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Geometric wavefront sets of genuine Iwahori-spherical representations

Fan Gao, Runze Wang

TL;DR

This work establishes a tight link between geometric wavefront sets of genuine Iwahori-spherical representations in $n$-fold covering groups and the covering Barbasch–Vogan duality, proving an upper bound for ${\rm WF}^{geo}({\rm AZ}(\pi))$ in terms of $d_{BV,G}^{(n)}(\mathcal{O}(\phi_{\pi}))$ for representations with positive real Satake parameters. It shows that equality holds in key families: covers of type $A$, certain exceptional-type cases, and particular $\pi_S$ arising from regular unramified principal series under non-autotomous conditions, with detailed, case-by-case verifications for classical types and algorithmic checks for exceptional types. The paper also provides a formula for the leading coefficients $c_{\Theta}(\mathcal{O})_{\psi}$ of Harish-Chandra characters for theta representations, expressing them via degenerate Whittaker models and BV-duality data, and clarifies when these coefficients match BV-predicted values. Overall, the results substantiate the conjectured BV-duality-guided framework in the covering setting, advancing understanding of wavefront geometry, KLR parameters, and the Theta/Levi-Theta structure within the Langlands program for covering groups.

Abstract

For Iwahori-spherical genuine representations of central covers with positive real Satake parameters, we prove the upper bound inequality for their geometric wavefront sets, formulated for general genuine representations in an earlier work by Gao--Liu--Lo--Shahidi. Meanwhile, we show the equality is attained for covers of type A groups and for some representations of covers of the exceptional groups. We also verify the equality for certain Iwahori-spherical representations occurring in regular unramified principal series; this uses and generalizes the earlier work of Karasiewicz--Okada--Wang on theta representations. Lastly, we determine the leading coefficients in the Harish-Chandra character expansion of a theta representation when its geometric wavefront set is of a special type.

Geometric wavefront sets of genuine Iwahori-spherical representations

TL;DR

This work establishes a tight link between geometric wavefront sets of genuine Iwahori-spherical representations in -fold covering groups and the covering Barbasch–Vogan duality, proving an upper bound for in terms of for representations with positive real Satake parameters. It shows that equality holds in key families: covers of type , certain exceptional-type cases, and particular arising from regular unramified principal series under non-autotomous conditions, with detailed, case-by-case verifications for classical types and algorithmic checks for exceptional types. The paper also provides a formula for the leading coefficients of Harish-Chandra characters for theta representations, expressing them via degenerate Whittaker models and BV-duality data, and clarifies when these coefficients match BV-predicted values. Overall, the results substantiate the conjectured BV-duality-guided framework in the covering setting, advancing understanding of wavefront geometry, KLR parameters, and the Theta/Levi-Theta structure within the Langlands program for covering groups.

Abstract

For Iwahori-spherical genuine representations of central covers with positive real Satake parameters, we prove the upper bound inequality for their geometric wavefront sets, formulated for general genuine representations in an earlier work by Gao--Liu--Lo--Shahidi. Meanwhile, we show the equality is attained for covers of type A groups and for some representations of covers of the exceptional groups. We also verify the equality for certain Iwahori-spherical representations occurring in regular unramified principal series; this uses and generalizes the earlier work of Karasiewicz--Okada--Wang on theta representations. Lastly, we determine the leading coefficients in the Harish-Chandra character expansion of a theta representation when its geometric wavefront set is of a special type.
Paper Structure (46 sections, 16 theorems, 226 equations, 2 figures, 2 tables)

This paper contains 46 sections, 16 theorems, 226 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgement
  4. Preliminaries
  5. Covering groups
  6. Dual groups
  7. Several Iwahori--Hecke type algebras
  8. Kazhdan--Lusztig--Reeder parameters
  9. Aubert--Zelevinsky duality
  10. Local character expansion and wavefront sets
  11. Truncated induction and Springer correspondence
  12. Covering Barbasch--Vogan duality
  13. Exceptional types
  14. Geometric wavefront sets for ${\rm Irr} _{\iota}(\overline{G})^I$
  15. A first reduction
  16. ...and 31 more sections

Key Result

Theorem 1.1

Assume $p$ is sufficiently large. Let $\pi:=\pi(s, u, \tau) \in {\rm Irr} _\iota(\overline{G})^I$ be an Iwahori-spherical genuine representation with positive real Satake parameter $s$. Then the inequality holds. Furthermore, if $\overline{G}$ is a cover of $G$ of type $A$, then the equality is attained. For covers of exceptional groups, the equality is also attained when $\tau=\mathbbm{1}$ and

Figures (2)

  • Figure 1: $W$ is of type $G_2$ and $x$ is hyperspecial
  • Figure 2: $W$ is of type $E_6$ and $W_x$ is of the type $3A_2$

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['T:main2']}
  • Theorem 1.3: Theorem \ref{['T:cO']}
  • Example 2.1
  • Theorem 3.1: KOW, Theorem 2.22, Lemma 2.23
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 23 more