PSU Mathematics Department - Meeting Details

For more information about this meeting, contact Yakov Pesin, Robin Enderle.

Abstract:
Title:	Contractions of Lie Groups and Representation Theory
Seminar:	Department of Mathematics Colloquium
Speaker:	N. Higson, Department of Mathematics, Penn State
The contraction of a Lie group G to a subgroup K is a new Lie group, usually easier to understand than G itself, that approximates G to first order near K. The name comes from the mathematical physicists, who examined the Galilean group as a contraction of the Poincare group of special relativity. My focus will be on a related but different class of examples: the prototype is the group of isometric motions of Euclidean space, viewed as a contraction of the group of isometric motions of hyperbolic space. It is natural to expect some sort of limiting relation between representations of the contraction and representations of G itself. But in the 1970s George Mackey discovered an interesting rigidity phenomenon: as the contraction group is deformed to G, the representation theory remains in some sense unchanged. In particular the irreducible representations of the contraction group parametrize the irreducible representations of G. I shall formulate a reasonably precise conjecture along these lines that was inspired by subsequent developments in C*-algebra theory and noncommutative geometry, and describe the evidence in support of it, which is by now substantial. However a conceptual explanation for Mackey's rigidity phenomenon remains elusive.