Zongzhu Lin

Support varieties for modules over Chevalley groups and classical Lie algebras

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.

Structure of cohomology of line bundles on GB for semisimple groups

Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field k of characteristic p > 0, T a maximal torus of G and B a Borel subgroup containing T . Each weight in X(T ) determines a line bundle on the flag varietyG/B. It turns out that cohomology of the line bundle is isomorphic to the derived functor of the induction functors from the category of B-modules to the category of G-modules for each 1-dimensional B-module defined by a weight in X(T ). Further, IndGBλ = H (G/B, λ) turns out to be the dual of a Weyl module. One of the main problems is to calculate the characters of the irreducible G-modules. For p = 0, the character of the irreducible G-module of highest weight λ is given by Weyl’s character formula and the G-module structures of the cohomology of line bundles are well understood. However, for p > 0, the story is quite different and many of the results remain conjectural. Since characters of Weyl modules are given by Weyl’s character formula, understanding the structure ofH(G/B, λ) turns out to be the main problem. It is also interesting to understand the structure of the higher cohomology, which might help us to understand H. H. Andersen has a series of papers toward the understanding of the higher cohomology, such as the simple socle of H and filtrations of H i [4]. In [5] , using the representations of infinitesimal subgroup schemes of G, he proved generically that H(G/B,w ·λ) has simple socle and simple head and their highest weights can be calculated. In [6], he proved that the socle series of H(λ) comes generically from the

Extensions between simple modules for Frobenius kernels

Introduction Let G be a simply connected and connected semisimple group over an algebraically closed field k of characteristic p > 0. T ⊂ G is a maximal torus and R is the root system relative to T . X(T ) is the weight lattice. Let B ⊃ T be a Borel subgroup corresponding to the negative roots R− = R. Denote by Gr the r-th Frobenius kernel of G. The socle and radical structures of the cohomology groups of line bundles on the flag variety G/B are determined by those structures of the GrT -modules Zr(λ) = Ind GrT BrTλ (cf. [12]). So the study of the GrT -structure of these modules turns out to be more interesting. Calculation of the extensions between simple modules plays an important rule in determining the socle structure. In this paper, we calculate the socle series of the Weyl modules with p-singular highest weight for the group of type G2 by studying the extensions between simple modules for Frobenius kernel. In the first section of this paper, we study the properties of Ext Gr(L(μ), H (λ)), which turns out to be semisimple and to have a good filtration for large p and prestricted weights μ and λ. Some of the vanishing properties of these modules are also studied. Then we use these properties to calculate Ext Gr(L(μ), L(λ)), which will lead a calculation of the extensions between simple GrT -modules. The results in this section will be used in Section 3 to calculate the socle series of Z1(λ) with p-singular weights λ for the group of type G2. The author ([13] ch3) used the method of Doty and Sullivan [7] and calculated the socle series of H(λ) for the p-singular weights λ in the bottom p-alcove for the groups of type A2 and B2. However when the multiplicities of simple modules in

Approach to Artinian algebras via natural quivers

Given an Artinian algebra

Representations of Tame Quivers and Affine Canonical Bases

A

Socle series of cohomology groups of line bundles on G / B

over a field

Representations of Algebraic Groups, Quantum Groups, and Lie Algebras

k

Extensions of modules over Hopf algebras arising from Lie algebras of Cartan type

, there are several combinatorial objects associated to

Good filtrations for representations of Lie algebras of Cartan type

A

Representations of Chevalley groups arising from admissible lattices

. They are the diagram