Homomorphisms and Extensions of Principal Series Representations

Catharina Stroppel

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 Bernstein-Gelfand-Gelfand-category $\cO$. 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 Bernstein-Gelfand-Gelfand-category $\cO$. 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), 245--285, the principal series representations correspond to Harish-Chandra 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 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 $\cO$ having a flag whose subquotients are principal series modules and for which a certain Ext-vanishing condition holds.

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