2007/04/05 by Xuhua He, He, Xuhua, Jesper Funch Thomsen +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT) #math.AC #math.AG #math.RT
paper · pdf · doi:10.48550/arxiv.0704.0778
Final version, 44 pages
openalex publication_date 2007/04/05 · arxiv created 2008/09/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be an equivariant embedding of a connected reductive group G over an algebraically closed field k of positive characteristic. Let B denote a Borel subgroup of G. A G-Schubert variety in X is a subvariety of the form \diag(G) ⋅ V, where V is a B × B-orbit closure in X. In the case where X is the wonderful compactification of a group of adjoint type, the G-Schubert varieties are the closures of Lusztig's G-stable pieces. We prove that X admits a Frobenius splitting which is compatible with all G-Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any G-Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that G-Schubert varieties have nice singularities we present an example of a non-normal G-Schubert variety in the wonderful compactification of a group of type G2. Finally we also extend the Frobenius splitting results to the more general class of \mathcal R-Schubert varieties.