| Résumé: | It is a classical question in abstract harmonic analysis to determine which locally compact groups are type I. Many well known classes of groups satisfy the type I property such as semisimple Lie groups and reductive algebraic groups over local fields. Recent research in the unitary representation theory of totally disconnected locally compact (tdlc) groups is interested in determining which automorphism groups of regular trees are type I. The focus so far has largely been on non-amenable automorphism groups of trees. I will talk about recent work that studies the unitary representation theory of scale groups. Scale groups are a class of amenable groups that arise from any non-uniscalar tdlc group and they naturally act on a regular tree as a fixator of a boundary point. I provide (non-)type I results for scale groups that are further assumed to be contractive. The results naturally contrast the results from the non-amenable setting. The arguments require heavy use of the structure theory of locally compact contraction groups and Mackey’s little subgroup analysis. I will give an introduction to all of these topics during the talk. |
| Résumé: | Scale theory, largely developed by George Willis and his collaborators, plays a major role in the theory of totally disconnected, locally compact groups. It is a powerful set of tools for analysing the dynamics of conjugation by a single element, or more generally by a flat subgroup, where 'flat' can be thought of as meaning 'approximately abelian'. Scale theory methods are quite general but also abstract, so it would be useful to find geometric interpretations of scale theory concepts when the group acts (in a nice way) on a (nice) metric space. In particular, since scale theory is vacuous for discrete groups, this ties in with geometric properties of actions that can only occur when the group is nondiscrete. I will present some work in progress, principally on the case where G acts properly and semisimply on a complete CAT(0) space. |
| Résumé: | We say that a group has Kazhdan’s property (T) if the trivial representation is isolated in its unitary dual. If a given group has this property, it is interesting to search for representations which are, in some sense, the closest to the trivial representation. In this talk, I will explain how to identify such a representation among boundary representations of groups acting on $\widetilde{A}_2$ buildings. |
| Résumé: |
Say we are given only the $R$-algebra structure of a group ring $RG$ of a finite group $G$ over a commutative ring $R$. Can we then find the isomorphism type of $G$ as a group? This so-called Isomorphism Problem has obvious negative answers, considering e.g. abelian groups over the complex numbers, but more specific formulations have led to many deep results and beautiful mathematics. The last classical open formulation was the so-called Modular Isomorphism Problem: Does the isomorphism type of $kG$ as a ring determine the isomorphism type of $G$ as a group, if $G$ is a $p$-group and $k$ a field of characteristic $p$?
Starting with an overview on the state of knowledge on general Isomorphism Problems and the modular one in particular, I will present a negative solution found in 2021 with D. García-Lucas and Á. del Río as well as a generalization obtained recently with T. Sakurai, but also positive structural results and several problems remaining open. |
| Résumé: | Let $\mathfrak{l}$ be a real form of a complex semisimple Lie algebra. If $\mathfrak{l}$ is of compact type, the usual (Drinfeld-Jimbo) quantum universal enveloping algebra $\mathcal{U}_q(\mathfrak{l})$ of $\mathfrak{l}$ has an associated quantum group $C^{\ast}$-algebra $C^{\ast}_q(L)$, and there is a nice interplay between the representation theories of these two objects. By contrast, when $\mathfrak{l}$ is non-compact, a path to such an associated object is so far unclear, especially from the representation theoretical viewpoint. In the first part of the talk, we sketch the general setting in the compact case. Subsequently, we use the example of $\mathfrak{sl}(2,\mathbb{R})$ to suggest a different way of quantising the universal enveloping algebra $\mathcal{U}(\mathfrak{sl}(2,\mathbb{R}))$, making the integration problem more approachable. We conclude by comparing the representations of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ to their classical analogues. |