Non-formal star-exponential on contracted one-sheeted hyperboloids

In this paper, we exhibit the non-formal star-exponential of the Lie group SL(2,R)SL(2,R) realized geometrically on the curvature contraction of its one-sheeted hyperboloid orbits endowed with its natural non-formal star-product. It is done by a direct resolution of the defining equation of the star-exponential and produces an expression with Bessel functions. This yields a continuous group homomorphism from SL(2,R)SL(2,R) into the von Neumann algebra of multipliers of the Hilbert algebra associated to this natural star-product. As an application, we prove a new identity on Bessel functions.