2010/10/05 by Richard Pink, Pink, Richard, Torsten Wedhorn +3 · 2 citations
Mathematics · #14L30 #20G15 #20G40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1010.0811
openalex publication_date 2010/10/05 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28
An algebraic zip datum is a tuple \CZ := (G,P,Q,ϕ) consisting of a reductive group G together with parabolic subgroups P and Q and an isogeny ϕ\colon P/RuP→ Q/RuQ. We study the action of the group E := \(p,q)∈ P×Q | ϕ(πP(p)) =πQ(q)\ on G given by ((p,q),g)↦ pgq-1. We define certain smooth E-invariant subvarieties of G, show that they define a stratification of G. We determine their dimensions and their closures and give a description of the stabilizers of the E-action on G. We also generalize all results to non-connected groups. We show that for special choices of \CZ the algebraic quotient stack [E \backslash G] is isomorphic to [G \backslash Z] or to [G \backslash Z'], where Z is a G-variety studied by Lusztig and He in the theory of character sheaves on spherical compactifications of G and where Z' has been defined by Moonen and the second author in their classification of F-zips. In these cases the E-invariant subvarieties correspond to the so-called "G-stable pieces" of Z defined by Lusztig (resp. the G-orbits of Z').