Base Change for the Iwahori-Hecke Algebra of $GL_2$

Abstract

Let F be a local non-Archimedean field of characteristic not equal to 2, let E/F be a finite unramified extension field, and let σ be a generator of Gal(E, F). Let f be an element of Z(H_{I_E}), the center of the Iwahori-Hecke algebra for GL2(E), and let b be the Iwahoric base change homomorphism from Z(H_{I_E}) to Z(H_{I_F} ), the center of the Iwahori-Hecke algebra for GL2(F) [8]. This paper proves the matching of the σ-twisted orbital integral over GL2(E) of f with the orbital integral over GL2(F) of bf. To do so, we compute the orbital and σ-twisted orbital integrals of the Bernstein functions z_μ. These integrals are computed by relating them to counting problems on the set of edges in the building for SL2. 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 study of Shimura varieties with Iwahori level structure at a prime p [7].