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Seminar on Arithmetic Algebraic Geometry
May 22, 2007 11:30, Moscow, Steklov Mathematical Institute, Room 540 (8 Gubkina)

Orbit method for totally disconnected groups

M. N. Sabitova

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Abstract: The classical orbit method due to A. A. Kirillov allows one to classify unitary irreducible representations of connected and simply connected nilpotent Lie groups in terms of the coadjoint orbits in their Lie algebras. The goal of this talk is to explain how the orbit method can be extended to certain totally disconnected groups for which an analogue of the Lie algebra can be defined. We will formulate a more general method, the so-called an abstract orbit method, for these groups. We will show how it can be used to prove a suitable analogue of Kirillov's orbit method for $p$-groups of nilpotence class $p$, as well as for uniform pro-$p$-groups. If time remains, we will present an application of this theory to a $p$-adic version of J. Brown's theorem, stating that the orbit method bijection between the unitary dual of a nilpotent $p$-adic Lie group and the space of coadjoint orbits in its Lie algebra is a homeomorphism with respect to the Fell topology on the former space and the natural quotient topology on the latter space. (We will try to make the talk accesible to people with different backgrounds, so all the necessary definitions will be provided.)

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