Toric principal bundles, Tits buildings and reduction of structure group

A toric principal $G$-bundle is a principal $G$-bundle over a toric variety together with a torus action commuting with the $G$-action. In a recent paper, extending the Klyachko classification of toric vector bundles, Chris Manon and the second author give a classification of toric principal bundles using "piecewise linear maps" to the (extended) Tits building of $G$. In this paper, we use this 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 number of Tits buildings of semisimple algebraic groups $G$.