Base Change Fundamental Lemma for Central Elements in Depth-Zero Hecke Algebras over Local Function Fields

Abstract

Let $G$ be an unramified group over a local function field $F$ of characteristic $p>0$. Let $E/F$ be an unramified extension of degree $r$. Let $\chi$ be a depth-zero character on the maximal torus $T$, which induces a character $\rho$ on an Iwahori subgroup $I$. Consider the corresponding Hecke algebras of depth-zero principal series block $\mathcal{H}(G(F),\rho)$ and $\mathcal{H}(G(E),\rho_E)$. One can define the base change homomorphism $b$ between the centers of these Hecke algebras, then base change fundamental lemma asks whether the functions $\phi$ and $b(\phi)$ have the same stable (twisted) orbital integrals for $\phi\in Z(\mathcal{H}(G(E),\rho_E))$, at corresponding semisimple elements.

In this article, we introduce an abstract norm map between the stable twisted conjugacy classes in $G(E)$ and stable conjugacy classes of $G(F)$ in the positive characteristics setting, prove the simple twisted trace formula then apply the process of stabilization of twisted trace formula to show the existence of local data and finally adapt Labesse elementary functions to prove the corresponding base change fundamental lemma for regular semisimple elements.