Holomorphic representations of nilpotent Lie groups

Let G be a connected, simply-connected complex nilpotent Lie group, and G_r ?( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of G_R to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of G_R, we study the convex cone of entire functions on G whose restrictions to G_R are positive-definite, and determine the minimal order of growth at infinity of such functions.