Differentiable vectors and analytic vectors in completions of certain representation spaces of a Kac-Moody algebra (Q5893761)

Let \({\mathfrak g}_ k\) be a Kac-Moody algebra over \(k={\mathbb{R}}\) or \({\mathbb{C}}\), and \({\mathfrak h}_ k\) the Cartan subalgebra of \({\mathfrak g}_ k\). Let \(\Lambda\) be a dominant integral element in \({\mathfrak h}^*_{{\mathbb{R}}}\), and L(\(\Lambda)\) the irreducible \({\mathfrak g}_{{\mathbb{C}}}\)-module with highest weight \(\Lambda\). Denote by H(ad) and H(\(\Lambda)\) the completions of \({\mathfrak g}_{{\mathbb{C}}}\) and L(\(\Lambda)\) with respect to the standard inner products ( \(| )_ 1\) and ( \(| )_{\Lambda}\), respectively. In this note the author defines the spaces of \(C^ m\)-vectors \((m=0,1,2,...,\infty,\omega)\) in H(ad) and H(\(\Lambda)\), and gives a simple characterization of them. Further he shows some properties of the exponential map.