The algebraic theory of non-archimedean intertwining periods

We develop an algebraic theory of local intertwining periods for symmetric pairs $(G,H,\theta )$ over non-archimedean local fields $F$ of characteristic different from $2$, and finite-length representations of Levi subgroups of $G$ with coefficients in an algebraically closed field $R$ of characteristic different from the residual characteristic of $F$. For open orbits this algebraic version is essentially contained in the work of Blanc and Delorme. The algebraic theory relies on a crucial new uniqueness result, extending a result of Blanc and Delorme for open periods, which asserts that an admissible intertwining period is the only $H$-equivariant linear form extending the corresponding integral on the space of well-supported sections. Moreover the algebraic construction does not require the unimodularity hypothesis imposed in the work of Matringe–Offen–Yang. When $R=\mathbb{C}$, we also remove this assumption in the classical non-archimedean analytic setting, and prove convergence of intertwining periods for all symmetric spaces. At the same time we extend the context of the whole theory in the following manner: we allow $F$ to be of positive odd characteristic, we allow twisting characters in the periods, and we allow $H$ to lie between $({\mathbb{G}}^{\theta }{)}^{\circ }(F)$ and $G^{\theta }$. Finally, in the algebraic setting, as in the work of Blanc and Delorme on open periods, we obtain explicit bounding polynomials for the singularities of the intertwining period. Such bounds are a key ingredient of a new proof of the cuspidal case of a conjecture of D. Prasad on quaternionic symplectic models.