Departmental Colloquium
Nigel Higson
Penn State
Contractions of Lie Groups and Representation Theory
Abstract: Let K be a closed subgroup of a Lie group G. The contraction of G to K is a Lie group, usually more elementary in structure than G itself, that approximates G to first order near K. The terminology is due to the mathematical physicists, who examined the group of Galilean transformations as a contraction of the group of Lorentz transformations. My focus will be on a related but different class of examples, the prototype of which 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. But in the 1970s George Mackey carried out a few calculations pointing to an interesting rigidity phenomenon: as the contraction group is deformed back 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 (partly 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.
Friday January 21, 2011 at 3:00 PM in SEO 636