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?



Why limit of discrete series representation?
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.


