Journal of Lie Theory Vol. 13, No. 1, pp. 193212 (2003) 

Homomorphisms and Extensions of Principal Series RepresentationsCatharina StroppelCatharina StroppelUniversity of Leicester University Road Leicester LE1 7RH (England) cs93@le.ac.uk Abstract: In this article we describe homomorphisms and extensions of principal series representations. Principal series are certain representations of a semisimple complex Lie algebra $\mg$ and are objects of the BernsteinGelfandGelfandcategory $\cO$. In this article we describe homomorphisms and extensions of principal series representations. Principal series are certain representations of a semisimple complex Lie algebra $\mg$ and are objects of the BernsteinGelfandGelfandcategory $\cO$. Verma modules and their duals are examples of such principal series representations. Via the equivalence of categories of Bernstein, I., and S. I. Gelfand, {\it Tensor products of finite and infinite di\men\sio\nal representations of semisimple Lie algebras}, Compositio math. {\bf 41} (1980), 245285, the principal series representations correspond to HarishChandra modules for $\mg\times\mg$ which arise by induction from a minimal parabolic subalgebra of $\mg\times\mg$. We show that all principal series have onedimensional endomorphism rings and trivial selfextensions. 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 $\cO$ having a flag whose subquotients are principal series modules and for which a certain Extvanishing condition holds. Full text of the article:
Electronic fulltext finalized on: 22 Nov 2002. This page was last modified: 3 Jan 2003.
© 2002 Heldermann Verlag
