  In this paper, we exhibit the non-formal star-exponential of the Lie group
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) into the von Neumann algebra of multipliers of the
Hilbert algebra underlied by this natural star-product. As an application, we
prove a new identity on Bessel functions.
