English

Spherical functions of symmetric forms and a conjecture of Hironaka

Number Theory 2026-02-11 v2 Representation Theory

Abstract

For all r1,r\ge1, we verify the following conjecture of Hironaka: for a pp-adic field FF with pp odd, the space of spherical functions of Symr×r(F)GLr(F)\mathrm{Sym}_{r\times r}(F)\cap\mathrm{GL}_r(F) is free of rank 4r4^r over the Hecke algebra.

Keywords

Cite

@article{arxiv.2311.00147,
  title  = {Spherical functions of symmetric forms and a conjecture of Hironaka},
  author = {Murilo Corato-Zanarella},
  journal= {arXiv preprint arXiv:2311.00147},
  year   = {2026}
}