The Frobenius direct image of line bundles and the structure of representations

In the representation theory of semisimple algebraic groups in positive characteristic, the induced line bundles, ${\cal L}$($\lambda$), on the flag variety G/B, are important objects of study. E.g., their global sections form representations of G which are dual to the Weyl modules. In this thesis, our central object of study is the direct image of the induced line bundles under the Frobenius morphism. I.e., $F\sb\*{\cal L}$($\lambda$). We study the homogeneous sheaf structure of $F\sb\*{\cal L}$($\lambda$) in the context of several celebrated representation theory problems. Specifically, our decomposition of the Frobenius direct image on the projective line provides a geometric interpretation of the structure of SL(2,k) Weyl modules. In addition, $F\sb\*{\cal L}$($\lambda$) is a homogeneous sheaf induced by a restricted Verma module. As a generalization of the decomposition of $F\sb\*{\cal L}$($\lambda$) on the projective line, we have discovered some remarkable complexes of restricted Verma modules for the group SL(3,k). These complexes are quite clearly related to the Bernstein-Gel'fand-Gel'fand resolution in characteristic zero. And as the Bernstein-Gel'fand-Gel'fand resolution provides an elegant proof of the Weyl character formula, our complexes provide a procedure for calculating the characters of the irreducible representations as sums of Weyl characters. Therefore, the question of the existence of such complexes for any semisimple simply connected algebraic group in positive characteristic is closely related to the search for a character formula for the irreducible representation.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Reed, Mary Lynn

Contributors dc:contributor

• Haboush, William J.

• Mathematics

Statement dc:rights

• Copyright 1995 Reed, Mary Lynn

Language dc:language

eng

Identifier

AAI9543702
(UMI)AAI9543702

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/21997