## Abstract

For a locally compact, totally disconnected group G, a subgroup H, and a character χ : H → ℂ× we define a Hecke algebra ℋχ and explore the connection between commutativity of ℋχ and the χ-Gelfand property of (G,H), that is, the property dimC(ρ∗)(H,χ-1) ≤ 1 for every ρ ∈ Irr(G), the irreducible representations of G. We show that the conditions of the Gelfand-Kazhdan criterion imply commutativity of ℋχ and verify in several simple cases that commutativity of ℋχ is equivalent to the χ-Gelfand property of (G,H). We then show that if G is a connected reductive group over a p-adic field F, and G/H is F-spherical, then the cuspidal part of ℋχ is commutative if and only if (G,H) satisfies the χ-Gelfand property with respect to all cuspidal representations ρ ∈ Irr(G). We conclude by showing that if (G,H) satisfies the χ-Gelfand property with respect to all irreducible (H\G, χ-1)-tempered representations of G then ℋχ is commutative.

Original language | American English |
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Pages (from-to) | 9203-9232 |

Number of pages | 30 |

Journal | International Mathematics Research Notices |

Volume | 2021 |

Issue number | 12 |

DOIs | |

State | Published - 1 Jun 2021 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- General Mathematics