Modular Gelfand Pairs and Multiplicity-Free Representations

Author:

Zhang Robin12

Affiliation:

1. Department of Mathematics, Columbia University , Room 509, MC 4406, 2990 Broadway, New York, NY 10027, USA

2. Department of Mathematics, Massachusetts Institute of Technology , Room 106, Simons Building (Building 2), 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Abstract

Abstract Over algebraically closed fields of arbitrary characteristic, we prove a general multiplicity-freeness theorem for finitely-generated modules with commutative endomorphism rings. Using more lenient versions of projectivity and injectivity for modules, this gives a generalization of Gelfand’s criterion on the commutativity of Hecke algebras for Gelfand pairs and multiplicity-free triples. 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 Gelfand–Graev representations in arbitrary characteristic and the uniqueness of modular trilinear forms on irreducible representations of quaternion division algebras over local fields.

Funder

National Science Foundation

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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