Generic spherical unitary dual for Chevalley groups over local fields

Abstract

We prove that a generic spherical parameter for the graded affine Hecke algebra with equal parameters is unitary if and only if its normalized intertwining forms are positive on the reflection representation and on the irreducible constituents of its second symmetric power. The proof combines a signature formula for the reflection representation, Yu's uniform two-wall operation, a simply-laced Jantzen recursion, and a root-poset analysis. In types B_n and C_n, we prove more sharply that the reflection representation V and the traceless-diagonal constituent of Sym^2(V) suffice. We also give a shorter second proof when arbitrary Weyl-group types are allowed. The consequence of the first proof (via Barbasch-Vogan petite K-types) is that the generic spherical unitary dual of a Chevalley group over the real numbers or a nonarchimedean local field, and for simply-laced groups and complex symplectic groups also over the complex numbers, is independent of the field. Moreover, the answer has a simple uniform description as the Weyl conjugates of a disjoint union of a power of 2 alcoves in the affine Weyl group arrangement in the fundamental Weyl chamber, where the exponent is the matching number of the Dynkin diagram of the subsystem of short coroots.