Compatibility of Kazhdan and Brauer homomorphism
Sabyasachi Dhar
TL;DR
The paper establishes a precise compatibility between Kazhdan’s mod-$l$ isomorphism of Hecke algebras over $m$-close local fields and the Brauer homomorphism arising from a prime-order automorphism, within the local base-change framework for connected split reductive groups over $\mathbb{Z}$. It proves that the Brauer map commutes with Kazhdan’s isomorphism after passing between $F$ and a closely related $F'$, both in unramified and totally ramified cases, by analyzing double coset decompositions and Galois actions. This compatibility yields a representation-theoretic analogue of Langlands functoriality in close local fields and applies to linkage via Tate cohomology, including finiteness results for $\widehat{H}^i(\Xi)$ in the GL$_n$ case. The results provide a framework to transfer base-change and linkage phenomena between close local fields, deepening understanding of how local Langlands correspondences behave under field close-ness and Brauer-type restrictions.
Abstract
Let $G$ be a connected split reductive group defined over $\mathbb{Z}$. Let $F$ and $F'$ be two non-Archimedean $m$-close local fields, where $m$ is a positive integer. D.Kazhdan gave an isomorphism between the Hecke algebras ${\rm Kaz}_m^F :\mathcal{H}\big(G(F),K_F\big) \rightarrow \mathcal{H}\big(G(F'),K_{F'}\big)$, where $K_F$ and $K_{F'}$ are the $m$-th usual congruence subgroups of $G(F)$ and $G(F')$ respectively. On the other hand, if $σ$ is an automorphism of $G$ of prime order $l$, then we have Brauer homomorphism ${\rm Br}:\mathcal{H}(G(F),U(F))\rightarrow \mathcal{H}(G^σ(F),U^σ(F))$, where $U(F)$ and $U^σ(F)$ are compact open subgroups of $G(F)$ and $G^σ(F)$ respectively. In this article, we study the compatibility between these two maps in the local base change setting. Further, an application of this compatibility is given in the context of linkage--which is the representation theoretic version of Brauer homomorphism.