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Spherical functions of symmetric forms and a conjecture of Hironaka

Murilo Corato-Zanarella

TL;DR

The paper proves Hironaka's conjecture that the space of spherical functions on X=Sym_{r×r}(F)∩GL_r(F) is a free module of rank 4^r over the Hecke algebra, for any r and p-adic F with p odd, provided 2 is invertible in the coefficient ring. The authors develop a unified framework based on lattice-counting in a relative Cartan setup, reduce the Hecke action to minuscule operators T_k, and encode the action via straightening relations Rel, enabling an explicit basis for S(X/K,R). This approach extends to the Hermitian and alternating variants, and yields explicit bases and modular freeness in all three cases, with potential connections to relative Langlands dual objects and inverse Satake transforms. The work provides concrete, combinatorial tools for analyzing spherical functions on symmetric and related varieties, with a broad method that applies to coefficient rings beyond the complex numbers and offers explicit spectral data via the Hecke action.

Abstract

For all $r\ge1,$ we verify the following conjecture of Hironaka: for a $p$-adic field $F$ with $p$ odd, the space of spherical functions of $\mathrm{Sym}_{r\times r}(F)\cap\mathrm{GL}_r(F)$ is free of rank $4^r$ over the Hecke algebra.

Spherical functions of symmetric forms and a conjecture of Hironaka

TL;DR

The paper proves Hironaka's conjecture that the space of spherical functions on X=Sym_{r×r}(F)∩GL_r(F) is a free module of rank 4^r over the Hecke algebra, for any r and p-adic F with p odd, provided 2 is invertible in the coefficient ring. The authors develop a unified framework based on lattice-counting in a relative Cartan setup, reduce the Hecke action to minuscule operators T_k, and encode the action via straightening relations Rel, enabling an explicit basis for S(X/K,R). This approach extends to the Hermitian and alternating variants, and yields explicit bases and modular freeness in all three cases, with potential connections to relative Langlands dual objects and inverse Satake transforms. The work provides concrete, combinatorial tools for analyzing spherical functions on symmetric and related varieties, with a broad method that applies to coefficient rings beyond the complex numbers and offers explicit spectral data via the Hecke action.

Abstract

For all we verify the following conjecture of Hironaka: for a -adic field with odd, the space of spherical functions of is free of rank over the Hecke algebra.
Paper Structure (15 sections, 26 theorems, 147 equations)

This paper contains 15 sections, 26 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Strategy and structure of the paper
  4. Notations
  5. Combinatorics over finite fields
  6. Lattice counting
  7. Explicit models of spherical functions
  8. Spherical functions and Hecke action
  9. Straightening relations
  10. Analysis of the Hecke module structure
  11. Explicit bases
  12. Case (uH)
  13. Case (S)
  14. Case (A)
  15. Relation with relative Satake isomorphism

Key Result

Proposition 2.1

The set of isomorphism classes of nondegenerate $(V,\langle\cdot,\cdot\rangle)$ is in bijection with $\{(0,1)\}\sqcup(\mathbb{Z}_{>0}\times\mathrm{Sign})$ via $\mathrm{typ}(V,\langle\cdot,\cdot\rangle)\vcentcolon=(d,\chi)$ where $\chi=\mathrm{sign}(\det(V,\langle\cdot,\cdot\rangle))$ and

Theorems & Definitions (74)

  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • ...and 64 more