I ntnat. J. Math.
Math. Si.
235
(1978)25-244
Vol.
A STONE-WEIERSTILASS THEOREM FOR GROUP REPRESENTATIONS
JOE REPKA
Department of Mathematics
University of Toronto
Toronto, Ontario, Canada MSS IAI
(Received January 16, 1978)
ABSTRACT.
It is well known that if G is a compact group and
a faithful
(unitary) representation, then each irreducible representation of G occurs in the
tensor product of some number of copies of
and its contragredlent.
We gener-
allze this result to a separable type I locally compact group G as follows:
let
be a faithful unitary representation whose matrix coefficient functions vanish
at infinity and satisfy an
isomorphism, the regular
appropriate integrabillty condition.
Then, up to
representation of G is contained in the direct sum of
all tensor products of finitely many copies of
and its contragredlent.
We apply this result to a symplectlc group and the Well representation assoclated to a quadratic form.
As the tensor products of such a representation are
also Well representations (associated to different forms), we see that any discrete
series representation can be realized as a subrepresentatlon of a Well represen-
tatlon.
JOE REPKA
236
i.
INTRODUCTION.
Let G be a separable locally compact group, and n a representation of G
("representation" will always mean a strongly continuous unitary representation
on a separable Hilbert
space).
For every pair of non-negative integers m, n,
not both zero, let
m,n
)
(
(
)
where n occurs m times, and its contragredient n occurs n times.
Our starting point is the following easy fact.
THEOREM i.
Let G be a compact group and n a faithful representation.
Then every irreducible representation of G occurs in
m,n>0
where the prime means we omit
PROOF.
(m, n)
m,n
(0, 0)
from the sum.
Consider the set of linear combinations of the matrix coefficient
functions of the representation
(.
These functions form an algebra, are
closed under complex conjugation, separate points of G,
vanish at any point of G.
and do not all
By the Stone-Weierstrass Theorem, they are sup-
norm dense in the continuous functions on G,
and hence dense in
L2(G)
But, by the Peter-Weyl Theorem, a representation whose coefficient functions
are dense in
REMARK.
L2(G)
must contain every irreducible representation.
Since, as is well known,
8 z
contains the trivial represent-
tation, it is easy to see that every irreducible representation occurs not
Just
once, but countably infinitely many times in
This theorem, in a slightly different form, is originally due to Burnside,
in the case of a finite group G
(see, e.g.,
Curtis and Reiner
[2],
Theorem
STONE-WEIERSTRASS THEOREM FOR GROUP REPRESENTATIONS
32.9
237
namely:
THEOREM 2.
tation.
(Burnside).
Let G be a finite group,
a faithful represen-
Then every irreducible representation of G occurs in the tensor
product of some number of copies of
Rather than prove Burnside’s theorem directly, we obtain it as an obvious
special case of the following.
THEOREM 3.
Let n be a positive integer.
of whose elements have order dividing n.
of G,
Let G be a compact group, all
If
is a faithful representation
then every irreducible representation of G occurs in the tensor
product of some number of copies of
PROOF.
The argument is the same as that for Theorem I; the only thing
that is not obvious is that the algebra spanned by coefficient functions is
closed under complex conjugation.
coefficients of
(g)
are
Just
But the complex conjugates of the matrix
(since w is unitary), and the matrix coefficients of
Just
(g)-I rearranged
?(g)-I (g )n-I are
the matrix coefficients of
sums of products of matrix coefficients of
(g)
so we are done.
We generalize Theorem 1 to a non-compact group.
2.
PRELIMINARIES.
In this section we establish two results needed for the proof of the main
theorem.
Let
As above, G is a separable locally compact group.
be a continuous square-integrable function on G.
open set where
is not zero.
speak of the closed subspace
vanish off
U
Let U be the
We may restrict Haar measure to
L2(U)
2
c
(G)
_1",
U
and
consisting of functions which
38
LEMMA
JO REPKA
.
A
Let
be an algebra of continuous functions on G which vanish
A
at infinity and separate points, and suppose
Let
Jugation.
Then
"
PROOF
U
A
Let
be as above.
{"
f
$
C (U)
A}
f
e > 0
the Stone-Weierstrass Theorem there exists
"
I
complete.
Now let
$/
Then
A
$’
A
Since
C
C
(U)
L
is dense in
Suppose the intersection of
functions of
spans a dense subspace of
G
L2(G)
L2(G)
PROOF.
If
is in
TV
The domain of
integrable; it is a
that
Tv
G-isomorphism
UV
2(U)
G
Iv
we define a linear map
the proof is
H
(i.e.
contained
SU,v(g)
from some subspace
SW,V
W
H
such that
SW,V
is square-
It is easy to check
U
G-map; hence we may apply Schur’s Lemma to conclude that
G-subspaces
/
V
W
whose span is dense in
L2(G)
V
v
c
H
L2(G)
W
2
c
v __L (G)
and a
surJective unitary
v contains the function SU,V
(countably many) coefficient functions
Moreover,
V
By hypothesis we may choose
dense subspace of
<
(up to isomorphism)
Then
G-invariant subspace, containing
V
II$’ $/II=
().
will be all vectors
is a closed
there are closed
and by
are such that the coefficient function
L2(G)
L2(G) by Iv(w)
<G(g),V>
H to
H
e
U
(U)
with the set of coefficient
the regular representation of G is quasi-contained in
in a direct sum of copies of
c
and we have approximated
be a representation of G on a Hilbert space
LEMMA 5.
of
C
such that
I ’)I12 < IIII 2 "e
I12
Consequently,
L2(U)
is dense in
and fix
c
by a function in
is closed under complex con-
W
The corresponding subspaces
and, sticking together the maps
W
v
U
1
thus span a
and
239
STONE-WEIERSTRASS THEOREM FOR GROUP REPRESENTATIONS
applying Schur’s Lemma again, we get a unitary
@ V c_C @
V
V
V
3.
H
L2(G)
G-isomorphism of
a countable direct sum of copies of
into
H
THE MAIN RESULT.
We shall prove two weaker versions of our main result as a preliminary step.
PROPOSITION 6.
Let G be a separable locally compact group, and
faithful representation of G.
of
n
LP(G)
is in
a
Suppose that a non-trivial coefficient function
for some finite p,
and that all the coefficient
functions vanish at infinity.
Then the regular representation of G is quasi-contained in
m,nZ0
By Lemma 5, it suffices to find coefficient functions of
PROOF.
L2(G)
span is dense in
is in
Lp(G)
Suppose
Then some power
coefficient function of
w
n,0
coefficient functions of
By Lemma
U
{g
h, A n
L2(G)
A n
Since
cn
is a coefficient function of
L2(G)
open, any element of
belonging to spaces
n. A
note
contains a dense set of
2
(
L2(U)
is invariant under translations by
L2(g.U)
L2(G)
2
L (g" U)
for any
n
Since
A
which
is a
spanned by the
w
contains
G
n,0
where
as desired.
g
E
G
G
we see that it
But since
U
is
can be approximated by a finite sum of elements
for appropriate choices of
A
g
G
Such a
2
L (G)
We have approximated an arbitrary element of
L (G)
2(G)
we see that the algebra
sum can be approximated by something in
A
L
is in
of
contains
contains a dense subset of
of
whose
O}
G: (g)
E
m,n
L2(G)
by a sum of elements
240
JOE EEPKA
In fact, we can strengthen this result slightly, as follows:
PROPOSITION 7.
of
G
u=
w
c be as above, and let
G
Then the regular representation of
PROOF.
n
Let
be any representation
T
is quasi-contained in
The proof is analogous to that of Proposition
0" on where 0
{G: *0"*() }
by
6;
is any coefficient function of
we
Just
T
e c
replace
and let
"r
|
We are now ready to prove the main result, which is essentially Proposition
7
with "quasi-contained" changed to "contained".
THEOREM 8.
Let
G be a separable type I locally compact group,
a
faithful representation whose coefficient functions vanish at infinity and
one of whose non-trivial
Let
p
T
cfficient functions is in
LP(G)
for some finite
be any representation.
Then the regular representation of
G
is
contained countably infinitely many times in
(up to
T
e
unitary isomorphism)
O
@’
T
m,nO m,n
)
We shall show that every subrepresentation of the regular
PROOF.
representation occurs not
Just
once but infinitely many times in
T @
c
It will suffice to consider multiplicity-free subrepresentations, i.e. those
of the form
d()
where
G
tions of
is the set of equivalence classes of unitary irreducible representa-
G
and
is some finite Radon measure on
continuous with respect to the Plancherel measure.
e > 0
By Proposition 7, we know that
G
which is absolutely
Fix such a
is contained in
and fix
c
so there
241
STONE-WEIERSTItASS THEOREM FOR GROUP REPRESENTATIONS
is a positive integer
M(1)
contains most of
namely,
measurable set
X1
c_ 8
such that
@
< a/4
(R)’
M(1)m,n m,n
By Proposition 4, this
M(1),M(1)
@
M(2)
and a positive integer
(R)’
M(1)m,n<M(2) m,n
I"
that
T
But
T @
Gk
G
@’
l
@
u2
@
uS
infinitely many copies of
(G \ X) <
4.
2
and since
1
XS
X2
X1
cG
M(1) < M(2) < M(3) <
and integers
M(k-l)m,n<M(k) m,n
U
@
X2
[
contains
X 2 c_ G with
d()
contains
< 2 -k.
(\Xk)
(
such that
Continuing inductively, we find measurable sets
such that
for some
so we may find a measurable set
representation contains
(\X2)
d()
< /2
(M(1),M(1))
(m,n)
the prime means omit
I’
X1
(\XI)
such that
| @’
@
1
contains
(1
Now consider the representation
[
@
Xk
contains
@
so
d()
where
such
dB()
u contains countably
T
X
Xk
Now
was arbitrary, the theorem is proved.
WElL REPRESENTATIONS AND THE DISCRETE SERIES.
The Weil representation has long been known as a fruitful place for
realizing various irreducible representations
Howe, [5]).
(see, e.g., Gelbart, [3]
and
We shall apply Theorem 8 to show that any discrete series
representation of a symplectic group can be so
realized.
In (Weil, [7]) there is constructed a representation of the symplectic
group associated to a quadratic form
representation with itself
and/or
The tensor products of this
its contragredient are again Weil represen-
tations, those associated to forms constructed by taking direct sums of
copies of
+
Since the discrete series representations of
G
are sub-
JO PKA
4
representations of the regular representation, we have as an immediate conse-
quence of Theorem 8"
THEOREM 9.
G
of
With
G
as above, any representation of the discrete series
is a subrepresentation of some Well representation of
(associated
G
to some quadratic form).
COMPLEMENTS.
5.
In this section we make some remarks and give some examples.
that its matrix coefficient functions must
The conditions on
vanish at infinity, and that at least one of them must be in some
seem
LP(G)
We remark that they are necessary, for
rather peculiar at first.
example, to exclude finite dimensional representations of a non-compact group
(indeed,
if
were finite dimensional, then
would be a direct sum of
finite dimensional representations, and could not possibly contain the
regular representation).
On the other hand, though something of this sort
is needed, the conditions we have given can be altered or weakened in
various ways; we give two examples.
Firstly, it is not necessary to know
It would suffice to
that the coefficient functions all vanish at infinity.
(e.g.
know that the ones belonging to some dense set of vectors did so
K-finite
vectors),
or even
Just
that one coefficient function achieves its
maximum absolute value on a compact set.
hold in these cases.
on
Suitable modifications of Lemma
Secondly, we note that in Theorem
could without difficulty be replaced with an
L
2
8
the
L
p
condition
condition on
x
Finally, we observe that neither of the conditions is actually as
unnatural as it may first seem.
They are both satisfied in a great many
cases, including all connected semi-simple Lie groups with finite centre
(see
Cowling,
[i]
and
Howe, []).
STONE-WEIERSTRASS THEORE FOR GROUP REPISENTATIONS
EXAMPLE i
Then
X1
Let
G be the circle group; let
Xn
be the character
are the only faithful irreducible representations
X_ 1
243
e
ine
(though
of
course there are many other faithful representations), and if, for example,
we let
m,n
then
X1
m,n Xm-n
Every
Xk
equals some
(in fact infinitely many).
EXAMPLE 2.
[61
and
X_ 1
Let
G
SL2(R)
or
PSL2(R)
is any non-trivial irreducible unitary representation,
show that if
then the regular representation is contained in
also shows that
Then the results of Repka,
g
(R)’
m,n
@’
m,n_<B m,n
This example
may contain subrepresentations which are not
contained in the regular representation (this happens if
is some
"comple-
mentary series" representations).
REFERENCES
i.
Cowling, M. Sur l’Algbre de Fourier-StieltJes d’un Groupe Semi-simple,
preprint (1976).
2.
Curtis, C.W. and Reiner, I. Representation Theory of Finite Groups and
As.sociative Algebras, Intersiene’ ’(i’966).’
3.
Gelbart, S. Holomorphic Discrete Series for the Real Symplectic Group,
Inv. Math. 19 (1973) h9-58.
Howe, R.
On the Asymptotic Behaviour of Matrix Coefficients, preprint
().
5.
Howe, R.
Invariant Theory and Duality for Classical Groups over Finite
Fields, preprint
(1976).
6.
Repka, J. Tensor Products of Unitary Representations of SL2(R)
Bull Amer. Math. Soc. 82 (1976) 930-932; also to appear, Amer. J.
Math.
7.
Weil, A.
Sur Certains Groupes d’Op6rateurs Unitaires, Acta Math. ll___l
196h lh 3-211.
JOE REPKA
244
KEY WORDS AND PHRASES.
Locy ompao. gu,
sentatn, teor product, reg
AMS(MOS) SUBJECT CLASSIFICATION (1910) COES.
22DI0, 22E45.