# Bernstein–Zelevinsky classification for classical groups

Bernstein and Zelevinsky classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations. The irreducible modules are parametrized by multi-segments.

Are there some references about parametrizations of irreducible representations of classical groups (types A, B, C, D) using something like multi-segments? Thank you very much.

• I may be wrong (I'm not an expert in this area), but I think there are no general results for classical groups of arbitrary rank other than GL(n), just specific small-rank groups. E.g. for Sp(4) and GSp(4) there is a classfication by Sally and Tadic (Mem. Soc. Math France, 1993). Searching MathSciNet also brings up a 2001 paper by Konno covering U(2, 2). – David Loeffler Oct 15 '16 at 20:14

You have such a classification for discrete series representations of classical groups by Moeglin and Moeglin-Tadic (and probably for tempered representations by Chris Jantzen) but it is not only using multisegment.

• Could you give more precise references or even links? – András Bátkai Oct 16 '16 at 20:39
• Colette Moeglin, Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité. J. Eur. Math. Soc. (JEMS) 4 (2002), no. 2, 143–200. Colette Moeglin, Marko Tadić, Construction of discrete series for classical p-adic groups. J. Amer. Math. Soc. 15 (2002), no. 3, 715–786. Chris Jantzen, Tempered representations for classical p-adic groups. (English summary) Manuscripta Math. 145 (2014), no. 3-4, 319–387. – kolmo Oct 16 '16 at 20:48
• Marko Tadic also did that for tempered representations in : On tempered and square integrable representations of classical p-adic groups. (English summary) Sci. China Math. 56 (2013), no. 11, 2273–2313. – kolmo Oct 16 '16 at 20:58