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

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