Galois representations attached to Hilbert–Siegel modular forms

Abstract

This article is a spinoff of the book of Harris and Taylor (HT), in which they prove the local Langlands conjecture for GL(n), and its companion paper by Taylor and Yoshida (TY) on local-global compatibility. We record some consequences in the case of genus two Hilbert-Siegel modular forms. In other words, we are concerned with cusp formson GSp(4) over a totally real field, such that �1 is regular algebraic (that is, � is cohomological). Whenis globally generic (that is, has a non-vanishing Fourier coefficient), andhas a Steinberg component at some finite place, we associate a Galois representation compatible with the local Langlands correspondence for GSp(4) defined by Gan and Takeda in a recent preprint (GT). Over Q, foras above, this leads to a new realization of the Galois representations studied previously by Laumon, Taylor and Weissauer. We are hopeful that our approach should apply more generally, once the functorial lift to GL(4) is understood, and once the so-called book project is completed. An application of the above compatibility is the following special case of a conjecture stated in (SU): Ifhas nonzero vectors fixed by a non-special maximal compact subgroup at v, the corresponding monodromy operator at v has rank at most one.