Abstract. Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p > 0, G 1 be the first Frobenius kernel, and G(F p ) be the corresponding finite Chevalley group. Let M be a rational G-module. In this paper we relate the support variety of M over the first Frobenius kernel with the support variety of M over the group algebra kG(F p ). This provides an answer to a question of Parshall. Applications of our new techniques are presented, which allow us to extend results of Alperin-Mason and Janiszczak-Jantzen, and to calculate the dimensions of support varieties for finite Chevalley groups.

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Let (g, [p]) be a restricted Lie algebra over an algebraically closed field of characteristic p > 0. The restricted nullcone of g, denoted by N 1 (g), consists of those x ∈ g such that x [p] = 0. In this paper the authors provide a concrete description of this variety (via closures of Richardson orbits) when g is the Lie algebra of a reductive group G and p a good prime. Various applications to representation theory and cohomology theory are provided.

There are several different ways to construct affine canonical bases, in addition to approaches by Lusztig and Kashiwara. In this paper we present a different approach to canonical bases via Hall algebras and representations of tame quivers over finite fields. The main idea is to tensor together integral bases constructed for cyclic quivers and Kronecker quivers with those from the preinjective and preprojective parts of tame quiver representations. Several different bases: a PBW type basis, a monomial basis, and a barinvariant basis are constructed and their relations to the canonical basis are discussed. The result also answers a question by Nakajima.

Abstract. Given an Artinian algebra A over a field k, there are several combinatorial objects associated to A. They are the diagram DA as defined in [DK], the natural quiver ∆A defined in [Li] (cf. Section 2), and a generalized version of k-species (A/r, r/r 2 ) with r being the Jacobson radical of A. When A is splitting over the field k, the diagram DA and the well-known ext-quiver ΓA are the same. The main objective of this paper is to investigate the relations among these combinatorial objects and in turn to use these relations to give a characterization of the algebra A.