# Global square integrability ensures local sq. integrability?

This might be a stupid question for expert in this area.

I am considering automorphic representation of algebraic group.

In studying it, local tempered, local square integrable representations occurs in the P-adic group case. So, I am wondering that what global automorphic representation gives rise to such representation at some finite set of places?

Does globally sq. int. auto. represenation give local sq. int reps at S, some finite set of places of F?

If not, would you tell me what kind of global automorphic representation would be locally sq. int for finite set of places?

One more notation question.

What is the meaning of tempered auto. reps. in global situation? For I learned its definition only in the local case, I suppose it should be locally tempered at finite ramified places. Is it ture?

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I'm voting to close as "not a real question", hoping that the OP will clarify the context. –  Andreas Blass Jan 4 '13 at 14:34
I think this is about as false as it's possible to be, isn't it? Any automorphic representation is unramified outside a finite set, and aren't unramified (local) representations essentially never square integrable? –  user30035 Jan 4 '13 at 22:22
Andreas Blass: this is a question about representation theory of adelic algebraic groups, but I agree that it would have been much better if the OP had said this in any way other than by adding tags... –  user30035 Jan 4 '13 at 22:23
Thanks wccanard. Then, is it true for ramified places? –  Jude Jan 5 '13 at 3:59
A) At ramified places the local representation could be a ramified principal series, which will still not be square integrable, and it is, in general, difficult to say when this does or does not occur. B) A tempered automorphic representation is indeed one that is locally tempered everywhere. –  B R Jan 5 '13 at 17:54

The naive form of Ramanujan-Peterssen-Selberg conjectures is that all the local repns attached to automorphic repns are tempered. This is not quite true, at least because of liftings, as observed by Roger Howe and I.I. Piatetski-Shapiro in the 1970s (maybe in the Corvallis proceedings). More recent conjectures of J. Arthur seem to offer a hypothetical accounting of such things. Clozel has written some surveys of consequences of Arthur's conjectures.

Square-integrable irreducibles of reductive Lie or p-adic groups are the exception rather than the rule. E.g., unitary principal series are never square-integrable. In an automorphic repn, almost all local repns are spherical, which generically ought to be thought of as unitary principal series, hence, certainly not square-integrable. At finite places, the Borel-Casselman-Matsumoto theorem that spherical repns are subrepns of unramified principal series makes this more precise... and there are few subquotients of unramified principal series are square-integrable, as is directly visible for small groups. For larger groups I do not know the classification, although Tadic, Moeglin and others have done a lot of work on square-integrable repns in recent years.

So, it seems a reasonable bet that only finitely-many local repns in an automorphic repn can be square-integrable, and that complete arguments exist, although I cannot give one.

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Thanks Paul. I have benefited from you this time too. I learned much from your reply. –  Jude Jan 6 '13 at 3:17
@anonymous29: you're welcome! :) –  paul garrett Jan 6 '13 at 3:47