# Which p-adic algebraic groups are type I?

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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Your first link doesn't work for me:( –  Marc Palm Apr 17 '13 at 12:08
@Marc: Yes there is something weird with the link, that I couldn't fix. I put the reference instead. –  Alain Valette Apr 18 '13 at 11:04
Or for those not able to speak french: ams.org/mathscinet/search/… –  Marc Palm Apr 18 '13 at 11:12
Thanks to David for reviving my question of 2 years ago. My reason for asking was a computation (joint with Henrik Petersen) of $L^2$-Betti numbers for locally compact groups, valid under a type I assumption; see arxiv.org/pdf/1307.0379.pdf –  Alain Valette May 16 at 17:11

## 1 Answer

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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Thanks David, welcome to MO! –  Alain Valette May 16 at 8:08
The first question would be whether unipotent $p$-adic groups are all of type I. –  YCor May 16 at 9:19
@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 –  m07kl Jun 11 at 16:15