Non-archimedean Infinite Hecke Algebra
Milo Bechtloff Weising
TL;DR
This work develops the representation theory of the infinite type A Hecke algebra over non-archimedean fields, focusing on almost-symmetric representations. It constructs and classifies irreducible modules hat{S}_{\lambda} as norm completions of stabilized Hecke-Specht modules, blending combinatorial and analytic methods to overcome non-archimedean obstacles. The paper then defines and analyzes regularized trace functionals Gamma_{\lambda} via carefully chosen inner products, proves integrality properties, and posits a rationality conjecture for Gamma_{\lambda}(1) with potential links to LLT/Kazhdan–Lusztig-type structures. Overall, it extends finite-type Hecke algebra techniques to an infinite, non-archimedean setting, enriching both the combinatorial and analytic toolbox for these algebras.
Abstract
We study the representation theory of the infinite type A Hecke algebra over a non-Archimedean field in the case where the parameter is a pseudo-uniformizer. Specifically, we consider a family of representations, called almost-symmetric, which satisfy additional topological and algebraic constraints. We give a full classification of the irreducible almost-symmetric representations. These turn out to be indexed by integer partitions arising as topological completions of specific direct limits of Hecke-Specht modules. We give detailed analysis of these representations and construct functionals analogous to finite Hecke algebra traces.