Jacquet module

In mathematics, the Jacquet module is an module used in the study of automorphic representations. The Jacquet functor is the functor that sends a linear representation to its Jacquet module. They are both named after Hervé Jacquet.

Definition

The Jacquet module J(V) of a representation (π,V) of a group N is the space of co-invariants of N; or in other words the largest quotient of V on which N acts trivially, or the zeroth homology group H0(N,V). In other words, it is the quotient V/VN where VN is the subspace of V generated by elements of the form π(n)v - v for all n in N and all v in V.

The Jacquet functor J is the functor taking V to its Jacquet module J(V).

Applications

Jacquet modules are used to classify admissible irreducible representations of a reductive algebraic group G over a local field, and N is the unipotent radical of a parabolic subgroup of G. In the case of p-adic groups, they were studied by Hervé Jacquet (1971).

For the general linear group GL(2), the Jacquet module of an admissible irreducible representation has dimension at most two. If the dimension is zero, then the representation is called a supercuspidal representation. If the dimension is one, then the representation is a special representation. If the dimension is two, then the representation is a principal series representation.

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References
  • Casselman, William A. (1980), "Jacquet modules for real reductive groups", in Lehto, Olli (ed.), Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Helsinki: Acad. Sci. Fennica, pp. 557–563, ISBN 978-951-41-0352-0, MR 0562655, archived from the original on 2017-11-14, retrieved 2011-06-21
  • Jacquet, Hervé (1971), "Représentations des groupes linéaires p-adiques", in Gherardelli, F. (ed.), Theory of group representations and Fourier analysis (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Montecatini Terme, 1970), Rome: Edizioni cremonese, pp. 119–220, doi:10.1007/978-3-642-11012-2, ISBN 978-3-642-11011-5, MR 0291360
  • Bump, Daniel (1997), Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, doi:10.1017/CBO9780511609572, ISBN 978-0-521-55098-7, MR 1431508
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