The algebraic theory of non-archimedean intertwining periods

Abstract

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 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. In the group case it recovers at once the results of Dat on algebraic intertwining operators. Our 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, we obtain explicit polynomial bounds for the singularities of intertwining periods, and compute them in a concrete case.

arXiv commentAI tools have been used; see the corresponding section for details. In this second version, we have replaced the application to quaternionic symplectic models, as we found a simpler proof of the conjecture it was used to establish. We also recover Dat's results on algebraic intertwining operators as a direct application of our algebraic theory