As the heat kernel plays an important role in many problems in harmonic analysis, an explicit usable expression is very much desirable. An explicit expression for the heat kernel for the Heisenberg group Hn = C ×R was obtained by Hulanicki [9] and by Gaveau [7]. Gaveau [7] also obtained the heat kernel for free nilpotent Lie groups of step two. Cygan [4] obtained the heat kernel for all nilpote...

Let G be a complex, connected, reductive algebraic group. In this paper we show analogues of the computations by Borho and MacPherson of the invariants and antiinvariants of the cohomology of the Springer fibres of the cone of nilpotent elements, N , of Lie(G) for the Steinberg variety Z of triples. Using a general specialization argument we show that for a parabolic subgroup WP ×WQ of W × W th...

Let G be a classical group and let g be its Lie algebra. For a nilpotent element X E g, the ring R(Ox) of regular functions on the nilpotent orbit Ox is a Gmodule. In this thesis, we will decompose it into irreducible representations of G for some spherical nilpotent orbits. Thesis Supervisor: David Alexander Vogan Title: Professor of Mathematics

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert seri...