@MISC{Mcgerty_modularrepresentation, author = {Kevin Mcgerty}, title = {MODULAR REPRESENTATION THEORY AND GEOMETRY}, year = {} }

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Abstract

The representation theory of algebraic groups is both a classical subject and one of active investigation. Over an algebraically closed field of characteristic zero we have a reasonably complete picture – the category of representations is semisimple, the simple objects are parametrized by highest weights, and their characters are given by Weyl’s formula. Indeed the story is an algebraic version of the even more classical theory of representations of compact Lie groups. Nevertheless, Lusztig’s discovery [5] of canonical bases a mere seventeen years ago showed that this theory still contained unexpected structure. Given that Chevalley’s classification of connected simply-connected simple algebraic groups over an algebraically closed field is, remarkably, independent of characteristic, it is natural to seek a description of the category of representations of an algebraic group over a field k of characteristic p> 0. Here, immediately, many properties familiar from characteristic zero collapse, as even elementary calculations with SL2 demonstrate. The category is no longer semisimple, and while the simple objects are again classified by highest weights, their character