Exponentials of certain completions of the unitary form of a Kac-Moody algebra
Let \({\mathfrak g}_ R\) be a real Kac-Moody algebra corresponding to a symmetrizable generalized Cartan matrix, \({\mathfrak g}\) be its complexification. Let \({\mathfrak k}\) be the unitary form of \({\mathfrak g}\) given by the author previously [Proc. Japan Acad., Ser. A 62, 392-395 (1986; Zbl 0617.22012)]. In Proc. Japan Acad., Ser. A 63, 225-228 (1987; Zbl 0625.17011) the author defined the spaces \(H_ m(\pi)\) of the \(C^ m\)-vectors \((m=0,1,2,...,\infty,\omega)\) for two kinds of representations (\(\pi\),V) of \({\mathfrak g}\), namely the adjoint representation (ad,\({\mathfrak g})\) and irreducible highest weight representation \((\pi_{\Lambda},L(\Lambda))\). Then he showed that the action of the analytic completion \({\mathfrak k}_{\omega}\) of \({\mathfrak k}\) can be exponentiated to exp \(\pi\) (x) (x\(\in {\mathfrak k}_{\omega})\), which leaves each \(H_ m(\pi)\) invariant. In this paper the author extends the above argument to a more precise form. Let \({\mathfrak g}_ m=H_ m(ad)\) and \({\mathfrak k}_ m\) be the closure of \({\mathfrak k}\) in \({\mathfrak g}_ m\). Then, by the generating theorem of semigroups, the action \(\pi\) (x) of \({\mathfrak k}_{m+2}\) can be exponentiated to an endomorphism exp \(\pi\) (x) on \(H_ m(\pi)\). Some properties of the exponential map are also discussed. All theorems and propositions are given without proofs.
scientific article; zbMATH DE number 4075351