Cuspidal representation explained

In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in

L ^{2} 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.\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.

Formulation

Let G be a reductive algebraic group over a number field K and let A denote the adeles of K. The group G(K) embeds diagonally in the group G(A) by sending g in G(K) to the tuple (g_p)_p in G(A) with g = g_p for all (finite and infinite) primes p. Let Z denote the center of G and let ω be a continuous unitary character from Z(K) \ Z(A)^× to C^×.Fix a Haar measure on G(A) and let L²₀(G(K) \ G(A), ω) denote the Hilbert space of complex-valued measurable functions, f, on G(A) satisfying

1. f(γg) = f(g) for all γ ∈ G(K)
2. f(gz) = f(g)ω(z) for all z ∈ Z(A)

\int _{Z(A)G(K)\setminusG(A)} |f(g)| ^{2dg} < inf ty \int _{U(K)\setminusU(A)} f(ug)du =0 for all unipotent radicals, U, of all proper parabolic subgroups of G(A) and g ∈ G(A).The vector space L²₀(G(K) \ G(A), ω) is called the space of cusp forms with central character ω on G(A). A function appearing in such a space is called a cuspidal function.

A cuspidal function generates a unitary representation of the group G(A) on the complex Hilbert space

V _{f} generated by the right translates of f. Here the action of g ∈ G(A) on V _{f} is given by(g ⋅ u)(x) = u(xg), u(x) = \sum _{j} c _{j} f(xg _{j)} \in V _{f} .

The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces

L ^{2} _{0 (G(K)\setminus} G( A ),\omega) = \widehat{ o plus} _{(\pi,V _{\pi)}} m _{\pi} V _{\pi} where the sum is over irreducible subrepresentations of L²₀(G(K) \ G(A), ω) and the m are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrepresentation (V) for some ω.

The groups for which the multiplicities m all equal one are said to have the multiplicity-one property.

See also

• Jacquet module

References

• James W. Cogdell, Henry Hyeongsin Kim, Maruti Ram Murty. Lectures on Automorphic L-functions (2004), Section 5 of Lecture 2.