Some remarks on quasi-invariant actions of loop groups and the group of diffeomorphisms of the circle
Let \(\text{Diff}^\infty\) denote the group of \(C^\infty\)-smooth orientation preserving diffeomorphisms of the circle \(S^1\), and \(\text{Diff}^1\) the group of orientation preserving diffeomorphisms of class \(C^1\). Let \(K\) be a compact Lie group. Denote by \(L^\infty (K)\) the group of \(C^\infty\)-smooth functions \(S^1 \to K\), and by \(L^0 (K)\) the group of functions of class \(C^0\). The paper is devoted to constructing the following dynamical systems: -- the series of \(\text{Diff}^\infty\)-quasi-invariant measures on the space \(\text{Diff}^1/T\), where \(T\) is the group of rotations of \(S^1\); -- the series of \(\text{Diff}^\infty\)-quasi-invariant measures on \(\text{Diff}^1\); -- \(\text{Diff}^\infty\)-quasi-invariant measures on the space of Cantor subsets of the circle; -- \(L^\infty (SO(n))\)-quasi-invariant measures on \(L^0 (SO (n))\); -- \(L^\infty (SO(n))\)-quasi-invariant measures on spaces of maps \(S^1 \to SO(n)/H\). The author's approach allows for a unification of the representation theories for the groups \(\text{Diff}^\infty\) and \(L^\infty (K)\) developed in the following works: [\textit{S. Albeverio}, \textit{R. Hoegh-Krohn}, \textit{D. Testard} and \textit{A. M. Vershik}, J. Funct. Anal. 51, 115-131 (1983; Zbl 0522.22013)], [\textit{R. S. Ismagilov}, Funct. Anal. Appl. 5, 209-216 (1971; Zbl 0248.22023)], [\textit{V. G. Kac}, Infinite dimensional Lie algebras, Boston-Basel-Stuttgart (1983; Zbl 0537.17001)], [\textit{A. A. Kirillov}, Sel. Math. Sov. 1, 351-372 (1981; Zbl 0515.58009)], [\textit{Yu. A. Neretin}, Russ. Math. Surv. 37, 229-230 (1983; Zbl 0533.22013)].
scientific article; zbMATH DE number 851931