AI 中文总结
本文确定了$\text{SL}_2(\boldsymbol{Q}_p)$的metaplectic二重覆盖在奇素数$p$处的Hecke子代数，研究其局部新形式，完成相关算子比较，为半整数权新形式理论提供奇素数局部对应。
AI 中文摘要
针对奇素数$p$，本文确定了$\text{SL}_2(\boldsymbol{Q}_p)$的metaplectic二重覆盖在同余子群$K_0(p^n)$处的显式紧Hecke子代数，并利用该子代数研究指定二次型的局部新形式。本文描述了对应的双陪集、生成元、关系、特征及相应的$\boldsymbol{\bar{K}}$-型，计算了所得Hecke算子对主系列、Weil表示、Steinberg表示和超尖点表示中$(K_0(p^n),\boldsymbol{\bar{\tau}})$同型向量的作用。因此，全文涉及的导子为$\boldsymbol{\bar{\tau}}$-导子，而非所有特征的最小值。在超尖点情形下，metaplectic计算被约化为对应的线性强尖点型，明确区分了非分歧和分歧$L$-包。本文还将算子$W_{m-1}$与Ishimoto给出的Ueda扭算子的局部实现进行比较：在固定水平压缩和提升的$\text{GL}_2$-共轭后，两个作用在显式标量和依赖奇偶性的型变换下一致。这些结果为半整数权新形式理论和减空间中使用的Hecke代数方法提供了奇素数情形的局部对应。
英文摘要
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.
Comments61 pages. Revised exposition and notation; updated Section 5 and references
