In his book on $\mathrm{SL}(2,\mathbb{R})$, Lang shows that any nontrivial irreducible unitary representation of this group is infinitesimally isomorphic to an irreducible admissible subrepresentation of an induced representation (Theorem 8, p. 123). This implies, regarding the classification of irreducible unitary representations of $\mathrm{SL}(2,\mathbb{R})$, that the so-called principal and mock discrete series exist since the related induced representations are unitary. However, this is not the case with discrete and complementary series. Is there an easy way to show the existence of those two types of representations? Lang seems to suggest so, at least for the complementary series, as he mentions the possibility of a unitarization by completing the space of $K$-finite vectors with respect to a certain scalar product (p. 123). I do not understand how this works. (One gets a different space after completion, thereby losing the original action of the group. What is the new action then?)
To rephrase: given that there exists a (nonunitary) irreducible admissible representation of $\mathrm{SL}(2,\mathbb{R})$ in a certain infinitesimal equivalence class like, say, discrete series of lowest weight $2$, can one find in a more or less straightforward way an irreducible unitary representation belonging to the same class?
Best Answer
For spherical principal series $I_s$ (non-normalized) induced from $\pmatrix{a & * \cr 0 & a^{-1}} \rightarrow |a|^{2s}$, the dual is $I_{1-s}$, with pairing given by integration over $K$. This is isomorphic to $I_{\bar{s}}$ if and only if $\bar{s}$ and $1-s$ are sent to each other by the Weyl group's action on these parameters. For $\Re(s)=1/2$, $\bar{s}=1-s$. For $s\in\mathbb R$ the "long" Weyl element has to be applied. For $s$ outside the interval $[0,1]$ the group-invariant pairing so-obtained fails to be positive-definite.
For holomorphic discrete series, it's probably simplest just to make a different model.
Certainly it's not possible to convert arbitrary not-unitary representations to unitary ones.