# Representations and support.

I am interested in the question: Does there are exist concept of support in representation theory?

When I say support I mean number of non-zero values of $f \in C[G]$. Do you know theorems which talks about the action of elements of $C[G]$ with small support in different representations?

The only example I know about is uncertainty principle which says that for abelian group $supp(f)supp(\hat{f})\geq |A|$.

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I think the question would benefit from being slightly more precise. It seems better to me if one asks for "theorems that show something like X happens" rather than "theorems which use the concept X". –  Yemon Choi Feb 28 '11 at 7:20

## 1 Answer

Although perhaps the question was directed more at finite groups: for reductive or semi-simple real Lie groups, (serious) results of Harish-Chandra (starting in the early 1950s) show that among regular semi-simple elements of the group, the supports of characters of principal series are the smallest, being confined to (maximally) split conjugacy classes, while at the opposite end the characters of discrete series have non-trivial support on all semi-simple conjugacy classes. The in-between repns, i.e., induced from discrete series on Levis, have in-between supports, in terms of split-ness.

I have the impression that suitable analogues hold for p-adic reductive groups, tho' the corresponding results are relatively much-newer (by a displacement of 15-20 years?), not to mention the complication in description of discrete series (supercuspidal repns) as induced from compact-open subgroups (Kutzko-Bushnell-et-alia). Early computations for the p-adic case go back to MacDonald (SL2) in the 1950s, and Shalika, Sally in the 1960s (SL2), Jacquet in the late 1960s, so far as I know.

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Thanks for your answer. Can you, please, give me a reference where I can read about this results? –  Klim Efremenko Jul 20 '11 at 7:17
Knapp's "Repn theory of semi-simple groups" is an extensive introduction, with huge biblio and historical notes, for the Lie case. AMS Proc Symp Pure Math 61, from Durham Conference. On-line, among many others, papers at the atlas-of-Lie groups site, liegroups.org/papers. Also the Langlands archive at IAS, the Arthur archive at Clay, and Casselman's archive at UBC. The p-adic business is currently very active, so googling will find many "live" things. –  paul garrett Jul 20 '11 at 14:05