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.
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 {{harvs|txt|authorlink=Hervé Jacquet| last=Jacquet|first=Hervé|year=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.
References
- {{Citation | last1=Casselman | first1=William A. | authorlink1=Bill Casselman (mathematician) | editor1-last=Lehto | editor1-first=Olli | editor1-link=Olli Lehto | title=Proceedings of the International Congress of Mathematicians (Helsinki, 1978) | chapter-url=http://mathunion.org/ICM/ICM1978.2/ | publisher=Acad. Sci. Fennica | location=Helsinki | isbn=978-951-41-0352-0 | mr=562655 | year=1980 | chapter=Jacquet modules for real reductive groups | pages=557–563 | access-date=2011-06-21 | archive-url=https://web.archive.org/web/20171114202000/http://www.mathunion.org/ICM/ICM1978.2/ | archive-date=2017-11-14 | url-status=dead }}
- {{Citation | last1=Jacquet | first1=Hervé | authorlink1=Hervé Jacquet| editor1-last=Gherardelli | editor1-first=F. | title=Theory of group representations and Fourier analysis (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Montecatini Terme, 1970) | publisher=Edizioni cremonese | location=Rome | isbn=978-3-642-11011-5 | doi=10.1007/978-3-642-11012-2 | mr=0291360 | year=1971 | chapter=Représentations des groupes linéaires p-adiques | pages=119–220}}
- {{Citation | last1=Bump | first1=Daniel | authorlink = Daniel Bump | title=Automorphic forms and representations | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-55098-7 |mr=1431508 | year=1997 | volume=55 | doi=10.1017/CBO9780511609572}}