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

Salem Ben Said (1)

1 IECN - Institut Élie Cartan de Nancy

The 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.

Domaines: Analyse fonctionnelle [math.FA]

Salem Ben Said. On the integrability of a representation of sl(2, R). 2006. ⟨hal-00095330⟩