Algebraic zip data

  • Richard Pink

  • Torsten Wedhorn

  • Paul Ziegler

    Richard Pink Torsten Wedhorn Dept. of Mathematics Dept. of Mathematics ETH Zürich University of Paderborn CH-8092 Zürich D-33098 Paderborn Switzerland Germany
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An algebraic zip datum is a tuple \CZ=(G,P,Q,φ)\CZ = (G,P,Q,\varphi) consisting of a reductive group GG together with parabolic subgroups PP and QQ and an isogeny φ ⁣:P/RuPQ/RuQ\varphi\colon P/R_uP\to Q/R_uQ. We study the action of the group E_\CZ := \bigl{ (p,q)\in P{\times}Q \bigm| \varphi(\pi_{P}(p)) =\pi_Q(q)\bigr} on GG given by ((p,q),g)pgq1((p,q),g)\mapsto pgq^{-1}. We define certain smooth E\CZE_\CZ-invariant subvarieties of GG, show that they define a stratification of GG. We determine their dimensions and their closures and give a description of the stabilizers of the E\CZE_\CZ-action on GG. We also generalize all results to non-connected groups. We show that for special choices of \CZ\CZ the algebraic quotient stack [E\CZ G][E_\CZ \ G] is isomorphic to [GZ][G \Z] or to [GZ][G \Z'], where ZZ is a GG-variety studied by Lusztig and He in the theory of character sheaves on spherical compactifications of GG and where ZZ' has been defined by Moonen and the second author in their classification of FF-zips. In these cases the E\CZE_\CZ-invariant subvarieties correspond to the so-called «GG-stable pieces» of ZZ defined by Lusztig (resp. the GG-orbits of ZZ').

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Richard Pink, Torsten Wedhorn, Paul Ziegler, Algebraic zip data. Doc. Math. 16 (2011), pp. 253–300

DOI 10.4171/DM/332