UB07

Random Walks on Groups and Representation Theory

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 Obiettivo del progetto (Objective)

'Locally compact groups have many unitary representations. For example, the regular representation, or more generally, the quasi-regular representations arising as function spaces over group actions. These representations can be written as direct integrals of irreducible ones. The space of all irreducible unitary representations is the 'unitary dual' of the group. It exists, and has a Borel structure and a funny topology that can be described abstractly, but the truth is, that for a general group, this space is a complete mystery. In fact, we don't know even a single non-trivial representation! The main objective of the proposed research is to study the unitary dual of a locally compact groups. We intend to do this by breaking it up into pieces, and study those pieces and the mutual relations between them. The 'pieces' that we propose to study are 'Generalized Principal Series' associated with Poisson Boundaries which stem from Random Walks on the group. The theory of Random Walks provides us spaces endowed with very strong ergodic properties - the Poisson boundaries of the random walks, and their factors. We conjecture that the associated quasi-regular representations are irreducible. One can use these spaces to construct not just one representation, but a series of such. These series could be called 'generalized principal series', as they form a generalization of the principal series arise in the representation theory of semi-simple groups. We propose certain character formula that, we believe, serves as an invariant associating the random walk with the corresponding representations. Aiming towards our goal, we propose various intermediate conjectures that could be studied independently, along with related subprojects concerning the study of the asymptotics of random walks, the ergodic properties of the boundary actions, the structure of the lattice of factors of the Poisson Boundary, and related topics'

Introduzione (Teaser)

In mathematics, a locally compact group G is a topological group. Its major characteristic is that spaces are preserved even under conditions of deformation such as stretching.

Descrizione progetto (Article)

A topological group is a mathematical object with an algebraic structure that allows for algebraic operations in the study of continuous symmetries, convergence and connectedness. Locally compact groups have many unitary representations where regular or quasi-regular representations come about as spaces of function rather than being affected by group action. Written as direct integrals of irreducible representations, the space of these representations is the unitary dual of the group. However, this space remains a complete mystery.

The 'Random walks on groups and representation theory' (UB07) project is investigating the unitary dual of locally compact groups by breaking them up into large 'pieces' to study the mutual relations among them. Specifically, the EU-funded project is relying on random walk theory to study those pieces termed generalised principal series. Random walks provides spaces with certain properties that allow for observation of the behaviours of related processes. Enabling this effort, researchers have proposed a certain formula that will link the random walk with corresponding representations.

Team members have to date succeeded in proving their hypotheses. The most important of these was their conjecture that the generalised principle series representations are irreducible. Their successful proofs relate to cases in which the particular group is the fundamental group of a negatively curved topological space or manifold.

In ongoing work, project partners are directing efforts to extend already obtained results above to the wider class of hyperbolic groups.