Jacobi descent charts and logarithmic quotient coordinates for split symmetric spaces

Abstract

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 $\theta$-split torus. Motivated by the singular contribution near the nilpotent fiber on the $\theta$-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 \[ Q(q)=\sum_{\alpha\in\Delta}(\gamma_\alpha n_{-\alpha}+q_\alpha n_\alpha) \] 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 \'etale. 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 $q_\alpha=z_\alpha^2$, the quotient Jacobian cancels the relative Weyl discriminant exactly: \[ \frac{ds}{|D_H^G|^{1/2}}=C\prod_{\alpha\in\Delta}d^\times z_\alpha. \] 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 $\operatorname{val}(z_\alpha)$. 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.