Kazhdan isomorphism over families and integrality under close local fields

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.