Pure Math colloquium

Nigel Higson, Penn State University

“Contractions of Lie Groups and Representation Theory”

The contraction of a Lie group G to a subgroup K is a Lie group, often simpler 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 trans- formations. In geometry, the group of isometric motions of Euclidean space may be viewed as a contraction of the group of isometric motions of hyperbolic space. It is natural to guess that there is some sort of limiting relationship between representations of the contraction group and represen- tations of the original group. But in the 1970s George Mackey made calculations that suggested a striking rigidity phenomenon: if K is the maximal compact subgroup of a semisimple group G, then representation theory is unchanged under contraction. 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 the- ory and noncommutative geometry, and I shall describe the evidence in support of it, which is by now substantial. However a conceptual explanation for Mackeys rigidity phenomenon remains elusive.

Refreshments will be served in MC 5046 at 3:30 p.m. Everyone is welcome to attend.