Hecke Subalgebras and Local Newforms for the Metaplectic Double Cover of SL2(ℚp)
Abstract
Let p be an odd prime, and let SL2(ℤp)̅ and K0(pn)̅ denote the inverse images of SL2(ℤp) and the congruence subgroup K0(pn) in the metaplectic double cover SL̃2(ℚp). For n≥2, we study the subalgebra of the genuine Hecke algebra H(SL̃2(ℚp)//K0(pn)̅,η) consisting of functions supported in SL2(ℤp)̅, where η is the genuine extension of either the trivial or the nontrivial quadratic character modulo p. We give an explicit basis and a presentation by generators and relations, and prove that this subalgebra is commutative of dimension 2n. We give the multiplicity-free decomposition of IndK0(pn)̅SL2(ℤp)̅η, determining the dimensions of its irreducible constituents and their associated Hecke eigenvalues. This Hecke subalgebra acts on the (K0(pn)̅,η)-fixed spaces in irreducible admissible genuine representations of SL̃2(ℚp). We compute its action explicitly on newvectors of prescribed η-type in principal series, Steinberg, even Weil, and supercuspidal representations. For supercuspidal representations, we construct the vectors by compact induction and use Ishimoto's conductor and dimension formulas. Finally, we relate our operator 𝒲n−1 to Ishimoto's local realization of Ueda's twisting operator.
Related
- Local newforms, Paškūnas-Stevens Whittaker functions, and explicit test vectors for GL(n)× GL(n)
- Calculating double-leaf coefficients in the Hecke category
- Double affine Frobenius Hecke algebras
- A bar operation on the double affine Hecke algebra
- Completed Iwahori–Hecke algebra for Kac–Moody groups over local fields
BibTeX
@misc{baruch2026hecke,
title = {{Hecke Subalgebras and Local Newforms for the Metaplectic Double Cover of $\mathrmSL_2(\mathbbQ_p)$}},
author = {Ehud Moshe Baruch and Markos Karameris and Soma Purkait},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29922},
eprint = {2608.29922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29922},
}
RIS
TY - UNPB
TI - Hecke Subalgebras and Local Newforms for the Metaplectic Double Cover of SL_2(ℚ_p)
AU - Ehud Moshe Baruch
AU - Markos Karameris
AU - Soma Purkait
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2608.29922
UR - https://arxiv.org/abs/2608.29922
AB - Let p be an odd prime, and let SL_2(ℤ_p)̅ and K_0(p^n)̅ denote the inverse images of SL_2(ℤ_p) and the congruence subgroup K_0(p^n) in the metaplectic double cover SL̃_2(ℚ_p). For n≥2, we study the subalgebra of the genuine Hecke algebra H(SL̃_2(ℚ_p)//K_0(p^n)̅,η) consisting of functions supported in SL_2(ℤ_p)̅, where η is the genuine extension of either the trivial or the nontrivial quadratic character modulo p. We give an explicit basis and a presentation by generators and relations, and prove that this subalgebra is commutative of dimension 2n. We give the multiplicity-free decomposition of Ind_K_0(p^n)̅^SL_2(ℤ_p)̅η, determining the dimensions of its irreducible constituents and their associated Hecke eigenvalues. This Hecke subalgebra acts on the (K_0(p^n)̅,η)-fixed spaces in irreducible admissible genuine representations of SL̃_2(ℚ_p). We compute its action explicitly on newvectors of prescribed η-type in principal series, Steinberg, even Weil, and supercuspidal representations. For supercuspidal representations, we construct the vectors by compact induction and use Ishimoto's conductor and dimension formulas. Finally, we relate our operator 𝒲_n−1 to Ishimoto's local realization of Ueda's twisting operator.
ER -