Global representations of the Heat and Schr\"odinger equation with singular potential

We study the $n$-dimensional Schr\"odinger equation with singular potential $V_\lambda(x)=\lambda |x|^{-2}$. Its solution space is studied as a global representation of $\widetilde{SL(2,\R)}\times O(n)$. A special subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. The space of $K$-finite vectors is calculated, obtaining conditions for $\lambda$ so that this space is non-empty. The direct sum of solution spaces, over such admissible values of $\lambda$ is studied as a representation of the $2n+1$-dimensional Heisenberg group

Distinguished orbits and the L-S category of simply connected compact Lie groups

We show that the Lusternik-Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of G.Comment: 10 pages, 2 figure

Global actions of Lie symmetries for the nonlinear heat equation

AbstractBy restricting to a natural class of functions, we show that the Lie point symmetries of the nonlinear heat equation exponentiate to a global action of the corresponding Lie group. Remarkably, in most of the cases, the action turns out to be linear

Relative extremal projectors

AbstractThis paper proves the existence of relative extremal projectors. An infinite factorization is given as well as a summation formula

L2(q) and the rank two lie groups : their construction, geometry, and character formulas

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994.Includes bibliographical references (leaves 111-113).by Mark R. Sepanski.Ph.D

Block-compatible metaplectic cocycles

Let F be a local field such that the group µ r (F) of r-th roots of unity in F × has cardinality r ≥ 1. Let G be the F-rational points of a simple Chevalley group defined over F. In his thesis, Matsumoto [5] gave a beautiful construction for the metaplectic cover G of G, a central extension of G by µ r (F) whose existence is intimately connected with the deep properties of the r-th order Hilbert symbol . Metaplectic groups figure prominently in the study of number theory, representation theory, and physics, arising naturally in the theory of theta functions, dual pair correspondences, Weil representations, and spin geometry. In this paper we study the class of central extensions of a simple Chevalley group over an arbitrary infinite field, of which the metaplectic groups form an important subclass. Metaplectic groups were constructed quite explicitly in Weil's memoir To summarize our results, let F be an infinite field, G the F-rational points of a simple Chevalley group defined over F, A an abelian group, and c : a Steinberg symbol that is bilinear if G is not symplectic (cf. §1). In this paper we describe an explicit 2-cocycle σ G in Z 2 (G; A) that represents the cohomology class in H