Compatibility of Kazhdan and Brauer homomorphism

Abstract Let G be a split connected 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 $\mathrm {Kaz}_m^F :\mathcal {H}\big (G(F),K_F\big ) \rightarrow \mathcal {H}\big (G(F'),K_{F'}\big )$ , 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 $\sigma $ is an automorphism of G of prime order l , then we have Brauer homomorphism $\mathrm {Br}:\mathcal {H}(G(F),U(F))\rightarrow \mathcal {H}(G^\sigma (F),U^\sigma (F))$ , where $U(F)$ and $U^\sigma (F)$ are compact open subgroups of $G(F)$ and $G^\sigma (F),$ 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.