Duality Theorem
  4. Conservation relations
  5. The invariant and distribution theoretic nature of local theta correspondence
  6. The doubling method and the invariant distribution theorem
  7. Non-occurrence
  8. Occurrence
  9. Correspondence of invariants (I): generalized Whittaker models
  10. Generalized Whittaker models associated to an $\sl_{2}$-triple
  11. Descent of $\sl_{2}$-triples
  12. A canonical symplectic subspace and the associated oscillator representation
  13. The twisted Jaquet modules of the oscillator representation
  14. Transition of generalized Whittaker models
  15. Correspondence of invariants (II): associated cycles
  16. ...and 9 more sections

Key Result

Theorem 1.2.1

The set $\operatorname{Irr} (\widetilde{G}\times \widetilde{G'}, \omega)$ is the graph of a bijection between (all of) $\operatorname{Irr} (\widetilde{G}, \omega )$ and (all of) $\operatorname{Irr} (\widetilde{G'}, \omega )$. Moreover, an element $\pi\widehat{\otimes} \pi '$ of $\operatorname{Irr}

Figures (1)

  • Figure 1: Diamond dual pairs

Theorems & Definitions (85)

  • Definition 1.1.1: Howe
  • Theorem 1.2.1: Howe, Ho89
  • Remark 1.2.2
  • Theorem 1.2.3: Howe, Ho89
  • Theorem 1.2.6: Ho89KV
  • Proposition 1.2.8: Ho89
  • Definition 1.2.10
  • Proposition 1.2.11: Ho89
  • Theorem 1.2.13: Ho89
  • Remark 1.2.14
  • ...and 75 more