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A Positive Characteristic Resolution Problem for SL(3, k)

✍ Scribed by Mary Lynn Reed

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 451 KB
Volume: 198
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1997.7102

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✦ Synopsis

In 1975, I. N. Bernstein, I. M. Gel'fand, and S. I. Gel'fand in ''Lie Groups and . Their Representations, '' pp. 21᎐64, Halsted, New York, 1975 resolved an irreducible representation of a complex semisimple Lie algebra by Verma modules indexed by the Weyl group. This resolution is now commonly referred to as the Ž . Bernstein ᎐Gel'fand᎐Gel'fand or BGG resolution. One consequence of the BGG resolution is a simple proof of the Weyl character formula. In this paper, we will describe an analogous resolution problem in positive characteristic: Is there a Ž resolution of a highest weight irreducible representation of a semisimple simply connected algebraic group over an algebraically closed field of positive characteris-. tic by restricted Verma modules? And if so, is it a generalization of the BGG Ž . resolution? This paper provides a complete answer to this problem for SL 3, k . Consequently, we are able to compute the formal character of the irreducible representation following a procedure similar to the BGG proof of the Weyl character formula. ᮊ 1997 Academic Press ᑿ restricted Lie algebra of B.

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A Positive Characteristic Resolution Problem for SL(3,&#xa0;k)

✦ Synopsis

A Positive Characteristic Resolution Problem for SL(3, k)