The Frobenius direct image of line bundles and the structure of representations
Reed, Mary Lynn
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https://hdl.handle.net/2142/21997
Description
Title
The Frobenius direct image of line bundles and the structure of representations
Author(s)
Reed, Mary Lynn
Issue Date
1995
Doctoral Committee Chair(s)
Haboush, William J.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
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.
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