Galois Representations Attached to Hilbert-Siegel Modular Forms

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 forms $\pi$ on $\GSp(4)$ 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 $\GSp(4)$ defined by Gan and Takeda in a recent preprint [GT]. Over $\Q$, 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 $\GL(4)$ 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.

2010 Mathematics Subject Classification: 11F33, 11F41, 11F70, 11F80

Keywords and Phrases: Galois representations, Hilbert-Siegel modular forms, monodromy

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