Jacquet module

In mathematics, the Jacquet module is a 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.

The Jacquet module J(V) of a representation (π,V) of a groupN 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/V_N where V_N 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).

Jacquet modules are used to classify admissible irreducible representations of a reductive algebraic groupG 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 HervéJacquet (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.

• 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 , archivado del original el 14-11-2017 , recuperado el 21-06-2011 
• Jacquet, Hervé (1971), "Représentations des groupes linéaires p-adiques", en Gherardelli, F. (ed.), Teoría de las representaciones de grupo y análisis de Fourier (Centro Internaz. Mat. Estivo (CIME), II Ciclo, Montecatini Terme, 1970) , Roma: Edizioni cremonese, págs. 119– 220, doi : 10.1007/978-3-642-11012-2 , ISBN  978-3-642-11011-5, MR 0291360 
• Bump, Daniel (1997), Formas y representaciones automórficas , Cambridge Studies in Advanced Mathematics, vol.  55, Cambridge University Press , doi : 10.1017/CBO9780511609572 , ISBN 978-0-521-55098-7, MR 1431508