Toric principal bundles, Tits buildings and reduction of structure group

Abstract

A toric principal $G$-bundle is a principal $G$-bundle over a toric variety together with a torus action commuting with the $G$-action. Recently, extending the Klyachko classification of toric vector bundles, Kaveh–Manon give a classification of toric principal $G$-bundles using piecewise linear maps to the (extended) Tits building of $G$. In this paper, we use the Kaveh–Manon classification to give a description of the (equivariant) automorphism group of a toric principal bundle as well as a simple criterion for (equivariant) reduction of structure group, recovering results of Dasgupta et al. Finally, motivated by the equivariant splitting problem for toric principal bundles, we introduce the notion of Helly’s number of a building and pose the problem of giving sharp upper bounds for Helly’s numbers of Tits buildings of semisimple algebraic groups $G$.