The Verma Modules In Representations Of Semisimple Algebraic Groups And Its Lie Algebras

Let G denote a connected, simply connected and semi-simple algebraic group over an algebraically closed field k of characteristic p > 0, and g = LieG be its Lie algebra. In this paper, we mainly study the Verma modules in representations of G and g. In this dissertation, The main results are listed below:1. When the highest weightλof Z₀(λ) lie in the fundamental alcove C₀, we can determine all the maximal weight vector of Z₀(λ) and they are monomials provided that p is bigger than a certain number. For general description of such maximal weight vectors, we give a sufficient condition to judge if a maximal weight vector of a Verma module in the Dist(G)-module category becomes a maximal vector of a baby Verma module in the U₀(g)-module category.2. We study irreducible non-restricted generalized baby Verma modules. We know that when p-characterχis zero, a baby Verma module is mostly reducible. But when p-characterχis not zero, the generalized baby Verma module may be irreducible. When the p-characterχhas standard Levi form and the highest weightλis included in the fundamental alcove C₀, we get an sufficient condition on generalized baby Verma module U_χ(g) (?)_{U0(P J)} L_J(λ) is irreducible. We partially answered an question addressed by Friedlander and Parshall in the reference [22, 5.1].3. We study support varieties and rank varieties for g. When the rank of p-characterχis 1,we prov the reduced enveloping algebra U_χ(g) and restricted enveloping algebra U₀(g) are (?)_g(χ)-equivariant isomorphism as left regular modules, so we get the relation between the rank variety of baby Verma module Z₀(λ) and the rank variety of baby Verma module Z_χ(λ):where (?)g(χ) = {X∈g |χ([X,g]) = 0}.