Algebraic zip data

TL;DR

This paper introduces algebraic zip data involving reductive groups and parabolic subgroups, studies the group actions and stratifications on the group, and connects these structures to known geometric objects like Lusztig's G-stable pieces and F-zip classifications.

Contribution

It generalizes the theory of algebraic zip data, describes the stratification and stabilizers of the group action, and links these to important geometric structures in representation theory and algebraic geometry.

Findings

Defined smooth E-invariant subvarieties and their stratification of G

Determined dimensions, closures, and stabilizers of these subvarieties

Connected algebraic quotients to Lusztig's G-stable pieces and F-zip classifications

Abstract

An algebraic zip datum is a tuple $\CZ := (G,P,Q,\phi)$ consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $\phi\colon P/R_uP\to Q/R_uQ$. We study the action of the group $E := \{(p,q)\in P{\times}Q | \phi(\pi_{P}(p)) =\pi_Q(q)\}$ on $G$ given by $((p,q),g)\mapsto pgq^{-1}$. 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 $[G \backslash Z']$, 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 $E$-invariant subvarieties correspond to the so-called "$G$-stable pieces" of $Z$ defined by Lusztig (resp. the $G$-orbits of $Z'$).