Equivariant vector bundles on group completions

Abstract

In 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.Comment: Final version. Improved Exposition. to appear in Crelle's J. 43page

Equivariant sheaves on some spherical varieties

“…Results in this spirit were first presented by Kato (cf. [Kat05]), and we recover and extend his results in several directions. Detailed proofs of our results will appear in [AP].…”

“…The crucial points are Definition 4.5 and Theorem 4.10. Section 5 indicates some specific cases to which our generalization applies, including those from [Kat05].…”

Equivariant Vector Bundles on Certain Affine G-Varieties

“…A concrete description of the category of equivariant vector bundles on affine or smooth complete toric varieties was obtained by Klyachko (see [Kly89] Thm 2.2.1). Recently, a paper of Syu Kato [Kat05] significantly generalizes Klyachko's results in a slightly different direction. For G a connected reductive group, Kato has given a description of the category of G × G-equivariant vector bundles on a class of spherical varieties consisting of certain smooth G × G-equivariant partial compactifications of the group G.…”

Extensions of real bounded symmetric domains

“…For a unitary representation (σ, V ) of K on the complex vector space V we let V denote the associated homogeneous vector bundle over D. Without loss of generality, we can assume that σ is irreducible, in which case we shall denote by µ a highest weight and, as before, by V µ its representation space. In [Br07] and [Ka05] homogeneous vector bundles over certain complex homogeneous spaces were shown to have an extension to natural compactifications, e.g. the wonderful compactification.…”