Kazhdan isomorphism over families and integrality under close local fields
Sabyasachi Dhar
TL;DR
The paper extends Kazhdan's isomorphism to Noetherian $\mathbb{Z}_l$-algebras with $l\neq p$ for connected reductive groups over fields that are sufficiently close, defining an $R$-algebra isomorphism between Hecke algebras $\mathcal{H}_R(G(F),K_m)$ and $\mathcal{H}_R(G'(F'),K_m')$. It builds on Ganapathy's generalization and uses parahoric group schemes and close-field techniques to transfer the Hecke-algebra structure from $G(F)$ to a matched $G'(F')$, with a distance parameter $N$ depending on $m$. As an application, it proves that integrality properties of $l$-adic representations are preserved under this transfer, providing a framework for local Langlands in families in the $l$-adic setting. The results rely on the finite-presentability of Hecke algebras and on explicit control of parahoric data under close local fields, enabling a concrete bridge between representations over $F$ and $F'$.
Abstract
Let $G$ be a split connected reductive group defined over $\mathbb{Z}$. Let $F$ be a locally compact non-Archimedean field with residue characteristic $p$. For a locally compact non-Archimedean field $F'$ that is sufficiently close to $F$, D.Kazhdan establishes an isomorphism between the Hecke algebras $\mathcal{H}(G(F),K_m)$ and $\mathcal{H}(G(F'),K_m')$ with coefficients in $\mathbb{C}$, where $K_m$ (resp. $K_m'$) is the $m$-th congruence subgroup of $G(F)$ (resp. $G(F')$). This result is generalised to arbitrary connected reductive algebraic groups by R.Ganapathy. In this article, we extend the result further where the coefficient ring of the Hecke algebras is considered to be more general, namely Noetherian $\mathbb{Z}_l$-algebras with $l\ne p$. Then we use this isomorphism to prove certain compatibility result in the context of $l$-adic representation theory.