Structure constants for Hecke and representation rings

TL;DR

This paper investigates the structure constants of the spherical Hecke algebra and the representation ring of the Langlands dual group, revealing their algebraic properties and interrelations in the context of reductive groups over local fields.

Contribution

It provides a detailed analysis of the structure constants for both the Hecke algebra and the dual group's representation ring, highlighting their connections and algebraic structures.

Findings

Explicit formulas for structure constants derived

Identification of algebraic relations between the rings

Insights into the duality between Hecke and representation rings

Abstract

We study the structure constants defining two related rings: the spherical Hecke algebra of a split connected reductive group over a non-Archimedean local field, and the representation ring of the Langlands dual group.