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Modular Gelfand pairs and multiplicity-free representations

Robin Zhang

Abstract

We give a generalization of Gelfand's criterion on the commutativity of Hecke algebras for Gelfand pairs and multiplicity-free triples over algebraically closed fields of arbitrary characteristic. Using more lenient versions of projectivity and injectivity for modules, we prove a general multiplicity-freeness theorem for finitely-generated modules with commutative endomorphism rings. For representations of finite and profinite groups, Gelfand pairs over the complex numbers are therefore also Gelfand pairs over the algebraic closure of any finite field. Applications include the uniqueness of Whittaker models of modular Gelfand-Graev representations and the uniqueness of modular trilinear forms on irreducible representations of quaternion division algebras over local fields.

Modular Gelfand pairs and multiplicity-free representations

Abstract

We give a generalization of Gelfand's criterion on the commutativity of Hecke algebras for Gelfand pairs and multiplicity-free triples over algebraically closed fields of arbitrary characteristic. Using more lenient versions of projectivity and injectivity for modules, we prove a general multiplicity-freeness theorem for finitely-generated modules with commutative endomorphism rings. For representations of finite and profinite groups, Gelfand pairs over the complex numbers are therefore also Gelfand pairs over the algebraic closure of any finite field. Applications include the uniqueness of Whittaker models of modular Gelfand-Graev representations and the uniqueness of modular trilinear forms on irreducible representations of quaternion division algebras over local fields.

Paper Structure

This paper contains 15 sections, 31 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Relative projectivity and relative injectivity
  3. Relatively-projective modules
  4. Relatively-injective modules
  5. Endomorphism rings
  6. Exact functors
  7. General criteria for multiplicity-freeness
  8. Finite and compact multiplicity-free triples
  9. Induction and restriction
  10. Commutative Hecke algebra criteria
  11. Compact groups
  12. Applications for finite and compact groups
  13. Gelfand--Graev and GL(n, q)
  14. Trilinear forms of quaternion division algebras
  15. Multiplicity-free restrictions

Key Result

Proposition 1.5

Let $G$ be a finite group, $H$ be a subgroup of $G$, and $\mathrm{triv}_H$ be the trivial representation of $H$. If the Hecke algebra $\mathcal{H}(G, H, \mathrm{triv}_H, \mathbb{C})$ is commutative, then $(G, H, \mathrm{triv}_H)$ is a multiplicity-free triple over $\mathbb{C}$.

Theorems & Definitions (64)

  • Remark 1.1
  • Definition 1.2: Multiplicity-free triple, GT1
  • Remark 1.3
  • Definition 1.4
  • Proposition 1.5: gelfandgelfand-graev
  • Theorem 1.6: Theorem \ref{['thm:gelfand-triple-2']}
  • Theorem 1.7: Theorem \ref{['thm:multiplicity-free-3']}
  • Corollary 1.8: Corollary \ref{['cor:Fq-2']}
  • Corollary 1.9: Corollary \ref{['cor:whittaker-2']}
  • Corollary 1.10: Corollary \ref{['cor:qda-2']}
  • ...and 54 more