Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings

We give a classification of rank r torus equivariant vector bundles E on a toric scheme X over a discrete valuation ring O , in terms of graded piecewise linear maps Φ from the fan of X to the (extended) building of GL ( r ) . This is an extension of Klyachko's classification of torus equivariant vector bundles on toric varieties over a field on one hand, and Mumford's classification of equivariant line bundles on toric schemes over O on the other hand. We also give a simple criterion for equivariant splitting of E into a sum of toric line bundles in terms of its piecewise linear map. Among other things, this work lays the foundations for study of arithmetic geometry of toric vector bundles.