Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

Question

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), 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). 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?

Answer 1

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 $SL_2(\mathbb{R})$.

Answer 2

You want to look at the paper: Atiyah, Michael; Schmid, Wilfried A geometric construction of the discrete series for semisimple Lie groups. Invent. Math. 42 (1977), 1–62.

Answer 3

Check an old paper of Joseph Wolf in Journal of Mechanics ... early sixties , there you find a detailed answer to your question. I do not have access to Mathscinet for a better answer. Best regards Jorge