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?

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  $$\f\_ n ^2 = n\int_{D}f(z)^2(1z^{2})^{(n1)}dxdy$$ 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 $\f\ _ r ^{2}=r\int_{D}f(z)^2(1z^{2})^{(r1)}dxdy$. 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. 


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 "limittaking 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 offhand, but probably HarishChandra and others used this term in the 1950s, also applied to more general (reductive and semisimple) Lie groups. 

