A quantum probabilistic approach to Hecke algebras for $\mathfrak{p}$-adic ${\rm PGL}_2$

TL;DR

This paper applies quantum probability to $p$-adic Hecke algebras for ${ m PGL}_2$, providing a new interpretation and a novel proof of the Fourier inversion formula in this setting.

Contribution

It introduces a quantum-probabilistic interpretation of the spherical Hecke algebra for ${ m PGL}_2(F)$, offering new insights and a proof of the Fourier inversion formula.

Findings

Quantum probability provides a new perspective on $p$-adic Hecke algebras.

A novel proof of the Fourier inversion formula for ${ m PGL}_2(F)$.

Enhanced understanding of harmonic analysis on $p$-adic groups.

Abstract

The subject of the present paper is an application of quantum probability to $p$-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for ${\rm PGL}_2(F)$, where $F$ is a $p$-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for ${\rm PGL}_2(F)$.