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Eigenweights for arithmetic Hirzebruch Proportionality

Tony Feng

TL;DR

This work completes the eigenweight computations for the Arithmetic Hirzebruch Proportionality principle across all classical types by linking the eigenweights to symmetric-group representation theory. It presents uniform, closed-form expressions in terms of Frobenius characters and MN-rule computations, expressed through Schur polynomials and associated integration formulas. The Type A, B, C, and D analyses reveal both expected alignments with prior FYZ4 results (e.g., for $m=1,2$ in Type A) and new phenomena (notably noncommutativity in type $D_{2m}$). The approach integrates algebraic combinatorics with geometric objects like shtukas and moduli stacks, offering explicit, algorithmically computable eigenweights that govern the differential operators in the arithmetic Hirzebruch framework, with potential implications for arithmetic volumes and Siegel–Weil-type formulas.

Abstract

Prior work of Feng--Yun--Zhang established a (Higher) Arithmetic Hirzebruch Proportionality Principle, expressing the arithmetic volumes of moduli stacks of shtukas in terms of differential operators applied to $L$-functions. This formula involves certain "eigenweights" which were calculated in simple cases by Feng--Yun--Zhang, but not in general. We document work of a (custom) AI Agent built upon Gemini Deep Think, which employs tools from algebraic combinatorics to connect these eigenweights to the representation theory of symmetric groups, and then determines them for all classical groups.

Eigenweights for arithmetic Hirzebruch Proportionality

TL;DR

This work completes the eigenweight computations for the Arithmetic Hirzebruch Proportionality principle across all classical types by linking the eigenweights to symmetric-group representation theory. It presents uniform, closed-form expressions in terms of Frobenius characters and MN-rule computations, expressed through Schur polynomials and associated integration formulas. The Type A, B, C, and D analyses reveal both expected alignments with prior FYZ4 results (e.g., for in Type A) and new phenomena (notably noncommutativity in type ). The approach integrates algebraic combinatorics with geometric objects like shtukas and moduli stacks, offering explicit, algorithmically computable eigenweights that govern the differential operators in the arithmetic Hirzebruch framework, with potential implications for arithmetic volumes and Siegel–Weil-type formulas.

Abstract

Prior work of Feng--Yun--Zhang established a (Higher) Arithmetic Hirzebruch Proportionality Principle, expressing the arithmetic volumes of moduli stacks of shtukas in terms of differential operators applied to -functions. This formula involves certain "eigenweights" which were calculated in simple cases by Feng--Yun--Zhang, but not in general. We document work of a (custom) AI Agent built upon Gemini Deep Think, which employs tools from algebraic combinatorics to connect these eigenweights to the representation theory of symmetric groups, and then determines them for all classical groups.
Paper Structure (44 sections, 8 theorems, 114 equations)

This paper contains 44 sections, 8 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Arithmetic Hirzebruch Proportionality
  3. An example of Arithmetic Hirzebruch Proportionality
  4. Arithmetic volume of Shimura varieties
  5. Definition of eigenweights
  6. Calculation of eigenweights
  7. Type A
  8. Type B
  9. Type C
  10. Type D
  11. Acknowledgments
  12. Declaration of AI Usage
  13. Eigenweights for type A
  14. Setup
  15. Minuscule coweight
  16. ...and 29 more sections

Key Result

Theorem 1.3.2

Let $G = \textup{GL}_n$ and fix $1 \leq m < n$. Consider the minuscule coweight $\mu = (1^m, 0^{n-m})$. Let $N = m(n-m)+1$ be the arithmetic dimension of $G/P_\mu$. For $\Omega := \frac{1}{2} \sum_{i=1}^n x_i^2$, the eigenweights areAs usual, the use of exponents in partitions indicates repetitions, where $\pi_j(k)$ is the partition $(k-j, 1^j)$ and $\nu_k$ is the partition $(k-1, 1^{N})$, and $\L

Theorems & Definitions (20)

  • Remark 1.1.2
  • Definition 1.2.1
  • Remark 1.2.2
  • Theorem 1.3.2
  • Remark 1.3.3
  • Theorem 1.3.6
  • Theorem 1.3.8
  • Remark 1.3.9
  • Lemma 3.2.3
  • proof
  • ...and 10 more