Analytic continuation of differential operators and applications to Galois representations

E. Eischen, M. Flander, A. Ghitza, E. Mantovan, A. McAndrew
The main goal of this paper is to describe the effect of certain differential operators on mod $p$ Galois representations associated to automorphic forms on unitary and symplectic groups. As intermediate steps, we analytically continue the mod $p$ reduction of certain $p$-adic differential operators, defined a priori only over the ordinary locus in some of the authors' earlier work, to the whole Shimura variety associated to a unitary or symplectic group; and we explicitly describe the commutation relations between Hecke operators and our differential operators. The motivation for this investigation comes from the special case of $GL(2)$, where similar operators were used to study the weight part of Serre's conjecture. In that case, one has a convenient description in terms of $q$-expansions, but such a formula-driven approach is not readily accessible in our settings. So, building on ideas of Gross and Katz, we pursue an intrinsic approach that avoids a need for $q$-expansions or analogues.