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Symmetric pairs and branching laws

Paul-Emile Paradan

Abstract

Let $G$ be a compact connected Lie group and let H be a subgroup fixed by an involution. A classical result assures that the action of the complex reductive group $H_C$ on the flag variety $F$ of $G$ admits a finite number of orbits. In this article we propose a formula for the branching coefficients of the symmetric pair $(G,H)$ that is parametrized by $H_C\backslash F$.

Symmetric pairs and branching laws

Abstract

Let be a compact connected Lie group and let H be a subgroup fixed by an involution. A classical result assures that the action of the complex reductive group on the flag variety of admits a finite number of orbits. In this article we propose a formula for the branching coefficients of the symmetric pair that is parametrized by .

Paper Structure

This paper contains 25 sections, 26 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Non abelian localization
  3. Matsuki duality
  4. Borel-Weil-Bott theorem
  5. Localization of the Riemann-Roch character
  6. Local model near $Hx\subset Z_\theta$
  7. Proof of the main theorem
  8. Computation of $Q_{Hx}(\lambda)$
  9. Step 1: holomorphic induction
  10. Step 2: cotangent induction
  11. Step 3: linear case
  12. Conclusion
  13. Another expression for $Q_{Hx}(\lambda)$
  14. Computation of the virtual module $\mathbb{M}_x(\lambda)$
  15. Examples
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\lambda\in\Lambda_+$. We have the decomposition where the terms $Q_{Hx}(\lambda)\in \widehat{R}(H)$ are defined by the following relation : Here ${\rm Sym}(\mathbb{E}^{\hbox{\scriptsize\rm nci}}_{x})$, which is the symmetric algebra of $\mathbb{E}^{\hbox{\scriptsize\rm nci}}_{x}$, is an admissible representation of $H_x$ and $\bigwedge \mathbb{E}^{\hbox{\scriptsize\rm ci}}_{x}= \bigwedge^{+

Figures (2)

  • Figure 1: Restriction from $U(3)$ to $U(2)$
  • Figure 2: Kostant decomposition

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 38 more