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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