In what sense is the limit of discrete series representation of $SL(2, \mathbb{R})$ a limit of discrete series representations? Where does the name origin from?
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1$\begingroup$ ... for what? $\:$ $\endgroup$– user5810Commented Jul 10, 2012 at 7:28
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$\begingroup$ Wikipedia says "Limit of discrete series representations are tempered representations, which means roughly that they only just fail to be discrete series representations." Does that answer the question? (I wouldn't know, just searched on Wikipedia to see what the term means). $\endgroup$– George LowtherCommented Jul 10, 2012 at 12:28
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1$\begingroup$ Dear Mrc, As far as I know, Paul Garret's explanation is correct; limits of discrete series have the same character formulas etc. as discrete series, but one has to allow the parameter to pass to a "limit" inside the wall of a Weyl chamber. Regards, $\endgroup$– EmertonCommented Jul 10, 2012 at 14:51
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1$\begingroup$ I vaguely recall something about a topology on the space of irreducible unitary representations - this would be similar to the topology on the Pontrjagin dual of a LCA group. Then perhaps the limits of discrete series representations are actually limits in the topological sense? $\endgroup$– MTSCommented Jul 10, 2012 at 17:40
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3$\begingroup$ @MTS, this topology is called the Fell topology, it turns out that this topology is evil (i.e. not Hausdorff). It turns out that in this topology, principal series reps. converge to limits of discrete series as $it\to 0$. There's a famous "picture" of the $SL_{2}$ reps. as a "double cross" in the complex plane which represents the unitary dual, see for example in Lang's SL2(R) book, p.124. $\endgroup$– AsafCommented Jul 10, 2012 at 19:02
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2 Answers
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Here is the explanation I know, just for $SL_2$.
The discrete series rep. have realizations in the Hardy spaces $H_n$ which have the norm - $$\|f\|_ n ^2 = n\int_{D}|f(z)|^2(1-|z|^{2})^{(n-1)}dxdy$$ notice this norm is scaled a bit differently than usual. The limit of discrete series is realized inside $H_2$ with the norm $\|f\| _ 2 ^{2}=\frac{1}{2\pi}\sup_{0\leq r<1} \int_{0}^{1}|f(re^{2\pi it})|^{2}dt$
So from what I know (which probably has nothing to do with rep. theory), one can consider the Hardy spaces with continuous parameter say $r$, with the norm $\|f\| _ r ^{2}=r\int_{D}|f(z)|^2(1-|z|^{2})^{(r-1)}dxdy$. It's not hard to show that for $H_{r}$ you have a complete orthonormal family $f_{n,r}=\[\frac{\Gamma(r+n+1)}{n!\Gamma(r+1)}\]^{1/2}z^n$. Then one can show that $H_{2}=\{f\in \cap_{r>0} H_{r} \mid \lim_{r\to 0}\|f\| _ r ^{2} \text{ exists and finite} \}$.
It might be interesting to try to work it out in different models for the representations.
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1$\begingroup$ This is a nice explanation and exactly in the spirit, I was looking for. $n$ is the highest weight going to one, $n$ real values makes sense representation theoretic, if we consider the universal covering of $\mathrm{SL}_2(\mathbb{R})$ and modular form people rather work with the upperhalfplane model for the Hardy-space. Thank you. $\endgroup$ Commented Jul 10, 2012 at 14:25
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These repns are not actually "discrete series", in that they do not appear in $L^2(G)$. Yet their construction/description is completely parallel to that of the discrete family of repns called "discrete series". Since the relevant parameter is discrete, it is hard to conjure up any "limit-taking process", indeed, in a mathematical sense. But in a colloquial sense, since the parameter (for $SL_2(\mathbb R)$ just the "weight") takes a more extreme value for these repns than for "genuine discrete series", it's not completely unreasonable to refer to them in the form " discrete series".
I couldn't give a citation off-hand, but probably Harish-Chandra and others used this term in the 1950s, also applied to more general (reductive and semi-simple) Lie groups.
answered Jul 10, 2012 at 12:43