Homomorphisms and Extensions of Principal Series Representations

C. Stroppel

Journal of Lie theory, Tome 13 (2003) no. 1, pp. 193-212.

Résumé

We describe homomorphisms and extensions of principal series representations. Principal series are certain representations of a semisimple complex Lie algebra  g  and are objects of the Bernstein-Gelfand-Gelfand-category  O . Verma modules and their duals are examples of such principal series representations.  Via the equivalence of categories of I. Bernstein and S. I. Gelfand ["Tensor products of finite and infinite dimensional representations of semisimple Lie algebras", Compositio Math. 41 (1980) 245--285], the principal series representations correspond to Harish-Chandra modules for  g times g  which arise by induction from a minimal parabolic subalgebra of  g times g.  We show that all principal series have one-dimensional endomorphism rings and trivial self-extensions. We also give an explicit example of a higher dimensional homomorphism space between principal series. As an application of these results we prove the existence of character formulae for "twisted tilting modules".  The twisted tilting modules are some indecomposable objects of  O  having a flag whose subquotients are principal series modules and for which a certain Ext-vanishing condition holds.