In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in
When the group is the general linear group
Formulation
Let G be a reductive algebraic group over a number field K and let A denote the adeles of K. Let Z denote the centre 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 measurable complex-valued functions, f, on G(A) satisfying
- f(γg) = f(g) for all γ ∈ G(K)
- f(gz) = f(g)ω(z) for all z ∈ Z(A)
-
∫ Z ( A ) G ( K ) ∖ G ( A ) | f ( g ) | 2 d g < ∞ -
∫ U ( K ) ∖ U ( A ) f ( u g ) d u = 0 for all unipotent radicals, U, of all proper parabolic subgroups of G(A).
This is called the space of cusp forms with central character ω on G(A). A function occurring in such a space is called a cuspidal function. This space is a unitary representation of the group G(A) where the action of g ∈ G(A) on a cuspidal function f is given by
The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces
where the sum is over irreducible subrepresentations of L20(G(K) \ G(A), ω) and mπ are positive integers (i.e. each irreducible subrepresentation occurs with finite multiplicity). A cuspidal representation of G(A) is such a subrepresentation (π, V) for some ω.
The groups for which the multiplicities mπ all equal one are said to have the multiplicity-one property.