We study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and we present a conjecture on the Deligne-Lusztig restriction of Gelfand-Graev representations. We prove the conjecture for restriction to a Coxeter torus. We deduce a proof of Broué's conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type $A_{n}$ and a Coxeter element. Our study is based on Lusztig's work in characteristic $0$ [Lu2].

We study some conjectures on the endomorphism algebras of the cohomology of Deligne-Lusztig varieties which are a refinement of those of [BMi].

We explain how A. Givental's mirror symmetric family [14] to the type $A$ flag variety and its proposed generalization [3] to partial flag varieties by Batyrev, Ciocan-Fontanine, Kim and van Straten relate to the Peterson variety $Y \subset SL_{n}/B$. We then use this theory to describe the totally nonnegative part of $Y$, extending a result from [30].

Let $\Lambda$ be the path algebra of a Dynkin quiver. A recent result of Bo Chen asserts that $\operatorname{Hom}(X, Y/X) = 0$ for any Gabriel-Roiter inclusion $X \subseteq Y$. The aim of the present note is to give an interpretation of this result in terms of Hall polynomials, and to extend it in this way to representation-directed split algebras. We further show its relevance when dealing with arbitrary representation-finite split algebras.

An equivalence relation is defined and studied on the set of $B$-stable ideals in the nilradical of the Lie algebra of a Borel subgroup $B$. Techniques are developed to compute the equivalence relation and these are carried out in the exceptional groups. There is a natural partial order on equivalence classes coming from inclusion of one ideal in another. A main theorem is that this partial order is a refinement of the closure ordering on nilpotent orbits.