Convergent Star Products on Cotangent Bundles of Lie Groups

For a connected real Lie group $G$ we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of $G$. This star product trivially converges on polynomial functions on $T^*G$ thanks to its homogeneity. We define a nuclear Fr\'echet algebra of certain analytic functions on $T^*G$, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter $\hbar$. This nuclear Fr\'echet algebra is realized as the completed (projective) tensor product of a nuclear Fr\'echet algebra of entire functions on $G$ with an appropriate nuclear Fr\'echet algebra of functions on $\mathfrak{g}^*$. The passage to the Weyl-ordered star product, i.e. the Gutt star product on $T^*G$, is shown to be preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on $\hbar$.