Question

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?

Answer 1

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.

Answer 2

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.