Exceptional Unitary Representations Of Semisimple Lie Groups

Abstract:

Let $G$ be a noncompact simple Lie group with finite center, let $K$ be a maximal compact subgroup, and suppose that $\text {rank }G=\text {rank }K$. If $G/K$ is not Hermitian symmetric, then a theorem of Borel and de Siebenthal gives the existence of a system of positive roots relative to a compact Cartan subalgebra so that there is just one noncompact simple root and it occurs exactly twice in the largest root. Let $\mathfrak {q}=\mathfrak {l}\oplus \mathfrak {u}$ be the $\theta$ stable parabolic obtained by building $\mathfrak {l}$ from the roots generated by the compact simple roots and by building $\mathfrak {u}$ from the other positive roots, and let $L\subseteq K$ be the normalizer of $\mathfrak {q}$ in $G$. Cohomological induction of an irreducible representation of $L$ produces a discrete series representation of $G$ under a dominance condition. This paper studies the results of this cohomological induction when the dominance condition fails. When the inducing representation is one-dimensional, a great deal is known about when the cohomologically induced representation is infinitesimally unitary. This paper addresses the question of finding Langlands parameters for the natural irreducible constituent of these representations, and also it finds some cases when the inducing representation is higher-dimensional and the cohomologically induced representation is infinitesimally unitary.