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