Inverse Satake transforms

TL;DR

This paper provides an explicit formula for the generating function of unramified L-functions related to split reductive groups over local fields, extending Satake transform concepts to spherical varieties.

Contribution

It introduces a new explicit formula for unramified L-functions and generalizes the Satake transform to a broad class of spherical varieties.

Findings

Derived an explicit formula for the generating function of unramified L-functions.

Extended Satake transform to spherical varieties.

Provided an alternative approach to existing solutions by Wen-Wei Li.

Abstract

Let H be a split reductive group over a local non-archimedean field, and let H^ denote its Langlands dual group. We present an explicit formula for the generating function of an unramified L-function associated to a highest weight representation of the dual group, considered as a series of elements in the Hecke algebra of H. This offers an alternative approach to a solution of the same problem by Wen-Wei Li. Moreover, we generalize the notion of "Satake transform" and perform the analogous calculation for a large class of spherical varieties.