Spectral transfer category of affine Hecke algebras
Seminar Room 1, Newton Institute
We introduce a notion of a ``spectral transfer morphism'' between affine Hecke algebras. Such a spectral transfer morphism from H_1 to H_2 is not given by an algebra homomorphism from H_1 to H_2 but rather by a homomorphism from the center Z_2 of H_2 to the center Z_1 of H_1 which is required to be ``compatible'' in a certain way with the Harish-Chandra \mu-functions on Z_1 and Z_2. The main property of such a transfer morphism is that it induces a correspondence between the tempered spectra of H_1 and H_2 which respects the canonical spectral measures (``Plancherel measures''), up to a locally constant factor with values in the rational numbers.
The category of smooth unipotent representations of a connected split simple p-adic group of adjoint type G(F) is Morita equivalent to a direct sum R of affine Hecke algebras. It is a remarkable fact that R admits an essentially unique ``spectral transfer morphism'' to the Iwahori-Matsumoto Hecke algebra of G. This fact offers a new perspective on Reeder's classification of unipotent characters for exceptional split groups which works in the general case, leading to an alternative approach to Lusztig's classification of unipotent characters of G(F).