Universal families of twisted cotangent bundles

Abstract

Given a complex algebraic group $G$ and complex $G$-variety $X$, one can study the affine Hamiltonian Lagrangian (AHL) $G$-bundles over $X$. Lisiecki indexes the isomorphism classes of such bundles in the case of a homogeneous $G$-variety $X=G/H$; the indexing set is the set of $H$-fixed points $(\mathfrak{h}^*)^H\subset\mathfrak{h}^*$, where $\mathfrak{h}$ is the Lie algebra of $H$. In very rough terms, one may regard $\psi\in(\mathfrak{h}^*)^H$ as labeling the isomorphism class of a $\psi$-twisted cotangent bundle of $G/H$. These twisted cotangent bundles feature prominently in geometric representation theory and symplectic geometry. We introduce and examine the notion of a universal family of AHL $G$-bundles over a $G$-variety $X$, as part of a broader program on Lie-theoretic and incidence-theoretic constructions of regular Poisson varieties. This family is defined to be a flat family $\pi:\mathcal{U}\longrightarrow Y$, in which $\mathcal{U}$ is a Poisson variety, the fibers of $\pi$ form a complete list of representatives of the isomorphism classes of AHL $G$-bundles over $X$, and other pertinent properties are satisfied. Our first main result is the construction of a universal family of AHL $G$-bundles over a homogeneous base $X=G/H$, for connected $H$. In our second main result, we take $X$ to be a conjugacy class $\mathcal{C}$ of self-normalizing closed subgroups of $G$. We associate to $\mathcal{C}$ a regular Poisson variety $\mathcal{U}_{\mathcal{C}}$, defined in incidence-theoretic terms. Attention is paid to the case of conjugacy classes of normalizers of symmetric subgroups. In the case of a connected semisimple group $G$ and conjugacy class $\mathcal{C}$ of parabolic subgroups, our third main result relates $\mathcal{U}_{\mathcal{C}}$ to the partial Grothendieck-Springer resolution for $\mathcal{C}$.