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The cohomology of $p$-adic Deligne-Luszitg schemes of Coxeter type

Alexander B. Ivanov, Sian Nie

Abstract

We determine the cohomology of the closed Drinfeld stratum of $p$-Deligne--Lusztig schemes of Coxeter type attached to arbitrary inner forms of unramified groups over a local non-archimedean field. We prove that the corresponding torus weight spaces are supported in exactly one cohomological degree, and are pairwisely non-isomorphic irreducible representations of the pro-unipotent radical of the corresponding parahoric subgroup. We also prove that all Moy--Prasad quotients of this stratum are maximal varieties, and we investigate the relation between the resulting representations and Kirillov's orbit method.

The cohomology of $p$-adic Deligne-Luszitg schemes of Coxeter type

Abstract

We determine the cohomology of the closed Drinfeld stratum of -Deligne--Lusztig schemes of Coxeter type attached to arbitrary inner forms of unramified groups over a local non-archimedean field. We prove that the corresponding torus weight spaces are supported in exactly one cohomological degree, and are pairwisely non-isomorphic irreducible representations of the pro-unipotent radical of the corresponding parahoric subgroup. We also prove that all Moy--Prasad quotients of this stratum are maximal varieties, and we investigate the relation between the resulting representations and Kirillov's orbit method.
Paper Structure (30 sections, 32 theorems, 121 equations)

This paper contains 30 sections, 32 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Notation and setup
  3. General notation
  4. Group-theoretic setup
  5. Filtration of the torus and affine roots
  6. Parahoric model and Moy--Prasad quotients
  7. Coxeter pairs
  8. A condition on $p$
  9. Homology
  10. Steinberg's cross-section
  11. Review and some properties of $X_{T,U}$
  12. Comparison with the definition in Ivanov_DL_indrep
  13. Integral decomposition of $X_{T,U}$
  14. Drinfeld stratification
  15. The minimal Drinfeld stratum
  16. ...and 15 more sections

Key Result

Theorem 1.1

Suppose that $(T, U)$ is a Coxeter pair. For a smooth character $\chi \colon \mathcal{T}\xspace^{+}(\mathcal{O}\xspace_k) \rightarrow \overline{\mathbb{Q}}_\ell^\times$ the following hold.

Theorems & Definitions (64)

  • Theorem 1.1
  • Conjecture 1.2
  • Corollary 1.3
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Remark 3.2
  • proof
  • Corollary 4.1
  • Theorem 4.2: Ivanov_Cox_orbits,Nie_23
  • ...and 54 more