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It was proved by Jacques Dixmier (Sur les représentations unitaires des groupes de Lie algébriques, Annales de l'institut Fourier, 7 (1957), p. 315-328, doi: 10.5802/aif.73, MR 20 #5820 , Zbl 0080.32101 )

that algebraic groups over the reals, are type I. Is a similar result known for algebraic groups over non-archimedean local fields (possibly of characteristic 0)? I am only aware of the result by Bernstein

http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/bernstein-P-tame-FAN.pdf

that reductive algebraic groups over non-archimedean local fields, are type I.

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Duflo gave a classification of the irreducible unitary representations of any algebraic group over a characteristic zero local field, in terms of the answer in the reductive case (Duflo, Michel Théorie de Mackey pour les groupes de Lie algébriques. (French) [Mackey theory for algebraic Lie groups] Acta Math. 149 (1982), no. 3-4, 153–213.) My guess would be that the type I result in general follows from Bernstein's type I result in the reductive case by Duflo's classification; but I don't know that for certain, and Duflo does not state such a result.

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    $\begingroup$ The first question would be whether unipotent $p$-adic groups are all of type I. $\endgroup$ – YCor May 16 '15 at 9:19
  • $\begingroup$ @YCor: The answer is yes. unipotent p -adic groups are all CCR, in particular type I. See Theorem 4 "Decomposition of Unitary Representations Defined by Discrete Subgroups of Nilpotent Groups" by Calvin C. Moore jstor.org/stable/1970567?seq=1#page_scan_tab_contents $\endgroup$ – m07kl Jun 11 '15 at 16:15

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