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 (
gp)
p in
G(
A) with
g =
gp 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 •
f(γ
g) =
f(
g) for all γ ∈
G(
K) •
f(
gz) =
f(
g)ω(
z) for all
z ∈
Z(
A) • \int_{Z(\mathbf{A})G(K)\,\setminus\,G(\mathbf{A})}|f(g)|^2\,dg • \int_{U(K)\,\setminus\,U(\mathbf{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 \cdot u)(x) = u(xg), \qquad 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(\mathbf{A}),\omega)=\widehat{\bigoplus}_{(\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==