The orbit method for profinite groups and a p-adic analogue of Brown’s theorem |
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Authors: | Mitya Boyarchenko Maria Sabitova |
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Institution: | (1) Department of Mathematics, University of Chicago, Chicago, IL 60637, USA;(2) Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA |
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Abstract: | We develop an approach to the character theory of certain classes of finite and profinite groups based on the construction
of a Lie algebra associated to such a group, but without making use of the notion of a polarization which is central to the
classical orbit method. Instead, Kirillov’s character formula becomes the fundamental object of study. Our results are then
used to produce an alternate proof of the orbit method classification of complex irreducible representations of p-groups of nilpotence class < p, where p is a prime, and of continuous complex irreducible representations of uniformly powerful pro-p-groups (with a certain modification for p = 2). As a main application, we give a quick and transparent proof of the p-adic analogue of Brown’s theorem, stating that for a nilpotent Lie group over ℚp the Fell topology on the set of isomorphism classes of its irreducible representations coincides with the quotient topology
on the set of its coadjoint orbits.
The research of M. B. was partially supported by NSF grant DMS-0401164. |
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