Some remarks on traces on the infinite-dimensional Iwahori--Hecke algebra

TL;DR

This paper explores the structure of traces on the infinite-dimensional Iwahori--Hecke algebra, classifies positive traces, and constructs related representations, linking algebraic traces to group representations.

Contribution

It provides a classification of positive traces on the infinite-dimensional Iwahori--Hecke algebra and constructs associated representations, extending understanding of these algebraic structures.

Findings

Classification of all indecomposable positive traces by Vershik and Kerov.

Construction of representations generated by these traces.

Relations established between group representations and convolution algebra representations.

Abstract

The infinite-dimensional Iwahori--Hecke algebras $\mathcal{H}_\infty(q)$ are direct limits of the usual finite-dimensional Iwahori--Hecke algebras. They arise in a natural way as convolution algebras of bi-invariant functions on groups $\mathrm{GLB}(\mathbb{F}_q)$ of infinite-dimensional matrices over finite-fields having only finite number of non-zero matrix elements under the diagonal. In 1988 Vershik and Kerov classified all indecomposable positive traces on $\mathcal{H}_\infty(q)$. Any such trace generates a representation of the double $\mathcal{H}_\infty(q)\otimes \mathcal{H}_\infty(q)$ and of the double $\mathrm{GLB}(\mathbb{F}_q)\times \mathrm{GLB}(\mathbb{F}_q)$. We present constructions of such representations; the traces are some distinguished matrix elements. We also obtain some (simple) general statements on relations between unitary representations of groups and representations of convolution algebras of measures bi-invariant with respect to compact subgroups.