2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63270In this paper, we describe the category of bi-equivariant vector bundles on a bi-equivariant smooth (partial) compactification of a reductive algebraic group with normal crossing boundary divisors. Our result is a generalization of the description of the category of equivariant vector bundles on toric varieties established by A.A. Klyachko [Math. USSR. Izvestiya, {\bf 35}, No.2 (1990)]. As an application, we prove splitting of equivariant vector bundles of low rank on the wonderful compactification of an adjoint semisimple group in the sense of C. De Concini and C. Procesi [Lecture Note in Math. {\bf 996} (1983)]. Moreover, we present an answer to a problem raised by B. Kostant in the case of complex groups.Final version. Improved Exposition. to appear in Crelle's J. 43pagesAlgebraic GeometryRepresentation Theory14J60; 14M17; 14F05; 18F99text