# Characters of Representations of Algebraic Groups in Small Characteristics

by Jens Kjærgaard Boldsen
PhD Dissertations | October 2012
Lusztig's conjecture describes the simple characters of certain algebraic groups, in sufficient large finite characteristic, as a linear combination of the Weyl characters with the Kazhdan-Lusztig polynomials (evaluated in 1) as the coefficients:
$\mbox{ch} L(w.\lambda_0) = \sum_{x \leq w} P_{w_0 x, w_0 w}(1) \chi(x.\lambda_0)$
By sufficient large characteristic we mean that it should be bigger than the Coxeter number of the Weyl group corresponding to the algebraic group. This dissertation considers what happens when the characteristic is smaller than the Coxeter number by calculating examples thereof, and suggesting new connections that might hold where Lusztig's formula fails.
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