Algebraic zip data

Abstract

An algebraic zip datum is a tuple $\mathcal{Z}=(G,P,Q,\phi )$ consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $\phi :P/R_{u}P\to Q/R_{u}Q$. We study the action of the group $E_{\mathcal{Z}}:=\{(p,q)\in P\times Q|\phi ({\pi }_P(p))={\pi }_Q(q)\}$ on $G$ given by $((p,q),g)\mapsto pg{q}^{-1}$. 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 $[G\mathbb{Z}]$ or to $[G{\mathbb{Z}}^{\prime }]$, 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 $E_{\mathcal{Z}}$-invariant subvarieties correspond to the so-called “$G$-stable pieces” of $Z$ defined by Lusztig (resp. the $G$-orbits of $Z^{\prime }$).