This paper studies the Poisson slices \(X_{\tau}:=\nu^{-1}\left( \mathcal{S}_{\tau}\right) \subset X\) determined by an \({sl}_{2}\)-triple \(\tau=\left( \xi,h,\eta\right) \in\mathfrak{g}^{\oplus3}\) and a momentum map \(\nu:X\rightarrow\mathfrak{g}\) of a Poisson Hamiltonian \(G\)-variety \(X\) for a complex semisimple Lie algebra \(\mathfrak{g}\) with adjoint group \(G\), where \(\mathcal{S}_{\tau}:=\xi+\mathfrak{g}_{\eta}\) for the centralizer \(\mathfrak{g}_{\eta}\) of \(\eta\), and \(\mathfrak{g}\) is identified with \(\mathfrak{g}^{\ast}\) via the Killing form. It also presents a systematic construction of a partial compactification of \(X_{\tau}\) using the Hamiltonian reduction \(\overline{X_{\tau}}:=\left( X\times\left( \overline {G\times\mathcal{S}_{\tau}}\right) \right) //G\) (when well defined), where the Poisson slice \(\overline{G\times\mathcal{S}_{\tau}}:=\overline{\rho}_{R}^{-1}\left( \mathcal{S}_{\tau}\right) \) for the momentum map \(\left( \overline{\rho}_{L},\overline{\rho}_{R}\right) :T^{\ast}\overline{G}\left( \log\left( D\right) \right) \rightarrow\mathfrak{g}\oplus\mathfrak{g}\) of the log cotangent bundle \(T^{\ast}\overline{G}\left( \log\left( D\right) \right) \) is shown to be irreducible and log symplectic, containing \(G\times\mathcal{S}_{\tau}\) as the unique open dense symplectic leaf.
In general, \(X_{\tau}\) is a Poisson transversal that is transverse to the \(G\)-orbits in \(X\), and when \(X\) is a log symplectic variety, each irreducible component of \(X_{\tau}\) is a log symplectic subvariety. Furthermore for an \({sl}_{2}\)-triple \(\tau\) of regular elements in \(\mathfrak{g}\), if both \(\overline{X_{\tau}}\) and \(X/G\) exist as well-defined geometric quotients then we get a well-defined canonical map \(\overline{X_{\tau}}\rightarrow X/G\) which gives a fiberwise compactification of the canonical map \(X_{\tau }\rightarrow X/G\). This covers some known examples, like the universal centralizers \(\mathcal{Z}_{\mathfrak{g}}^{\tau}\) and the hyper-Kähler slices \(G\times\mathcal{S}_{\tau}\).

Reviewer: Albert Sheu (Lawrence)