Algebraic zip data

Abstract

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 $\varphi :P/R_{u}P\to Q/R_{u}Q$. We study the action of the group $E:=\{(p,q)\in P\times Q|\varphi ({\pi }_P(p))={\pi }_Q(q)\}$ on $G$ given by $((p,q),g)\mapsto pg{q}^{-1}$. 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 $[G\backslash Z^{\prime }]$, where $Z$ is a $G$-variety studied by Lusztig and He in the theory of character sheaves on spherical compactifications of $G$ and where $Z^{\prime }$ 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 $Z^{\prime }$).

42 pages, added some references, to appear in Doc. Math