Algebraic zip data

By R. Pink, T. Wedhorn, Paul Ziegler. Documenta Mathematica, 2010

74 citations recorded by the source corpus.

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 $Z'$).

Subfields: Computational Theory and Mathematics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/4e91b252d4e17ee545d00459a4d55b3a11bb31b2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Chao Zhang
2. Daniel Wortmann
3. T. Wedhorn
4. Paul Ziegler
5. R. Pink
6. W. Goldring
7. Jean-Stefan Koskivirta
8. L. Taelman

In Molecular Biology

1. Chao Zhang
2. Daniel Wortmann
3. T. Wedhorn
4. Paul Ziegler
5. R. Pink

Citations in the corpus

• The \mu-ordinary locus for Shimura varieties of Hodge type (2013, methodology): “ ≃JW from Prop. 5.12 in this situation: The Frobenius isogeny gives a morphism σ: P+/U+ ∼=M−→M(σ) ∼=P(σ) − /U (σ) − , such that the tuple (GF,P+,P (σ) − ,σ) is an algebraic zip datum in the sense of ([PWZ1], 3.1.). The associated zip group is defined as Eχ := (mu+,σ(m)u −) |u+ ∈U+,u− ∈U (σ),m∈M ⊆P+ ×”
• Strata Hasse invariants, Hecke algebras and Galois representations (2015, background): “rtain subset IWof the Weyl group Wof the reductive group attached to the PEL-structure. In the third stage, in a series of papers, Moonen, Wedhorn, Pink and Ziegler defined the algebraic stack G-Zipµ, [82, 107, 87, 88], whose k-points parametrize isomorphism classes of BT1’s with G-structure over k. ”
• $F$-zips with additional structure (2012, methodology): “To G and χ we can also associate a natural algebraic zip datum in the sense of [18] (see Definition 3.”
• Ekedahl-Oort Strata for Good Reductions of Shimura Varieties of Hodge Type (2013, methodology): “By the same reason as in [24] Theorem 12.”