| Résume | Résumé : A locally compact group G is called "type I" if every unitary representation of G generates a type I von Neumann algebra. It is well known that being type I is equivalent to the following statements:
- the unitary dual of G is a standard Borel space when equipped with the Mackey-Borel structure;
- every unitary representation of G uniquely decomposes into a direct integral of irreducible unitary representations.
In the case that G is not type I, and hence the above properties do not hold, it is essentially impossible to classify the (irreducible) unitary representations of G. The question of which groups are type I has been studied extensively in the literature, especially for Lie/algebraic groups over local fields, where the situation is well understood. One exception to this, however, is that it is still unknown whether unipotent groups over positive characteristic local fields are type I. In this talk I will discuss ongoing work regarding determining which contraction groups are type I. Contraction groups are an important class of groups which arise in the theory of totally disconnected locally compact groups, and, for example, include unipotent groups over local fields. In this work, joint with Pierre-Emmanuel Caprace, we prove that many two-step nilpotent unipotent groups over positive characteristic local fields are type I. We also exhibit an uncountable class of two-step nilpotent torsion contraction groups which are not type I. The results rely on an algebraic characterisation of the type I property for two-step nilpotent groups that we produce. |
