Cuspidal representation

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In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L2 spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups.

When the group is the general linear group \operatorname{GL}_2, the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation.

In detail, let G be a connected reductive algebraic group over an algebraic number field F. Let \mathbb{A} be the adele of F. Let Z be the center of G. Let ω be a Hecke character. The character can be regarded as a character over

Z(\mathbb{A}).

Let

L^2(Z(\mathbb{A}) G(F) \backslash G(\mathbb{A}), \omega) be the space

of measurable functions on

G(F) \backslash G(\mathbb{A})

satisfying

\varphi(zx) = \omega(z) \varphi(x), z \in Z(\mathbb{A}), x \in G(\mathbb{A})

and

\int_{Z(\mathbb{A}) G(F) \backslash G(\mathbb{A})} |\varphi(x)|^2 dx < \infty.

There is a representation R of G(\mathbb{A}) (called right regular representation) on the space

L^2(Z(\mathbb{A}) G(F) \backslash G(\mathbb{A}), \omega).

The representation is defined by

R(g)\varphi(x) = \varphi(xg).

A function in

L^2(Z(\mathbb{A}) G(F) \backslash G(\mathbb{A}), \omega)

is a cuspidal function if for every parabolic subgroup P of G

\int_{N(F) \backslash N(\mathbb{A})} f(nx) dn = 0.

for almost every x, where N is the unipotent radical of P.

Denote the space of cuspidal functions by

L^2_0(\omega).

It is easily shown that the space is stable under the right regular representation.

It can be shown that the subspace of cuspidal functions can be decomposed discretely, i.e. it can be written as a direct sum of represntations of G(\mathbb{A}). Those irreducible subrepresentations of L^2_0(\omega) are called cuspidal representations.