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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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  • $\begingroup$ Your first link doesn't work for me:( $\endgroup$
    – Marc Palm
    Commented Apr 17, 2013 at 12:08
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    $\begingroup$ @Marc: Yes there is something weird with the link, that I couldn't fix. I put the reference instead. $\endgroup$
    – Alain Valette
    Commented Apr 18, 2013 at 11:04
  • $\begingroup$ archive.numdam.org/ARCHIVE/AIF/AIF_1957__7_/AIF_1957__7__315_0/… $\endgroup$
    – Marc Palm
    Commented Apr 18, 2013 at 11:08
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    $\begingroup$ @Alain: Note that Joseph Bernstein's name has an extra "n", though there is also a mathematician named Berstein. (To add to the name confusion, early English translations of Russian papers co-authored by Joseph Bernstein gave his initials as I.N.) $\endgroup$
    – Jim Humphreys
    Commented May 16, 2015 at 14:07
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    $\begingroup$ 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 $\endgroup$
    – Alain Valette
    Commented May 16, 2015 at 17:11

2 Answers 2

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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
    Commented May 16, 2015 at 9:19
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    $\begingroup$ @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
    Commented Jun 11, 2015 at 16:15
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Theorem 1.2.3 (p18) of

MOHAMED HACHMI SLIMAN Théorie de Mackey pour les groupes adéliques Astérisque, tome 115 (1984) Link at SMF site

asserts that the $F$-points of linear algebraic groups over characteristic zero local fields $F$ are of type 1. The author states the result in a slightly more general setting that allows for some covering groups.

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    $\begingroup$ This is far from holding in such a setting. In this theorem there are many assumptions on the field and on the algebraic group which you're omitting. $\endgroup$
    – YCor
    Commented Jan 17, 2021 at 12:12
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    $\begingroup$ I am confused by this. In the original question the field is local so I didn't say that explicitly before. I don't see how you can't recover the statement I gave above by taking $G=G(F)$, $F=1$, and $\underline{\underline{G}}=G$ in the notation of the reference. $\endgroup$
    – jaycegetz
    Commented Jan 18, 2021 at 23:08
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    $\begingroup$ Thanks for clarifying. Indeed I had the impression that more hypotheses were missing, but it seems indeed to hold in this setting (and "local field" in this book means "nondiscrete locally compact field"). $\endgroup$
    – YCor
    Commented Jan 18, 2021 at 23:16

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