Seminarium KMMF "Teoria Dwoistości"

Let K be a closed subgroup in a connected Lie group G. The contraction of G to K isthe semi-direct product group associated to the adjoint action of K on the quotientof the Lie algebras of G and K. The terminology is due to Inonu and Wigner; in morecurrent terms, the group G is a deformation of its contraction. Inonu and Wignerexamined the group of Galilean transformations as a contraction of the group ofLorentz transformations. Our focus will be on a related but different class ofexamples: the prototype is the group of isometric motions of the Euclidean plane,viewed as a contraction of the group of isometric motions of the hyperbolic plane;in general one can consider any Riemannian symmetric space of noncompact type inplace of the hyperbolic plane, and its tangent space at any point in place of theEuclidean plane. It is natural to expect some sort of limiting relation betweenrepresentations of the contraction and representations of G. In the 1970s Mackeymade some calculations pointing to an interesting rigidity phenomenon: as thecontraction group is deformed back to G, the representation theory remains in somesense unchanged. In particular the irreducible representations of the contractiongroup parametrize the irreducible representations of G. I shall formulate areasonably precise conjecture (partly inspired by subsequent developments inC*-algebra theory and noncommutative geometry) and describe the evidence in supportof it, which is by now substantial. However a conceptual explanation for Mackey'srigidity phenomenon remains elusive.