I’m studying the paper of [(Baum-Connes-Higson, ex 4.25)](http://www.mmas.univ-metz.fr/~gnc/bibliographie/BaumConnes/Baum-Connes-Higson.pdf), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R).


The idea is that each non-trivial representation of the compact circle group $K$ induces a discrete series representation. I have found a description of the discrete series in the book of Knapp:


$$
\{ f: \text{analytic for Im}~z > 0 ~:~ \| f \|^2 = \int |f(z)^2| y^{n-2} dx dy < \infty \}.
$$
with action
$$
\begin{pmatrix} a & b \\ c & d \end{pmatrix} f(z) = \left(-bz + d\right)^{-n} f \left( \frac{az -c}{-bz +d } \right).
$$


I would like to show explicit how to realize a discrete series representation as the kernel of a Dirac operator, as it discussed in [(Atiyah-Schmid)](http://dx.doi.org/10.1007/BF01389783). I think that the idea is that the kernel of the Dirac operator ensures that $f$ is analytic, and that the $y^{n-2}$ comes from the $G$-invariant metric on the vector bundle $S \otimes V$ over $G/K$, where $S$ is the spinor bundle and $V$ an irreducible representation of $K$.


I don’t know how to compute the metric on the twisted vector bundle $S \otimes V$, and how to show that the sections of this vector bundle give a discrete series representation.


Is there any reference where they show how to construct the discrete series of SL(2,R) as the kernel of twisted Dirac operators?



Construct discrete series of SL(2,R) as kernel of twisted Dirac operators
For $G=Spin(2n,1)$ (the double cover of $SO(2n,1)$) and $K=Spin(2n)$, the fact that Dirac induction $R(K)\rightarrow K\_0(C^\*\_r(G))$ is an isomorphism, is checked by hand, explicitly, in section 3 of my old paper "K-theory for the reduced C\*-algebra of a semi-simple Lie group with real rank 1 and finite centre", Quart. J. Math. Oxford 35 (1984), 341-359. Observe that $Spin(2,1)$ is isomorphic to $SL\_2(\mathbb{R})$.


