Holomorphic functions of exponential type on connected complex Lie groups

For each connected complex Lie group $G$, we explicitly describe the algebra of holomorphic functions of exponential type $\mathcal{O}_{exp}(G)$ introduced by Akbarov. The key point is a decomposition of the maximal length function into three summands, which yields a decomposition of $\mathcal{O}_{exp}(G)$ into the projective tensor products of three factors. We also essentially use the exponential radical of $G$. As a corollary, we show that the polynomial functions on $G$ are contained in $\mathcal{O}_{exp}(G)$ if and only if $G$ is linear.
This is a prequel of our talk of 28.09.2018.