Functors for Unitary Representations of Classical Real Groups and Affine Hecke Algebras

We define exact functors from categories of Harish-Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they preserve unitarity. In the case of split classical groups, we thus obtain a functorial inclusion of the real spherical unitary dual (with “real infinitesimal character”) into the corresponding p-adic spherical unitary dual.