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

TL;DR

The paper advances harmonic analysis for $GL_n(F)$ when $F$ is a two-dimensional local field by building an $\mathbb{R}((X))$-valued measure (via Fesenko) and a compatible integration framework (via Morrow) to define a $\mathbb{C}((X))$-valued Hecke algebra on $GL_n(F)$. It introduces the function space $\mathcal{L}^M(GL_n(F))$ of measurable, $\mathbb{C}((X))$-valued functions and proves a well-defined convolution that yields an associative algebra $\mathcal{H}$ acting as the Hecke algebra. The work then defines measurable $\mathbb{C}((X))$-representations and proves that $\mathcal{L}^M(GL_n(F))$ furnishes a natural measurable representation, linking Hecke modules to representation theory in the higher local field setting. This establishes a foundational bridge for harmonic analysis on $GL_n(F)$ over two-dimensional local fields and lays groundwork for further structural results, such as possible Satake-type correspondences in this context.

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.

Theorems & Definitions (41)

• Example 2.1
• Proposition 2.2
• Example 2.3
• Remark 2.4
• Theorem 2.5
• Proposition 2.6
• proof
• Remark 2.7
• Remark 2.8
• Lemma 2.9