Reduction of cocycles and groups of diffeomorphisms of the circle

The subgroup $PSL(2,R)$ is spherical in the group of diffeomorphisms of the circle

We show that the group $PSL(2,R)$ is a spherical subgroup in the group of $C^3$-diffeomorphisms of the circle. Also, the group of automorphisms of a Bruhat--Tits tree is a spherical subgroup in the group of hierarchomorphisms of the tree.

The group of diffeomorphisms of the circle: reproducing kernels and analogs of spherical functions

The group $Diff$ of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups $U(p,q)$, $Sp(2n,R)$, $SO^*(2n)$; the space $Ξ$ of univalent functions is an analog of the corresponding classical complex Cartan domains. We present explicit formulas for realizations of highest weight representations of $Diff$ in the space of holomorphic functionals on $Ξ$, reproducing kernels on $Ξ$ determining inner products, and expressions ('canonical cocycles') replacing spherical functions.

Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective unitary representation of Diff(S^1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms D^s(S^1) for any real s>3, and in particular to C^k-diffeomorphisms Diff^k(S^1) with k>=4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S^1 is covariant with respect to D^s(S^1), s > 3. Moreover every direct sum of irreducible representations of…

Actions of semisimple Lie groups on circle bundles

Suppose G is a connected, simple, real Lie group with real rank at least two, M is an ergodic G-space with invariant probability measure, and f is a Homeo(T)-valued Borel cocycle, where Homeo(T) denotes the group of homeomorphisms of the circle T. We use an argument of E.Ghys to show that there is a G-invariant probability measure on the skew product of M and T. Furthermore, if the image of f consists of diffeomorphisms, then there is an invariant measure that is equivalent to the product measure; therefore, f is cohomologous to a cocycle with values in the isometry group of T.

圆圈的解析微分同胚与圆纤维的拓扑Riemann-Roch定理

Analytic diffeomorphisms of the circle and topological Riemann-Roch theorem for circle fibrations

本文通过构造圆圈的解析微分同胚群的中心扩张，推导出新的拓扑Riemann-Roch定理，用于复线丛在圆纤维化上的应用。

带有断裂点的微分同胚的维拉索罗扩展

Virasoro extensions for diffeomorphisms with breaks

研究带有有限个断裂点的圆圈微分同胚，构建维拉索罗群的扩展，并描述相关的断裂维拉索罗李括约束。