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 on \( \GSp(4) \) over a totally real field, such that is regular algebraic (that is, is cohomological). When is globally generic (that is, has a non-vanishing Fourier coefficient), and 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 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 has nonzero vectors fixed by a non-special maximal compact subgroup at , the corresponding monodromy operator at has rank at most one.
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Claus M. Sorensen, Galois representations attached to Hilbert-Siegel modular forms. Doc. Math. 15 (2010), pp. 623–670DOI 10.4171/DM/309