分裂对称空间的雅可比下降图与对数商坐标
Jacobi descent charts and logarithmic quotient coordinates for split symmetric spaces
AI总结：
本文针对等秩分裂对称对，构造了趋近幂零纤维的正则族的有限下降图，利用库默理论分类相关覆盖，得到嘉当提升的显式形式，证明了商密度的乘法哈尔测度性质。
AI中文摘要：
设F是特征为零的非阿基米德局部域，其余特征异于2；(G,H)是等秩分裂对称对，其中G为分裂半单伴随群，且具有F-分裂的极大θ-分裂环面。受无穷小局部相对迹公式中θ-固定侧幂零纤维附近奇异贡献的启发，我们为h中趋近该纤维的正则族构造有限下降图。在每个图上，一个定义在F上的嘉当提升将稀疏雅可比族Q(q)=∑_{α∈Δ}(γ_α n_{-α}+q_α n_α)共轭到h中。该族对所有q均正则，且取主幂零值Q(0)。伴随商在该族上的限制的一般秩等于G的奇数指数个数；对分裂对称对，此秩等于H的秩，故等秩情形下该映射一般为有限平展映射。库默理论对定义嘉当提升的扭曲平方根覆盖进行分类，这些覆盖穷尽有理分支。在每个分支上，令q_α=z_α²，商雅可比恰好抵消相对外尔判别式：ds/|D_H^G|^{1/2}=C∏_{α∈Δ}d^×z_α。因此，归一化的H°-商密度是参数环面上的乘法哈尔测度，在所有有理扭转上一致。经有限开闭细化后，相关岩泽高度在赋值val(z_α)上分段仿射。对于(Sp_{2n},GL_n)，该构造在赫维茨坐标中显式，且嘉当提升通过可交换长根SL₂子群分解。
英文摘要：
Let $F$ be a non-Archimedean local field of characteristic zero and residue characteristic different from $2$, and let $(G,H)$ be an equal-rank split symmetric pair with $G$ split semisimple and adjoint and with an $F$-split maximal $θ$-split torus. Motivated by the singular contribution near the nilpotent fiber on the $θ$-fixed side of the infinitesimal local relative trace formula, we construct finite descent charts for regular families in $\mathfrak h$ approaching that fiber. On each chart, a Cartan lift defined over $F$ conjugates the sparse Jacobi family \[ Q(q)=\sum_{α\inΔ}(γ_αn_{-α}+q_αn_α) \] into $\mathfrak h$. The family $Q(q)$ is regular for every $q$ and takes the principal nilpotent value $Q(0)$. The restriction of the adjoint quotient to this family has generic rank equal to the number of odd exponents of $G$. For split symmetric pairs this rank equals $\operatorname{rank}H$; hence in equal rank the map is generically finite étale. Kummer theory classifies the twisted square-root covers on which the Cartan lifts are defined, and these covers exhaust the rational branches. On every branch, with $q_α=z_α^2$, the quotient Jacobian cancels the relative Weyl discriminant exactly: \[ \frac{ds}{|D_H^G|^{1/2}}=C\prod_{α\inΔ}d^\times z_α. \] Thus the normalized $H^\circ$-quotient density is a multiplicative Haar measure on the parameter torus, uniformly across all rational twists. After a finite clopen refinement, the associated Iwasawa heights are piecewise affine in the valuations $\operatorname{val}(z_α)$. For $(\operatorname{Sp}_{2n},\operatorname{GL}_n)$, the construction is explicit in Hurwitz coordinates, and the Cartan lift factors through commuting long-root $\operatorname{SL}_2$-subgroups.
