Entire Functions on Lie Groups

Every Lie group $G$ carries a distinguished algebra of particularly well-behaved real-analytic mappings: The entire functions $\mathcal{E}(G)$. They were introduced for the purposes of strict deformation quantization. This paper establishes a one-to-one correspondence between entire functions and holomorphic mappings $\mathcal{H}(G_\mathbb{C})$ on the universal complexification $G_\mathbb{C}$ of $G$ as Fr\'{e}chet algebras. Methodically, this is achieved by porting aspects of classical complex analysis into a left-invariant guise and by studying the geometry of $G_\mathbb{C}$. As a byproduct, we obtain a strict deformation quantization of the holomorphic cotangent bundle of any universal complexification.