arXiv2026
Let $p$ be an odd prime, and let $\overline{\mathrm{SL}_2(\mathbb{Z}_p)}$ and $\overline{K_0(p^n)}$ denote the inverse images of $\mathrm{SL}_2(\mathbb{Z}_p)$ and the congruence subgroup $K_0(p^n)$ in the metaplectic double cover $\widetilde{\mathrm{SL}}_2(\mathbb{Q}_p)$. For $n\geq2$, we study the subalgebra of the genuine Hecke algebra $H(\widetilde{\mathrm{SL}}_2(\mathbb{Q}_p)//\overline{K_0(p^n)},η)$ consisting of functions supported in $\overline{\mathrm{SL}_2(\mathbb{Z}_p)}$, 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 $\operatorname{Ind}_{\overline{K_0(p^n)}}^{\overline{\mathrm{SL}_2(\mathbb{Z}_p)}}η$, determining the dimensions of its irreducible constituents and their associated Hecke eigenvalues. This Hecke subalgebra acts on the $(\overline{K_0(p^n)},η)$-fixed spaces in irreducible admissible genuine representations of $\widetilde{\mathrm{SL}}_2(\mathbb{Q}_p)$. 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 $\mathcal{W}_{n-1}$ to Ishimoto's local realization of Ueda's twisting operator.