Mathematical Sciences: Symmetry and Orthogonal Polynomials

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Several areas within the theory of special functions will be investigated. Particular emphasis will be placed on the development of a mathematical theory of orthogonal polynomials of several variables in which there will be a concentration on functions defined on the n-sphere. These polynomials are taken to be orthogonal with respect to a weight function which is invariant under a finite reflection group. The weight functions are powers of products of linear functions whose zero sets are the mirrors for the reflections in the group. A form of Selberg's integral, corresponding to the octahedral group, is an important special case. By means of a commutative algebra of first-order differential-difference operators, which have an action analogous to that of partial derivatives in the theory of spherical harmonics, work will focus on three problems. The first is to construct a nonabelian fractional integral operator which intertwines ordinary differentiation and the differential- difference operators. This gives a mapping from harmonic functions and other eigenfunctions of the Laplacian to the functions under investigation. Secondly, by use of these operators and their adjoints, work will be done constructing explicit orthogonal bases and norms for functions on the sphere. Emphasis will be placed first on three and four dimensional examples. Thirdly, work will be done extending a recent positivity result on the Poisson kernel for the ball to finding this kernel explicitly for the general reflection group. This work is expected to have application to theoretical physics. It has already demonstrated its utility by playing a major role in the grinding of mirrors for the new Keck 10-meter telescope.