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A canonical dimension estimate for non-split semisimple \(p\)-adic Lie groups. (English) Zbl 1397.22013

Summary: We prove that the canonical dimension of an admissible Banach space or a locally analytic representation of an arbitrary semisimple \( p\)-adic Lie group is either zero or at least half the dimension of a non-zero coadjoint orbit. This extends the results of Ardakov, Wadsley, and Schmidt in the split semisimple case.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F85 \(p\)-adic theory, local fields
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16S35 Twisted and skew group rings, crossed products
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20E18 Limits, profinite groups
20F40 Associated Lie structures for groups
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References:

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