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Geometry and cohomology of compactified Deligne--Lusztig varieties

Yingying Wang

Abstract

For connected reductive groups together with a Frobenius root $F$, we show that the cohomology of the structure sheaf and respectively the canonical sheaf for compactified Deligne--Lusztig varieties associated to an element in the free monoid generated by the simple reflections is isomorphic to that of a minimal length element in an $F$-conjugacy class in the Weyl group.

Geometry and cohomology of compactified Deligne--Lusztig varieties

Abstract

For connected reductive groups together with a Frobenius root , we show that the cohomology of the structure sheaf and respectively the canonical sheaf for compactified Deligne--Lusztig varieties associated to an element in the free monoid generated by the simple reflections is isomorphic to that of a minimal length element in an -conjugacy class in the Weyl group.
Paper Structure (17 sections, 10 theorems, 48 equations)

This paper contains 17 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Compactified Deligne--Lusztig varieties
  5. Definitions and basic properties
  6. The Lang map
  7. Criterion for irreducibility
  8. $F_p$-regularity
  9. Braid relations in the Weyl group and birational morphisms
  10. Smoothness via braid relations
  11. Higher direct images of the structure sheaf for birational morphisms
  12. $\mathbb{P}^1$-fibrations and $F$-conjugacy classes
  13. $\mathbb{P}^1$-fibrations over Deligne--Lusztig varieties
  14. $F$-conjugacy classes in $W$
  15. Cohomology of compactified Deligne--Lusztig varieties
  16. ...and 2 more sections

Key Result

Lemma 2.2.1

For any $w_1,\ldots,w_r\in W$, there are $G^F$-equivariant isomorphisms and Moreover, $G^F$ acts transitively on the irreducible components of $X(w_1,\ldots ,w_r)$ (resp. $X(\underline{w_1},\ldots, \underline{w_r})$). In particular, $X(w_1,\ldots ,w_r)$ and $X(\underline{w_1},\ldots, \underline{w_r})$ are equidimensional.

Theorems & Definitions (28)

  • Definition 2.1.1
  • Lemma 2.2.1
  • proof
  • Remark 2.2.2
  • Definition 2.3.1
  • Remark 2.3.2
  • Proposition 2.3.3
  • proof
  • Proposition 2.4.1
  • proof
  • ...and 18 more