Non-unitarity outside the fundamental parallelepiped

Abstract

One of the challenges of the unitary dual problem is the daunting number of possible representations to consider. This article describes an approach to narrowing the search space for minimal principal series representations, in terms of the ``fundamental parallelepiped'' (or ``FPP''): the set of linear combinations of fundamental weights with coefficients in the interval $[0,1]$. An earlier conjecture of the author, which was rooted in work of Barbasch and then subsequently vastly generalized by Vogan (and recently proven by Davis and Mason-Brown), asserts that the FPP houses all dominant infinitesimal characters for which minimal principal series have a unitarizable quotient. We present a technique to prove this ``FPP inequality'' in specific examples, demonstrated here for the split real form $E_{8(8)}$, by introducing a limiting theory of intertwining operators as $\nu$ approaches $\infty$ in directions of fundamental weights. This limiting theory is inspired by Wilfried Schmid's work on variation of Hodge structure. As an application, we show that the unitary set for a particular minimal principal series consists of the closure of a single open alcove.