2005-10-21
Wai Ling Yee, University of Alberta
Signatures of Invariant Hermitian Forms
Classifying the irreducible unitary representations of a real
reductive Lie group may be formulated as the algebraic problem of
classifying the irreducible Harish-Chandra modules which admit a
positive definite invariant Hermitian form. It is thus of interest to
study signatures of invariant Hermitian forms and to understand how
positivity can fail. Because Harish-Chandra modules may be
constructed through cohomological induction, in which Zuckerman
functors are applied to generalized Verma modules, we wish to compute
the signature of the Shapovalov form on irreducible Verma modules
$M(\lambda)$. Careful consideration of Gabber and Joseph's proof of
Kazhdan and Lusztig's inductive formula for computing Kazhdan-Lusztig
polynomials gives us a means of computing signatures of invariant
Hermitian forms on irreducible highest weight modules $L(\lambda)$.