On the integrability of a representation of sl(2, R)

Abstract

Soumis à Journal of functional analysisThe Dunkl operators involve a multiplicity function $k$ as parameter. For positive real values of this function, we consider on the Schwartz space $\mathcal S(\R^N)$ a representation $\omega_k$ of \s\l(2,\R) defined in terms of the Dunkl-Laplacian operator. By means of a beautiful theorem due to E. Nelson, we prove that $\omega_k$ exponentiates to a unique unitary representation $\Omega_k$ of the universal covering group \GG of ${SL(2,\R)}.$ Next we show that the Dunkl transform is given by $\Omega_k(g_\circ),$ for an element g_\circ \in \GG. Finally, the representation theory is used to derive a Bochner-type identity for the Dunkl transform