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.