Hecke algebra of $\mathrm{GL}_n$ over a 2-dimensional local field

TL;DR

This paper develops a new Hecke algebra framework for $ ext{GL}_n$ over two-dimensional local fields using $ ext{R}((X))$-measures, introducing convolution structures and candidate representations.

Contribution

It introduces a novel Hecke algebra construction for $ ext{GL}_n$ over 2D local fields using $ ext{R}((X))$-measures and defines associated measurable representations.

Findings

Defined a convolution product on $ ext{C}((X))$-valued functions

Constructed the Hecke algebra of $ ext{GL}_n(F)$

Proposed a candidate for measurable $ ext{C}((X))$-representations

Abstract

Using the $\mathbb{R}((X))$-measure, we define and study certain $\mathbb{C}((X))$-valued functions on $\mathrm{GL}_n(F)$ for $F$ a two-dimensional local field. In particular, we define a convolution product on such suitable functions, which leads us to define the Hecke algebra of $\mathrm{GL}_n(F)$. We then define the measurable $\mathbb{C}((X))$-representations of $\mathrm{GL}_n(F)$, and prove that function space is a candidate for such representations.