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Letbe a connected reductive algebraic group defined over an algebraically closed field. Letbe a maximal torus ofand letbe a Borel subgroup ofcontaining. LetWdenote the Weyl group ofand letẇdenote an element in the preimage of the natural map fromtoW. We investigate what conditions are necessary and sufficient for the double cosets(called Bruhat cells) to contain only regular elements. An application is provided on how this information about Bruhat cells can be used to determine character values for certain Hecke algebras. [ABSTRACT FROM PUBLISHER]

This article provides new, relatively simple proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive the description of the natural partial order of unipotent classes in the general linear group in terms of their Jordan blocks. This result was originally due to Gerstenhaber. Finally, we obtain a proof of the Bala-Carter Theorem, which holds even in certain bad characteristics. This proof requires the prior knowledge of the number of unipotent classes, unlike the original proofs due to Bala, Carter and Pommerening. [ABSTRACT FROM AUTHOR]