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Pseudo-dual pairs and branching of Discrete Series

Bent Ørsted, Jorge A. Vargas

Abstract

For a semisimple Lie group $G$, we study Discrete Series representations with admissible branching to a symmetric subgroup $H$. This is done using a canonical associated symmetric subgroup $H_0$, forming a pseudo-dual pair with $H$, and a corresponding branching law for this group with respect to its maximal compact subgroup. This is in analogy with either Blattner's or Kostant-Heckmann multiplicity formulas, and has some resemblance to Frobenius reciprocity. We give several explicit examples and links to Kobayashi-Pevzner theory of symmetry breaking and holographic operators. Our method is well adapted to computer algorithms, such as for example the Atlas program.

Pseudo-dual pairs and branching of Discrete Series

Abstract

For a semisimple Lie group , we study Discrete Series representations with admissible branching to a symmetric subgroup . This is done using a canonical associated symmetric subgroup , forming a pseudo-dual pair with , and a corresponding branching law for this group with respect to its maximal compact subgroup. This is in analogy with either Blattner's or Kostant-Heckmann multiplicity formulas, and has some resemblance to Frobenius reciprocity. We give several explicit examples and links to Kobayashi-Pevzner theory of symmetry breaking and holographic operators. Our method is well adapted to computer algorithms, such as for example the Atlas program.
Paper Structure (43 sections, 22 theorems, 68 equations)

This paper contains 43 sections, 22 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and some notation
  3. Duality Theorem, explicit isomorphism
  4. Statement and proof of the duality result
  5. Kernel of the restriction map
  6. The map $r_0^D$ is injective
  7. Previous work on duality formula and Harish-Chandra parameters
  8. Computing Harish-Chandra parameters from Theorem \ref{['prop:Rwelldefgc']}
  9. Gross-Wallach multiplicity formula
  10. Duflo-Vargas multiplicity formula, DV
  11. Harris-He-Olafsson multiplicity formula, HHO
  12. Completion of the Proof of Theorem \ref{['prop:Rwelldefgc']}, the map $r_0^D$ is surjective
  13. Duality Theorem, proof of dimension equality
  14. Dimension equality Theorem, statement
  15. Conclusion proof of Theorem \ref{['prop:equaldim']}
  16. ...and 28 more sections

Key Result

Theorem 3.1

We assume $(G,H)$ is a symmetric pair and $res_H(\pi_\lambda)$ is admissible. We fix a irreducible factor $V_\mu^H$ for $res_H(\pi_\lambda)$. Then, the following statements hold. i) The map $r_0 : H^2(G,\tau) \rightarrow L^2(H_0 \times_\tau W)$ restricted to $\mathrm{Cl}(\mathcal{U}(\mathfrak h_0)W is a linear isomorphism.

Theorems & Definitions (39)

  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.6
  • proof
  • Corollary 3.7
  • Corollary 3.8
  • Corollary 3.9
  • ...and 29 more