In mathematics, the Jacquet module is a

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

used in the study of

automorphic representations In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

. The Jacquet functor is the

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

that sends a

linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...

to its Jacquet module. They are both named after

Hervé Jacquet Hervé Jacquet is a French American mathematician, working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his results play a central role in modern numb ...

Definition

The Jacquet module ''J''(''V'') of a representation (''π'',''V'') of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

''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 In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topol ...

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'').

Applications

Jacquet modules are used to classify admissible irreducible representations of a

reductive algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

''G'' over a

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compa ...

, and ''N'' is the

unipotent radical In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipote ...

of a

parabolic subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...

of ''G''. In the case of ''p''-adic groups, they were studied by . For the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible ...

GL(2), the Jacquet module of an admissible irreducible representation has

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

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=Bump , first1=Daniel , authorlink = Daniel Bump , title=Automorphic forms and representations , publisher=

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

, series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-55098-7 , mr=1431508 , year=1997 , volume=55 , doi=10.1017/CBO9780511609572 Representation theory