Algebraic zip data
Documenta mathematica, Tome 16 (2011), pp. 253-300
An algebraic zip datum is a tuple Z=(G,P,Q,φ) consisting of a reductive group G together with parabolic subgroups P and Q and an isogeny φ:P/RuP→Q/RuQ. We study the action of the group EZ:={(p,q)∈P×Q∣φ(πP(p))=πQ(q)} on G given by ((p,q),g)↦pgq−1. We define certain smooth EZ-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 EZ-action on G. We also generalize all results to non-connected groups. We show that for special choices of Z the algebraic quotient stack [EZ G] is isomorphic to [GZ] or to [GZ′], 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 EZ-invariant subvarieties correspond to the so-called “G-stable pieces” of Z defined by Lusztig (resp. the G-orbits of Z′).
Classification :
14L30, 20F55, 20G15, 20G40
Mots-clés : generalized G-stable pieces, zip data, F-zips
Mots-clés : generalized G-stable pieces, zip data, F-zips
@article{10_4171_dm_332,
author = {Richard Pink and Torsten Wedhorn and Paul Ziegler},
title = {Algebraic zip data},
journal = {Documenta mathematica},
pages = {253--300},
year = {2011},
volume = {16},
doi = {10.4171/dm/332},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/332/}
}
Richard Pink; Torsten Wedhorn; Paul Ziegler. Algebraic zip data. Documenta mathematica, Tome 16 (2011), pp. 253-300. doi: 10.4171/dm/332
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