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Generic decompositions of Deligne--Lusztig representations

Daniel Le, Bao V. Le Hung, Brandon Levin, Stefano Morra

Abstract

Let $G_0$ be a reductive group over $\mathbb{F}_p$ with simply connected derived subgroup, (geometrically) connected center and Coxeter number $h+1$. We extend Jantzen's generic decomposition pattern from $(2h-1)$-generic to $h$-generic Deligne--Lusztig representations, which is optimal. We also prove several results on the ``obvious'' Jordan--Hölder factors of general Deligne--Lusztig representations. As an application we improve the weight elimination result of arXiv:1610.04819 [math.NT]

Generic decompositions of Deligne--Lusztig representations

Abstract

Let be a reductive group over with simply connected derived subgroup, (geometrically) connected center and Coxeter number . We extend Jantzen's generic decomposition pattern from -generic to -generic Deligne--Lusztig representations, which is optimal. We also prove several results on the ``obvious'' Jordan--Hölder factors of general Deligne--Lusztig representations. As an application we improve the weight elimination result of arXiv:1610.04819 [math.NT]
Paper Structure (9 sections, 23 theorems, 47 equations)

This paper contains 9 sections, 23 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. A number theoretic application
  3. Acknowledgments
  4. Notation
  5. Lemmata
  6. Ingredients
  7. Generic decompositions of Deligne--Lusztig representations
  8. Deligne--Lusztig reductions containing a simple module
  9. Applications to weight elimination

Key Result

Theorem 1.1

Suppose that $G$ has connected center, $\mu-\eta$ is $h$-deep in the base $p$-alcove, and $\lambda$ is a $p$-restricted dominant weight. Then $[\overline{R}_s(\mu):F(\lambda)] \neq 0$ if and only if there exist $\widetilde{w}$ and $\widetilde{w}_\lambda$ in the extended affine Weyl group $\widetilde Moreover, in this case: Here, $\widehat{Z}(1,-)$ and $\widehat{L}(1,-)$ are the baby Verma/standar

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 34 more