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$U$-Operators Acting on Harmonic Cocycles for $\mathrm{GL}_3$ and Their Slopes

Gebhard Boeckle, Peter Mathias Graef, Theresa Kaiser

TL;DR

The paper advances the study of $U$-operators in a rank-$3$ Drinfeld-type setting by translating the problem to the combinatorics of harmonic cocycles on the Bruhat–Tits building and exploiting explicit isomorphisms with representation spaces. It provides concrete formulas for $U$ and Hecke operators through double cosets, and implements these in Magma to compute slopes, reporting novel periodicities and multiplicity patterns in the $GL_3$ context. These results constitute the first systematic slope analysis in the $GL_3$ function-field setting and offer data that support conjectural links to Drinfeld cusp forms via a residue map, while suggesting new phenomena absent in rank $2$. The work thus provides both a concrete computational platform and initial structural insights that may guide future theoretical developments and comparisons with cusp-form theory in higher rank.

Abstract

In this article, we describe a computational study of the action of the two natural $U$-operators acting on $Γ$-invariant spaces of harmonic cocycles for $\mathrm{GL}_3$ for certain congruence subgroups $Γ$, in a positive characteristic setting. The cocycle spaces we consider are conjecturally isomorphic to spaces of Drinfeld cusp forms of rank $3$ and level $Γ$ via an analogue of Teitelbaum's residue map. We give explicit descriptions of the spaces of harmonic cocycles as subspaces of the vector space of coefficients, and of the resulting $U$- and Hecke operators acting on these. We then implement these formulas in a computer algebra system. Using the resulting data of slopes (and characteristic polynomials) for the Hecke actions, we observe several patterns and interesting phenomena present in our slope tables. This appears to be the first such study in a $\mathrm{GL}_3$ setting.

$U$-Operators Acting on Harmonic Cocycles for $\mathrm{GL}_3$ and Their Slopes

TL;DR

The paper advances the study of -operators in a rank- Drinfeld-type setting by translating the problem to the combinatorics of harmonic cocycles on the Bruhat–Tits building and exploiting explicit isomorphisms with representation spaces. It provides concrete formulas for and Hecke operators through double cosets, and implements these in Magma to compute slopes, reporting novel periodicities and multiplicity patterns in the context. These results constitute the first systematic slope analysis in the function-field setting and offer data that support conjectural links to Drinfeld cusp forms via a residue map, while suggesting new phenomena absent in rank . The work thus provides both a concrete computational platform and initial structural insights that may guide future theoretical developments and comparisons with cusp-form theory in higher rank.

Abstract

In this article, we describe a computational study of the action of the two natural -operators acting on -invariant spaces of harmonic cocycles for for certain congruence subgroups , in a positive characteristic setting. The cocycle spaces we consider are conjecturally isomorphic to spaces of Drinfeld cusp forms of rank and level via an analogue of Teitelbaum's residue map. We give explicit descriptions of the spaces of harmonic cocycles as subspaces of the vector space of coefficients, and of the resulting - and Hecke operators acting on these. We then implement these formulas in a computer algebra system. Using the resulting data of slopes (and characteristic polynomials) for the Hecke actions, we observe several patterns and interesting phenomena present in our slope tables. This appears to be the first such study in a setting.

Paper Structure

This paper contains 38 sections, 42 theorems, 116 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Bruhat-Tits Building
  3. Basic definitions
  4. Simplices as equivalence classes of matrices
  5. The action of congruence subgroups
  6. Definitions
  7. The action of $\mathop{\mathrm{GL}}\nolimits_3(A)$
  8. The action of $\Gamma_1(t)$
  9. The origin
  10. The sides of $\mathcal{W}$
  11. The interior
  12. $\Gamma_1(t)$-stabilizers
  13. Harmonic Cocycles
  14. Some definitions
  15. Harmonic cocycles for some congruence subgroups
  16. ...and 23 more sections

Key Result

Theorem 1

The following hold:

Figures (4)

  • Figure 1: The standard apartment $\mathcal{A}$ with the standard sector $\mathcal{W}$ colored in blue.
  • Figure 2: Illustration of the standard sector $\mathcal{W}$
  • Figure 3: Orbit of an interior chamber with vertex $[j, k]$ under the action of $W = \mathfrak{S}_3$
  • Figure 4: An edge $e = \{[j-1,k-1],[j,k]\}$ with its two adjacent simplices in $\mathcal{A}$

Theorems & Definitions (95)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Definition 2.8
  • ...and 85 more