Appears in collection : 2025 - T1 - WS2 - Tempered representations and K-theory

Let $G$ be a compact connected semisimple Lie group. We will describe the Langlands dual group~$G^\vee$. We now have two extended affine Weyl groups, one for $G$ and one for $G^\vee$. We will compare the C*-algebras of these two discrete groups, and show that they have the same K-theory. In this sense, Langlands duality is an invariant of K-theory.

With the aid of the equivariant Chern character of Baum-Connes, we will compute this K-theory for $\mathrm{SU}(n)$ and the exceptional Lie group $E_6$. As an application, we will compute the K-theory of the Iwahori-spherical C_-algebra of the $p$-adic version of $E_6$. The spectrum of this C_-algebra comprises irreducible tempered representations of $E_6$ which admit a nonzero Iwahori-fixed vector. From the point of view of noncommutative geometry, we are computing the K-theory of this spectrum.