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 $\mathrm{GL}(n)$, 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 forms $\pi$ on $\mathrm{GSp}(4)$ over a totally real field, such that ${\pi }_{\infty }$ is regular algebraic (that is, $\pi$ is cohomological). When $\pi$ is globally generic (that is, has a non-vanishing Fourier coefficient), and $\pi$ has a Steinberg component at some finite place, we associate a Galois representation compatible with the local Langlands correspondence for $\mathrm{GSp}(4)$ defined by Gan and Takeda in a recent preprint [GT]. Over $\mathbb{Q}$, for $\pi$ as 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 $\mathrm{GL}(4)$ 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]: If $\pi$ has nonzero vectors fixed by a non-special maximal compact subgroup at $v$, the corresponding monodromy operator at $v$ has rank at most one.