Parahoric Spherical Whittaker Functions
Abstract
Let F be a non-archimedean local field and let Kμ be a standard parahoric subgroup of GLn(F). We express the diagonal values of normalized parahoric Whittaker eigenfunctions with trivial central character using blockwise Casselman--Shalika formulas with the appropriate modulus factor. Every irreducible generic Iwahori-spherical representation has a one-dimensional fixed space for a suitable parahoric, with no invariants under a strictly larger parahoric. The full Hecke character of this line determines the representation. We compute its adjacent-block Hecke eigenvalues and, under additional irreducibility hypotheses, obtain an explicit finite formula on every Bruhat cell using an opposite induced model, Whittaker-preserving intertwiners, and Demazure--Whittaker operators. We apply these formulas to local Rankin--Selberg integrals for GLn× GL1 and for compatible parahorics in GLn× GLm. In the latter case, the lowest Laurent coefficient and the equal-rank functional equation determine the integral. Explicit translations and rescalings then give test vectors for the local L-factor.
Related
- Spin Toda systems and N-point spherical Whittaker functions
- Characteristic Functions of Parahoric Character Sheaves
- Bounds on Whittaker Functions for GL(n)
- Reflection equations and Littlewood identities for spin Hall-Littlewood and q-Whittaker functions
- Iwahori Whittaker functions for unramified quasi-split groups
BibTeX
@misc{karameris2026parahoric,
title = {{Parahoric Spherical Whittaker Functions}},
author = {Markos Karameris},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40021},
eprint = {2610.40021},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40021},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Parahoric Spherical Whittaker Functions
AU - Markos Karameris
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40021
UR - https://arxiv.org/abs/2610.40021
AB - Let F be a non-archimedean local field and let K_μ be a standard parahoric subgroup of GL_n(F). We express the diagonal values of normalized parahoric Whittaker eigenfunctions with trivial central character using blockwise Casselman--Shalika formulas with the appropriate modulus factor. Every irreducible generic Iwahori-spherical representation has a one-dimensional fixed space for a suitable parahoric, with no invariants under a strictly larger parahoric. The full Hecke character of this line determines the representation. We compute its adjacent-block Hecke eigenvalues and, under additional irreducibility hypotheses, obtain an explicit finite formula on every Bruhat cell using an opposite induced model, Whittaker-preserving intertwiners, and Demazure--Whittaker operators. We apply these formulas to local Rankin--Selberg integrals for GL_n× GL_1 and for compatible parahorics in GL_n× GL_m. In the latter case, the lowest Laurent coefficient and the equal-rank functional equation determine the integral. Explicit translations and rescalings then give test vectors for the local L-factor.
ER -