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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 .

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

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