How does one show the existence of discrete and complementary series for SL(2,R)?

Question

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?

Answer 1

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.

Answer 2

Fix a complex number $u$ with positive real part. Denote by $\pi _u$ the space of $K$ finite functions on $G=SL_2({\mathbb R})$ which under left action by the group $B$ of upper triangular matrices $p= \begin{pmatrix}a & b \cr 0 & a^{-1}\end{pmatrix}$ transform as $f(pg)=\mid a \mid a^{1+s}f(g)$ for all $g\in G$. One can define an integral (the $G$-intertwining operator) from $\pi _u$ into $\pi _{-u}$ by sending $f$ into the function $I(f)(g)$ defined by $$ I(f)(g)=\int _{\mathbb R} dx \quad f(w_0\begin{pmatrix} 1 & x \cr 0 & 1\end{pmatrix}g) .$$ Now, as Garrett says, $\pi _u$ and $\pi _{-u}$ are dual to each other by integrating the product over $K$. Hence, combining the intertwiner with the duality we have a $G$-invariant pairing from $\pi _u \otimes \pi _u$ into $\mathbb C$. Now, if $u$ is taken to be a real number between $0$ and $1$, then this pairing defines an inner product and the resulting completion is the complementary series for $SL_2({\mathbb R})$.

Answer 3

I found Traces of Hecke operators by Knightly and Li very readable. They treat Gl(2,R) by a similar method. Knapp or Wallach is also nice to read and more general. They have chapters for Sl(2,R) and Sl(2,C), Bumps Automorphic forms as well but is closer to Knightly-Li.

The main ingredient are the intertwiner. At their poles and zeros you find the discrete series inside non-unitarizable representations.

Ruling out unitarizability is often achieved by studying the growth of the matrix coefficients.

Answer 4

For complementary series, I'd recommend §V.4 of Sugiura (available on the internet) for a very careful description of their embedding in nonunitary principal series. In particular, Prop. 4.6 explains how the original group action extends to the completed space, as you asked.

For discrete series I'd recommend §4 of Sally, where "a study of the intertwining operators relating two unitarily equivalent members of the principal series (...) leads naturally to global realizations of all the irreducible unitary representations of SL(2,R), in particular the discrete series".