On density of polynomials in algebra of holomorphic functions of exponential type on linear Lie group

The question of the density of the algebra of polynomials (regular functions) in the algebra of holomorphic functions of exponential type on a complex Lie group arose in the study of duality for Hopf algebras of holomorphic functions. It was shown by the author in [J . Lie Theory, 29:4, 1045–1070 (2019)] that the answer is affirmative in the connected linear case. However, the argument is quite involved and here we present a short proof. It contains two ingredients. The first is the existences of a finite–dimensional faithful holomorphic representation with closed range. To prove it, we use an approach developed by Ðjoković. The second is a lower bound for the norm of a one–parameter matrix subgroup, which is based on some elementary linear algebra consideration. The rest of the proof is close to the original one and uses a decomposition of the group into a semidirect product of a simply connected solvable and linearly complex reductive factors.