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Homological aspects of branching laws

Dipendra Prasad

Abstract

In this mostly expository article, we consider certain homological aspects of branching laws for representations of a group restricted to its subgroups in the context of $p$-adic groups. We follow our earlier paper, ICM 2018 proceedings, updating it with some more recent works. In particular, following Chan and Chan-Savin, see many of their papers listed in the bibliography, we have emphasized in this work that the restriction of a (generic) representation $π$ of a group $G$ to a closed subgroup $H$ (most of the paper is written in the context of GGP) turns out to be a projective representation on most Bernstein blocks of the category of smooth representations of $H$. Further, once $π|_H$ is a projective module in a particular Bernstein block, it has a simple structure.

Homological aspects of branching laws

Abstract

In this mostly expository article, we consider certain homological aspects of branching laws for representations of a group restricted to its subgroups in the context of -adic groups. We follow our earlier paper, ICM 2018 proceedings, updating it with some more recent works. In particular, following Chan and Chan-Savin, see many of their papers listed in the bibliography, we have emphasized in this work that the restriction of a (generic) representation of a group to a closed subgroup (most of the paper is written in the context of GGP) turns out to be a projective representation on most Bernstein blocks of the category of smooth representations of . Further, once is a projective module in a particular Bernstein block, it has a simple structure.
Paper Structure (14 sections, 36 theorems, 104 equations)

This paper contains 14 sections, 36 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Branching laws from ${\rm GL}_{n+1}(F)$ to ${\rm GL}_n(F)$
  3. An erratum to Kunneth Theorem from Pr3
  4. Bessel subgroup
  5. What does the restriction really looks like!
  6. A theorem of Roche and some consequences
  7. Euler-Poincaré characteristic for classical groups
  8. Local constancy of the Euler-Poincaré characteristic
  9. Euler-Poincaré characteristic for the group case: Kazhdan orthogonality
  10. An integral formula of Waldspurger
  11. Conjectured EP formula
  12. The Schneider-Stuhler duality theorem
  13. An example: triple products for ${\rm GL}_2(F)$
  14. Template from algebraic geometry

Key Result

Theorem 1.2

(Aizenbud-Sayag) For $\pi$ an irreducible admissible representation of ${\rm GL}_{n+1}(F)$, the restriction of $\pi$ to ${\rm GL}_n(F)$ is locally finite. More generally, if $(G,H)$ with $H\subset H \times G$ is a spherical pair, i.e., $H$ has on open orbit on the flag variety of $G\times H$, where

Theorems & Definitions (66)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • ...and 56 more