Weyl Group Representations and Signatures of Intertwining Operators

For G real split, I studied the (non)-unitarity of a spherical principal series I(1, ν) of G by means of Weyl group representations. For a good choice of the parameter ν, the spherical principal series I(1, ν) admits an invariant hermitian form. To discuss the unitarity of I(1, ν), one needs to compute the signature of such a form. Computations can be done separately on the isotypic of each K−type, and when the K−type is petite they can be reduced to Weyl group calculations. It follows that, to compute the signatures on the isotypic of a petite K−type (ρ,Eρ), one only needs to understand the representation σρ of the Weyl group on the space of M−invariants in (ρ,Eρ). This work is meant to describe the set of Weyl group representations that can arise from this construction. The result is an “almost minimal” subset of Weyl group representations on which we have to test the signature of the “algebraic” intertwining operator in order to detect the non-unitarity of the principal series. In the classical case, this also detects unitarity. For SL(n,R), this “almost minimal” set consists of those Weyl group representations whose restriction to any W (SL(3)) does not contain the sign representation. If you identify the Weyl group with the symmetric group Sn, the “almost minimal” set of Weyl group representations is identified with the set of partitions of n in at most two parts. For each Weyl group representation σ satisfying this condition, I give an explicit construction for a petite representation μσ such that σ is a submodule of the representation of W on the space of M -fixed vectors of μσ. When σ varies in the set of permutations in at most two parts, this construction produces exactly Barbasch’s list of K-types for detecting unitarity in SL(n,R).