Talk:Tempered representation

Tempered representations

Hello again, and thanks for the tip on how to avoid edit conflicts! I am a little uneasy about your recent edits of tempered representation. The concerns are two-fold: factual and expository. Which sources are you using for that article? I do not have Knapp, Wallach or Borel-Wallach close at hand, but usually tempered representations are defined in terms of the rate of growth of their matrix coefficients, not of their Harish-Chandra character, which is a highly non-trivial generalization of a character of a representation of a finite group. Likewise, in Harish-Chandra's work, the fact that the matrix coefficients of tempered representations belong to L^2+ε(G) comes after a fair amount of work, since a priori it's unclear that this would be a good condition, and in particular, that this property holds for the matrix coefficients of the (spherical) unitary principal series. The converse is also a non-trivial fact, since the 2+ε growth rate is a 'softer' condition than the what you get from an explicit comparison with the $\Xi$-function. Even if someone does define temperedness in terms of the Harish-Chandra character, we should really give the definition in terms of the matrix coefficients. Firstly, it explicitly occurs in most of the literature, and secondly, it does not rely on rather non-trivial theory of the Harish-Chandra character (note, by the way, that it is still a red link). Of course, in Properties it can be mentioned that the three conditions on an irreducible admissible representation of a real reductive Lie group are equivalent: one K-finite matrix coefficient is asymptotically bounded by the $\Xi$-function, the same holds for all K-finite matrix coefficients, and finally, the Harish-Chandra character satisfies the growth estimate, making it a tempered distribution. The same should also be true for semisimple p-adic groups (I can't find it in Harish-Chandra; cf. Silberger's book, which I also don't have).

I do not want to make changes myself, for a couple of reasons. Firstly, you seem to be working rather actively on that article, and also, I do not have any of the standard references and having already made one rather questionable substitution (I've put locally compact group in the beginning, since the L^2+ε condition is meaningful in this generality, although I can't recall if anyone refers to representations of non-reductive groups with this property as 'tempered'), I'd rather not introduce any more hastily made statements. Arcfrk 04:11, 13 May 2007 (UTC)

I put the comment about Harish-Chandra characters in just as a historical note to explain where the word "tempered" came from. I think this was how H-C originally defined them, though I ought to go back and check exactly what he did as I was writing this from memory. As you say, no-one seems to use this definition of tempered representation any more; in fact it is not even mentioned in most books.

I have not seen the L^2+ε condition used for non-reductive groups either. R.e.b. 13:32, 13 May 2007 (UTC)