Langlands parameters of derived functor modules and Vogan diagrams

Abstract

Let $G$ be a linear reductive Lie group with finite center, let $K$ be a maximal compact subgroup, and assume that $\mathrm{rank } G = \mathrm {rank } K$. Let $\ g= l\oplus u$ be a $\theta$ stable parabolic subalgebra obtained by building $l$ from a subset of the compact simple roots and form $A_g(\lambda)$. Suppose $\Lambda=\lambda+2\delta( u\cap p)$ is $K$-dominant and the infinitesimal character, $\lambda+\delta$, of $A_{g}(\lambda)$ is nondominant due to a noncompact simple root. By interpreting these conditions on the level of Vogan diagrams, a conjecture by Knapp is (essentially) settled for the groups $G=SU(p,q),\, Sp(p,q)$, and $SO^*(2n)$, thereby determining the Langlands parameters of natural irreducible subquotient of $A_{ g}(\lambda)$. For the remaining classical groups, simple supplementary conditions are given under which the Langlands parameters may be determined.