Base Change for the Iwahori-Hecke Algebra of GL_2

TL;DR

This paper proves a fundamental lemma relating orbital integrals in the Iwahori-Hecke algebra for GL_2 over local fields, using building combinatorics, crucial for understanding Shimura varieties with Iwahori level structure.

Contribution

It establishes the matching of twisted orbital integrals under base change for the Iwahori-Hecke algebra of GL_2, employing geometric counting methods on the building.

Findings

Orbital integrals relate to counting edges in the building for SL_2.

Integrals are surprisingly independent of conjugacy class.

Matching of integrals confirms the fundamental lemma for base change.

Abstract

Let $F$ be a non-Archimedean local field of characteristic not equal to 2, let $E/F$ be a finite unramified extension field, and let $\sigma$ be a generator of $\text{Gal}(E, F )$. Let $f$ be an element of $Z(H_{I_E})$, the center of the Iwahori-Hecke algebra for $GL_2(E)$, and let $b$ be the Iwahori base change homomorphism from $Z(H_{I_E})$ to $Z(H_{I_F})$, the center of the Iwahori-Hecke algebra for $GL_2(F)$ [8]. This paper proves the matching of the $\sigma$-twisted orbital integral over $GL_2(E)$ of $f$ with the orbital integral over $GL_2(F)$ of $bf$. To do so, we compute the orbital and $\sigma$-twisted orbital integrals of the Bernstein functions $z_{\mu}$. These integrals are computed by relating them to counting problems on the set of edges in the building for $SL_2$. Surprisingly, the integrals are found to be somewhat independent of the conjugacy class over which one is integrating. The matching of the integrals follows from direct comparison of the results of these computations. The fundamental lemma proved here is an important ingredient in the study of Shimura varieties with Iwahori level structure at a prime $p$ [7].