Modular Gelfand pairs and multiplicity-free representations (Q6623603)

Let \(G\) be a finite group and let \(H \leq G\) be a subgroup of \(G\). The couple \((G,H)\) is a Gelfand pair if the induced representation \(\mathrm{Ind}^{G}_{H}(\mathrm{triv}_{H}) = \mathbb{C}[G] \otimes_{\mathbb{C}[H]} \mathrm{triv}_{H}\) of the trivial representation of \(H\) is multiplicity-free. The theory of Gelfand pairs has been extended by replacing the induced trivial representation \(\mathrm{Ind}^{G}_{H}(\mathrm{triv}_{H})\) with \(\mathrm{Ind}^{G}_{H}(\eta)\) for an arbitrary irreducible representation \(\eta\) of \(H\). This generalization is the study of multiplicity-free triples \((G,H, \eta)\) (see [\textit{D. Bump}, Lie groups. 2nd ed. New York, NY: Springer (2013; Zbl 1279.22001)], \S 45. Such objects are also known as Gelfand triples, but according to the reviewer, this name is particularly unfortunate, as it overlaps with the term \textit{Gelfand triples} used for a rigged Hilbert space in functional analysis).\N\NIn this paper the author proves a general multiplicity-freeness theorem for finitely-generated modules with commutative endomorphism rings. Using more flexible 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. In particular the author proves (Theorem 1.6): Let \(F\) be an algebraically closed field, \(\eta\) be an irreducible representation of \(H\). If the Hecke algebra \(\mathcal{H}(G, H, \eta, F)\) is commutative, then \((G,H,\eta)\) is a multiplicity-free triple over \(F\).\N\NApplications 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.