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The wavefront set over a maximal unramified field extension

Emile Okada

Abstract

Let $(π,X)$ be a depth-$0$ admissible smooth complex representation of a $p$-adic reductive group that splits over an unramified extension. In this paper we develop the theory necessary to study the wavefront set of $X$ over a maximal unramified field extension of the base $p$-adic field. In the final section we then apply these methods to compute the geometric wavefront set of spherical Arthur representations of split $p$-adic reductive groups. In this case we see how the wavefront set over a maximal unramified extension can be computed using perverse sheaves on the Langlands dual group.

The wavefront set over a maximal unramified field extension

Abstract

Let be a depth- admissible smooth complex representation of a -adic reductive group that splits over an unramified extension. In this paper we develop the theory necessary to study the wavefront set of over a maximal unramified field extension of the base -adic field. In the final section we then apply these methods to compute the geometric wavefront set of spherical Arthur representations of split -adic reductive groups. In this case we see how the wavefront set over a maximal unramified extension can be computed using perverse sheaves on the Langlands dual group.

Paper Structure

This paper contains 40 sections, 64 theorems, 217 equations, 3 figures.

Table of Contents

  1. The Wavefront Set
  2. Basic Notation
  3. Buildings, Parahorics and Associated Notation
  4. Fourier Transforms
  5. The Harish-Chandra-Howe Local Character Expansion
  6. The Wavefront Set of Representations of p-adic Groups
  7. Lifting Nilpotent Orbits and Closure Relations
  8. The Lifting Map
  9. Closure relations
  10. The Wavefont Set of Representations of Finite Groups of Lie Type
  11. Generalised Gelfand--Graev Representations
  12. The Kawanaka Wavefront Set
  13. Relating the Qp and Fp Wavefront Sets
  14. Inflated Generalised Gelfand--Graev Representations
  15. Local Wavefront Sets
  16. ...and 25 more sections

Key Result

theorem \oldthetheorem

[Theorem lem:liftwf] Let $(\pi,X)$ be a depth-$0$ representation of ${\mathbf G}(k)$. Then In fact one can restrict $c$ to range over the faces (or vertices) of any fixed chamber of $\mathcal{B}({\mathbf G},k)$.

Figures (3)

  • Figure 1: The chamber complex for $V$ is displayed alongside the coroot lattice (in grey), the coroots $\check \alpha_0$,$\check \alpha_1$,$\check \alpha_2$ (in blue), and the fundamental domain for the torus $\mathbb T$ (in red).
  • Figure 2: The colored disks (grey, blue, yellow) are all the lifts of the vanishing set of $\lbrace\alpha_0,\alpha_1\rbrace$ in $\mathbb T$. Having the same color indicates having the same image in $\mathbb T$.
  • Figure 3: The colored disks (grey, red, cyan, green) are all the lifts of the vanishing set of $\lbrace\alpha_0,\alpha_2\rbrace$ in $\mathbb T$. Having the same color indicates having the same image in $\mathbb T$.

Theorems & Definitions (123)

  • theorem \oldthetheorem
  • corollary \oldthetheorem
  • theorem \oldthetheorem
  • conjecture \oldthetheorem
  • theorem \oldthetheorem: Theorem \ref{['thm:thetabar']}
  • theorem \oldthetheorem: Theorem \ref{['thm:arthurwf']}
  • lemma \oldthetheorem: Pommerening,Pommerening2
  • remark \oldthetheorem
  • theorem \oldthetheorem
  • remark \oldthetheorem
  • ...and 113 more