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 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).
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.
- 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, vol. 55, Cambridge University Press, doi: 10.1017/CBO9780511609572, ISBN 978-0-521-55098-7, MR 1431508
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 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).
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.
- 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, vol. 55, Cambridge University Press, doi: 10.1017/CBO9780511609572, ISBN 978-0-521-55098-7, MR 1431508