Scalar generalized Verma modules

Details Scalar generalized Verma modules Scalar generalized Verma modules

2011-01-17

arxiv.org/abs/1101.3134v1

Mathematics

Representation Theory

Scientific paper

In this paper we study the scalar generalized Verma module $M$ associated to a character of a parabolic subalgebra of $sl(E)$. Here $E$ is a finite dimensional vector space over an algebraically closed field $K$ of characteristic zero. The Verma module $M$ has a canonical simple quotient $L$ with a canonical filtration $F$. In the case when the quotient $L$ is finite dimensional we use left annihilator ideals in $U(sl(E))$ and geometric results on jet bundles to generalize to an algebraically closed field of characteristic zero a classical formula of W. Smoke on the structure of the jet bundle of a line bundle on an arbitrary quotient $SL(E)/P$ where $P$ is a parabolic subgroup of $SL(E)$. This formula was originally proved by Smoke in 1967 using analytic techniques.

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