Differentiable vectors and analytic vectors in completions of certain representation spaces of a Kac-Moody algebra (Q5916458)

Let \({\mathfrak g}\) be a symmetrizable Kac-Moody Lie algebra over \({\mathbb{C}}\) and let \({\mathfrak k}\subset {\mathfrak g}\) be the standard unitary form. For any dominant integral weight \(\Lambda\), let \(L(\Lambda)\) be the irreducible highest weight (integrable) \({\mathfrak g}\)-module with highest weight \(\Lambda\). Let \(\underline{\mathfrak g}\) (resp. \(\underline L(\Lambda)\)) be the direct product of all the root spaces of \({\mathfrak g}\) including the 0-root space \({\mathfrak h}\) (resp. all the weight spaces of \(L(\Lambda)\)). Then clearly \(\underline{\mathfrak g}\) and \(\underline L(\Lambda)\) have natural \({\mathfrak g}\)-module structures. The author defines (for any \(m\in {\mathbb{Z}}_+=\{0,1,2,...\})\) the space of vectors of class \(C^ m\), \(H_ m(ad)\subset \underline {\mathfrak g}\) (resp. \(H_ m(\Lambda)\subset \underline L(\Lambda))\) by first defining \(H_ 0(ad)\) (resp. \(H_ 0(\Lambda))\) as the completion of \(\underline{\mathfrak g}\subset g\) (resp. \(L(\Lambda)\subset \underline L(\Lambda))\) with respect to the invariant inner product. Then, inductively, set (for any \(m\geq 1)\) \[ H_ m(ad):=\{y\in H_{m-1}(ad):\quad [x,y]\in H_{m- 1}(ad),\quad for\quad any\quad x\in {\mathfrak g}\} \] \[ and\quad H_ m(\Lambda):=\{\nu \in H_{m-1}(\Lambda):\quad x\nu \in H_{m- 1}(\Lambda),\quad for\quad any\quad x\in {\mathfrak g}\}. \] Further he defines the space of vectors of class \(C^{\infty}\) by \(H_{\infty}(ad):=\cap_{m\geq 0}H_ m(ad)\), and \(H_{\infty}(\Lambda):=\cap_{m\geq 0}H_ m(\Lambda)\). He also defines the space of analytic vectors \(H_{\omega}(ad)\subset H_{\infty}(ad)\) and \(H_{\omega}(\Lambda)\subset H_{\infty}(\Lambda).\) As his first main theorem, he proves that it suffices to check the condition in the above definitions of \(H_ m(ad)\) and \(H_ m(\Lambda)\) (for any \(m\in {\mathbb{Z}}_+\cup \{\infty,\omega \})\) for any single ``strictly dominant'' element of the Cartan subalgebra \({\mathfrak h}\), instead of the whole \({\mathfrak g}\). This enables him to define certain topologies on all the above spaces. He further proves that under this topology \(H_{\infty}(ad)\) and \(H_{\omega}(ad)\) become topological Lie algebras denoted respectively by \(g_{\infty}\) and \(g_{\omega}.\) Let \(H^ u_ 1(ad)\) (resp. \({\mathfrak k}_{\omega})\) be the closure of the unitary form \({\mathfrak k}\) in the Hilbert space \(H_ 1(ad)\) (resp. \({\mathfrak g}_{\omega})\). The author shows that the elements of \(H^ u_ 1(ad)\) can be exponentiated as operators on the Hilbert spaces \(H_ 0(ad)\) and \(H_ 0(\Lambda)\). (On \(H_ 0(\Lambda)\) this was considered earlier by Goodman-Wallach). Moreover \({\mathfrak k}_{\omega}\) keeps all the spaces \(H_ m(ad)\) and \(H_ m(\Lambda)\) stable, for any \(m\in {\mathbb{Z}}_+\cup \{\infty,\omega \}.\) In the end the author also investigates the convergence of the Campbell- Hausdorff formula.