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Let F be a nonarchimedean local field and let G D GL.n/ D 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 GL.doi:10.4171/jncg/9 fatcat:gf5qov3zqrdethmiaed2lzbwwy