The Springer Morphism, Polynomial Representation Rings, and the Cohomology Ring of Grassmannians

To any almost faithful representation of a complex, connected, reductive algebraic group $G$ of highest weight $\\lambda$ one can associate a dominant morphism from the group to its Lie algebra $\\fg$. This map enjoys many nice properties. In particular, when restricted to a maximal torus it maps to the Cartan subalgebra. This map can be used to give a natural definition of polynomial representations for the classical groups of types B, C, and D. Given a parabolic subgroup $P\\subset G$, Kumar showed there is a surjective algebra homomorphism from the polynomial representations of a Levi subgroup of P to the cohomology of G/P which extends a classical result relating the polynomial representations of GL(r) and the cohomology of the Grassmannian of r-planes in n-space $H^*(Gr(r,n))$. In this work we give an explicit determination of the map $\\theta_\\lambda$ for simple groups and consider Kumar's map for types B, C, and G.Doctor of Philosoph