UB07 · Random Walks on Groups and Representation Theory
FP7 — People (Marie Curie Actions)
- Duration
- 2008-03-01 → 2012-02-29
- EU contribution
- €100,000
- Participants
- 1
- Scheme
- MC-IRG
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Results in brief
Random Walks on Groups and Representation Theory
The project enables us to develop a surprising link between random walk theory on groups and representation theory. Whereas this is a new theory and many questions and basic problems still remain a mystery, we were able to give a set of axioms to boundary theory and to associate with every group a pair of boundaries, and in particular a (smaller) group of symmetries, to be called 'the Weyl group' with representation theory which controls the representation theory of the ambient group. This is with an obvious similarity to the classical semi-simple theory. Our project resulted in publishing various professional papers, as listed below.
Data: CORDIS, © European Union
Project 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""
Original text from CORDIS.
Participants
- TECHNION - ISRAEL INSTITUTE OF TECHNOLOGY · HaifaCoordinatorIsrael
Links
Data: CORDIS, © European Union