With D. Nadler. Affine Hecke Algebras and Vogan Duality.
We apply ideas from the geometric Langlands program to study the representation theory of real Lie groups. Our main result may be interpreted as an affine version of Vogan's character duality for representations of real Lie groups, or as a real version of Kazhdan and Lusztig's construction of the affine Hecke algebra. More precisely, we give a geometric description of the K-group of sheaves on a real form of the moduli space of bundles on P^1 (or of Harish-Chandra modules for a real loop group), as a module for the affine Hecke algebra. This result, combined with equivariant localization, implies Vogan's duality and provides it with a conceptual proof, linking the real local Langlands program (as studied by Adams-Barbasch-Vogan) and the geometric Langlands program. This provides strong evidence for a program to prove Soergel's categorical real local Langlands conjecture, of which this is the K-theoretic shadow.