Base change and $K$-theory for $\mathrm{GL}(n)$

Abstract

Let F be a nonarchimedean local field and let $G=\mathrm{GL}(n)=\mathrm{GL}(n,F)$. Let $E/F$ be a finite Galois extension. We investigate base change $E/F$ at two levels: at the level of algebraic varieties, and at the level of $K$-theory. We put special emphasis on the representations with Iwahori fixed vectors, and the tempered spectrum of $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$. In this context, the prominent arithmetic invariant is the residue degree $f(E/F)$.