Analysis and Invariant Theory for Gelfand Pairs

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Proposal: DMS-9970552Principal Investigators: Gail Ratcliff, Chal BensonAbstract. This research project concerns analysis on compact extensions of nilpotent Lie groups. One says that the action of a compact Lie group K on a nilpotent Lie group N yields a Gelfand pair when the integrable K-invariant functions on N commute under convolution. The investigators will continue their study of such Gelfand pairs, in particular their study of the associated space of spherical functions, especially when N is a Heisenberg group. Specific goals include further development of algorithms for computing spherical functions and their eigenvalues, study of identities for generalized binomial coefficients, a description of the topology of the Gelfand space, and applications to questions concerning hull minimal ideals in Schwartz spaces.The techniques of Fourier analysis, which traces its roots to the study of problems in heat conduction, are among the most powerful tools in the mathematician's arsenal. An important construction in Fourier analysis is the so-called convolution product. In the classical setting, convolution has the important property of commutativity. That is, one can rearrange the order of the factors in the product without affecting the result of the multiplication. There is, however, a modern and far-reaching generalization of Fourier analysis, like its ancestor linked to important problems in physics, in which convolution often fails to be commutative. In many problems in this type of analysis, which is done in the framework of objects known as "Lie groups," noncommutativity arises quite naturally and cannot be avoided, leading to great technical difficulties. Gelfand pairs provide a device to recover the simplicity of commutativity in certain noncommutative settings. Each Gelfand pair provides an environment with enough symmetry to ensure commutativity of the convolution product. Such pairs are named in honor of the Russian mathematician I. M. Gelfand and have been studied since the 1950s. Their usefulness is now well established in connection with analysis in the noncommutative setting of semi-simple Lie groups. In the past decade, it has been shown that Gelfand pairs also arise in another non-commutative setting, that of nilpotent Lie groups. The goals of this research concern the systematic study and mathematical analysis of this new class of Gelfand pairs, including the development of algorithms for explicit computations.