Equivariant principal bundles over toric Deligne--Mumford stacks

Abstract

Let $\mathcal{X}(\mathbf{\Sigma })$ be the toric Deligne--Mumford stack with stacky torus $\mathcal{T}$ associated to the stacky fan $\mathbf{\Sigma }=(N,\Sigma ,\beta )$. A toric principal $H$-bundle over $\mathcal{X}(\mathbf{\Sigma })$ is a principal $H$-bundle $\mathcal{P}$ equipped with a $\mathcal{T}$-action lifting the $\mathcal{T}$-action on $\mathcal{X}(\mathbf{\Sigma })$, such that the $\mathcal{T}$-action and the $H$-action on $\mathcal{P}$ commute. For a connected reductive algebraic group $H$ over $\mathbb{C}$, we classify the isomorphism classes of framed toric principal $H$-bundles over $\mathcal{X}(\mathbf{\Sigma })$ in terms of piecewise linear maps from $|\Sigma |$ to the cone over the Tits building of $H$. Using this classification, we describe the equivariant automorphism group of a toric principal $H$-bundle, and give a necessary and sufficient condition for an equivariant reduction of the structure group. We also show that every toric principal $\mathrm{GL}(r)$-bundle over the weighted projective stack $\mathbb{P}(w_0,\dots ,w_n)$ splits equivariantly whenever $r<n$, and that every toric principal $H$-bundle over the weighted stacky projective line $\mathbb{P}(a,b)$ splits equivariantly when $H$ is a connected reductive algebraic group.

arXiv comment32 pages