Algebraic zip data
Richard Pink, Torsten Wedhorn, Paul K. Ziegler · Documenta Mathematica · 2011
An algebraic zip datum is a tuple \mathcal Z = (G,P,Q,\varphi) consisting of a reductive group G together with parabolic subgroups P and Q and an isogeny \varphi\colon P/R_uP\to Q/R_uQ . We study the action of the group E_{\mathcal Z} := \{ (p,q)\in P{\times}Q | \varphi(\pi_{P}(p)) =\pi_Q(q)\} on G given by ((p,q),g)\mapsto pgq^{-1} . We define certain smooth E_{\mathcal Z} -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_\mathcal Z -action on G . We also generalize all results to non-connected groups. We show that for special choices of \mathcal Z the algebraic quotient stack [E_{\mathcal Z} \ G] is isomorphic to [G \Z] or to [G \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_{\mathcal Z} -invariant subvarieties correspond to the so-called “ G -stable pieces” of Z defined by Lusztig (resp. the G -orbits of Z' ).