Langlands parameters of derived functor modules and Vogan diagrams - Friedman, Paul D. (MATHEMATICA SCANDINAVICA, 2003)
Let $G$ be a linear reductive Lie group with finite center, let $K$ be a maximal compact subgroup, and assume that $rank G = rank K$. Let $\ g= l\oplus u$ be a $θ$ stable parabolic subalgebra obtained by building $l$ from a subset of the compact simple roots and form $A_g(λ)$. Suppose $Λ=λ+2δ( u\cap p)$ is $K$-dominant and the infinitesimal character, $λ+δ$, of $A_{g}(λ)$ 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 $A_{ g}(λ)$. For the remaining classical groups, simple supplementary conditions are given under which the Langlands parameters may be determined.