Heisenberg varieties and the existence of de Rham lifts

Abstract: Let $F$ be a $p$-adic field. For certain non-abelian nilpotent algebraic groups $U$ over $\bar{\mathbb{Z}}p$ equipped with $\operatorname{Gal}_F$-action, we study the associated {\it Heisenberg varieties} which model the non-abelian cohomology set ``$H^1(\operatorname{Gal}_F, U)$''. The construction of Heisenberg varieties involves the Herr complexes including their cup product structure. Write $U_n$ for a quasi-split unitary group and assume $p\ne 2$. We classify mod $p$ Langlands parameters for $U_n$ (quasi-split), $\operatorname{SO}{2n+1}$, $\operatorname{SO}{2n}$, $\operatorname{Sp}{2n}$, $\operatorname{GSpin}{2m}$ and $\operatorname{GSpin}{2m+1}$ (split) over $F$, and show they are successive Heisenberg-type extensions of elliptic Langlands parameters. We employ the Heisenberg variety to study the obstructions for lifting a non-abelian cocycle along the map $H^1(\operatorname{Gal}_F, U(\bar{\mathbb{Z}}_p))\to H^1(\operatorname{Gal}_F, U(\bar{\mathbb{F}}_p))$. We present a precise theorem that reduces the task of finding de Rham lifts of mod $p$ Langlands parameters for unitary, symplectic, orthogonal, and spin similitude groups to the dimension analysis of specific closed substacks of the reduced Emerton-Gee stacks for the corresponding group. Finally, we carry out the dimension analysis for the unitary Emerton-Gee stacks using the geometry of Grassmannian varieties. The paper culminates in the proof of the existence of potentially crystalline lifts of regular Hodge type for all mod $p$ Langlands parameters for $p$-adic (possibly ramified) unitary groups $U_n$. It is the first general existence of de Rham lifts result for non-split (ramified) groups, and provides evidence for the topological Breuil-M\'ezard conjecture for more general groups.