MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
    
This way you get only "half of" the discrete series of $SL_2(\mathbb{R})$, namely the holomorphic discrete series, corresponding to characters $z\mapsto z^n$ of $K$ with $n>0$. You miss the anti-holomorphic discrete series, corresponding to $n<0$. – Alain Valette Jan 24 at 22:58
up vote 3 down vote accepted

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 $SL_2(\mathbb{R})$.

share|cite|improve this answer

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.

share|cite|improve this answer

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

share|cite|improve this answer
    
Thanks for your answer. Here is a list of his papers on his homepage: https://math.berkeley.edu/~jawolf/publications.html Is the paper you are referring to on that list? – Didier Collard Jan 30 at 12:42
    

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.