On Matrix-Valued Square Integrable Positive Definite Functions

0.15 MB·English

by  Hongyu He

1 2
ON MATRIX VALUED SQUARE INTEGRABLE POSITIVE DEFINITE
FUNCTIONS
9 HONGYUHE
0
0
2 Abstract. In this paper, we study matrix valued positive definite functions on a unimodular group.
We generalize two important results of Godement on L2 positive definite functions to matrix valued
y L2 positive definite functions. We show that a matrix-valuedcontinuous L2 positivedefinite function
a can always be written as a convolution of a L2 positive definite function with itself. We also prove
M that, given two L2 matrix valued positive definite functions Φ and Ψ, RGTrace(Φ(g)Ψ(g)t)dg ≥ 0.
In addition this integral equals zero if and only if Φ∗Ψ = 0. Our proofs are operator-theoretic and
1
independent ofthegroup.

]
A
O
. 1. Introduction
h
t
a About 60 years ago, Godement published a paper on square integrable positive definite functions on
m a locally compact group ( [1]). In his paper, Godement proved that every continuous square integrable
[ positivedefinitefunctionhasanL2-positivedefinitesquareroot. Healsoproved,amongothers,thatthe
inner product between two positive definite L2-functions must be nonnegative. Godement’s results and
1
proofs werequite elegant. The purpose ofthis paper is to extend Godement’s theoremto matrix-valued
v
9 positive definite functions on unimodular groups. Obviously, the diagonal of matrix-valued positive
6 definite functions must all be positive definite. Yet, there is not much to say about the off-diagonal
1 entries and their relationship with diagonal entries. So Godement’s results do not carry trivially to the
0
matrix-valued case. The generalization of Godement’s result will likely involve revisiting and general-
.
5 izing Godement’s ideas to the matrix-valued case. This is what is done in this paper. We essentially
0 examine Godement’s argument as presented in [1] and Ch 13. of Dixmier [2] for matrix-valued positive
9
definite functions. Necessary modifications are made. Our results, we believe, are new.
0
:
v Let M (C) be the set of n n matrices. For a matrix A, let [A] be the (i,j)-th entry of A. Let
n ij
Xi G be a unimodular group. A×continuous function φ : G Mn(C) is said to be positive definite if for
any C Cn l and x G l , →
ar { i ∈ }i=1 { i ∈ }i=1
l
(C )tΦ(x−1x )C 0.
i i j j ≥
X
i,j=1
t
Take x = e and x = g. The above inequality implies that Φ(g) = Φ(g−1) (See for example, Prop.
1 2
2.4.6[6]). Whenn=1,ourdefinitionagreeswiththedefinitionofcontinuouspositivedefinitefunctions.
We denote the set of continuous matrix-valued positive definite functions by (G,M ).
n
P
Definition 1. Let L1 (G,M ) be the set of M -valued locally integrable functions on G. Let Φ
L1 (G,M ) acts on luoc C (nG,Cn) by [λ(Φ)(un)(x)] = n [Φ(g)] [u(g−1x)] dg. We will writ∈e
loc n ∈ c i j=1 ij j
P R
1ThisresearchispartiallysupportedbyNSFgrantDMS-0700809.
2
Keyword: Positivedefinitefunction,unimodulargroup,moderatedfunctions,squareintegrablefunctions,convolution
algebra,unitaryrepresentations.
1
2 HONGYUHE
λ(Φ)(u)(x)= Φ(g)u(g−1x)dg. Clearly, λ(Φ)u is continuous. We say that Φ is positive definite if
R
n
λ(Φ)(u),u = [λ(Φ)u(g)] [u(g)] dg 0
h i Z i i ≥
Xi=1 G
for all u C (G,Cn).
c
∈
We denote the set of matrix-valued positive definite functions by P(G,M ). Clearly, P(G,M )
n n
⊃
(G,M ) (see Prop 13.4.4 [2]).
n
P
Definition 2. A matrix-valued function Φ(x) is said to be square integrable, or simply L2 if [Φ] is in
ij
L2(G) for all (i,j). We denote the set of matrix-valued square integrable function by L2(G,M ). Define
n
t
Φ,Ψ = TrΦΨ dg.
h i Z
G
Put 2(G,M )=L2(G,M ) (G,M ) and P2(G,M )=L2(G,M ) P(G,M ).
n n n n n n
P ∩P ∩
Let Φ,Ψ L1 (G,M ). Define the convolution
∈ loc n
n
[Φ Ψ] = [Φ] [Ψ]
ij ik kj
∗ ∗
X
k=1
whenever the latter is well-defined.
Theorem [A] Let G be a unimodular locally compact group. Let Φ 2(G,M ). Then there ex-
n
∈ P
ists a Ψ P2(G,M ) such that Φ=Ψ Ψ.
n
∈ ∗
Theorem [B] Let G be a unimodular locally compact group. Let Φ,Ψ P2(G,M ). Then Φ,Ψ 0.
n
∈ h i≥
Theorem [C] Let G be a unimodular locally compact group. Let Φ,Ψ P2(G,M ). Then Φ,Ψ = 0
n
∈ h i
if and only of Φ Ψ=0.
∗
Our motivation comes from the theory of unitary representations of Lie groups. There are representa-
tionsthatappearasaspaceof“invariantdistributions”inaunitaryrepresentation(π, ). Toconstruct
π
H
a Hilbert inner product for the invariant distributions, one is often led to investigate whether
(1) (π(g)u,u)dg 0
Z ≥
G
when (π(g)u,u) is L1, u and G is unimodular. This problem is originally brought out in Gode-
π
∈ H
ment’sthesis[1]. NowitisknownthatforGamenable,theInequality(1)isalwaystrue(Prop18.3.6[2]).
AnamenablegroupischaracterizedbythefactthattheunitarydualisweaklycontainedinL2(G). Now
consider the other extreme, namely, G semisimple and noncompact. The inequality (1) is false in its
full generality. Yet, applying the result of this paper, we show that Inequality (1) holds if can be
π
H
written as a tensor product of two L2-representations of G and u is finite in the tensor decomposition.
In Cor. 2, we give a result about a certain integral related to Howe’s correspondence ( [5]). It is more
general than the results given in [4].
I should mention that matrix-valued positive definite functions have been studied for special classes
of locally compact groups, mostly Abelian groups. I would like to thank Prof. Kadison for his lectures
given at LSU in the Spring of 2008, for stimulating conversations and for encouraging me to write this
down.
ON MATRIX VALUED SQUARE INTEGRABLE POSITIVE DEFINITE FUNCTIONS 3
2. Convolution Algebras
A matrix-valued function on G is said to be in Lp if each entry is in Lp(G). Let Φ L1(G,M ).
n
∈
Define Φ L1 = [Φ]ij L1. Then L1(G,Mn) becomes a Banach algebra. We have
k k k k
L1(GP,M ) L1(G,M ) L1(G,M ); C (G,M ) C (G,M ) C (G,M ).
n n n c n c n c n
∗ ⊆ ∗ ⊆
For each u,v L2(G,Cn), define the standard inner product u,v = [u(g)] [v(g)] dg. obviously
∈ h i G i i i
L1(G,M ) acts on L2(G,Cn). Then map R P
n
λ:L1(G,M ) (L2(G,Cn))
n
→B
defines a bounded Banachalgebra isomorphism. Notice that if Φ L2(G,M ) and u L1(G,Cn), then
n
λ(Φ)u L2(G,Cn). ∈ ∈
∈
We define the operation on L (G,M ) by letting
loc n
∗
[Φ∗(g)] =[Φ(g−1)] .
ij ji
For u,v C (G,Cn), we have λ(Φ)u,v = u,λ(Φ∗)v . If Φ is positive definite, then
c
∈ h i h i
λ(Φ)u,u = λ(Φ)u,u = u,λ(Φ)u = λ(Φ∗)u,u .
h i h i h i h i
So λ(Φ Φ∗)u,u =0. Let u=v+tw(t R). Then
h − i ∈
0= λ(Φ Φ∗)(v+tw),v+tw = λ(Φ Φ∗)v,tw + λ(Φ Φ∗)tw,v .
h − i h − i h − i
We obtain λ(Φ Φ∗)v,w = λ( Φ+Φ∗)w,v = w,λ( Φ∗ +Φ)v = λ(Φ Φ∗)v,w . Hence λ(Φ
Φ∗)v,w muhst be−real for aill vh,w−C (G,Cn).iIt fohllows−that Φ∗ =iΦ. h − i h −
c
i ∈
Lemma 1. Let Ψ,Φ P2(G,M ). Then Φ=Φ∗, Ψ=Φ∗ and Φ,Ψ =Trace(Φ Ψ(e)).
n
∈ h i ∗
Proof: Our lemma follows from
(2)
Trace(Φ Ψ(e))= [Φ] (g)[Ψ] (g−1)dg = [Φ] (g)[Ψ] (g)dg = TrΦ(g)Ψ(g)tdg = Φ,Ψ .
∗ Z ij ji Z ij ij Z h i
Xi,j G Xi,j G G
Definition 3. Let Φ L2(G,M ). We say that Φ is moderated if λ(Φ) on C (G,Cn) is a bounded
n c
∈
operator in the L2 norm, i.e., there is a M such that
λ(Φ)u M u
k k≤ k k
for any u C (G,Cn).
c
∈
WhenΦismoderated,λ(Φ)|Cc(G,Cn) canbeextendedtoaboundedoperatoronL2(G,Cn)whichcoin-
cides with λ(Φ)L2(G,Cn). To see this, let ui u under the L2-norm with ui Cc(G,Cn). Then λ(Φ)ui
| → ∈
converges as a Cauchy sequence to v L2(G,Cn) since λ(Φ) is bounded. In particular, (λ(Φ)u )(g)
i
∈
converges in L2-norm to v(g) on any compact subset K. On the other hand, (λ(Φ)u )(g) converges to
i
λ(Φ)u uniformly on G, in particular, on K. So (λ(Φ)u)(g) = v(g) for g K almost everywhere. It
follows that v(g) = λ(Φ)u(g) almost everywhere. So λ(Φ)u L2(G,Cn). ∈In short, if Φ is moderated,
and u L2(G,Cn), then λ(Φ)(u) L2(G,Cn). The following∈is obvious.
∈ ∈
Lemma 2. Φ is moderated if and only if [Φ] are all moderated in L2(G).
ij
Let M(G,M ) be the space of moderated L2 functions on G. Lemma 13.8.4 [2] asserts that M(G)
n
∗
M(G) M(G) and λ :M(G) (L2(G)) is an algebra homomorphism. Therefore, we obtain
M(G)
⊆ | →B
Lemma 3. Let Φ,Ψ L2(G,M ). If Φ and Ψ are moderated, then Φ Ψ is also moderated. In addition
n
∈ ∗
λ(Φ Ψ)=λ(Φ)λ(Ψ).
∗
So M(G,M ) is a Hilbert algebra.
n
4 HONGYUHE
3. Positive Definite Functions
Let us recall some basic result from [2]. Let Φ,Ψ P(G,M ). We define an ordering Φ Ψ if
n
∈ (cid:22)
Ψ Φ P(G,M ). FortwoboundedoperatorX andY in ( ),wesaythatX Y ifY X ispositive
n
− ∈ B H (cid:22) −
(Ch 2.4 [6]). Y X is positive implies that Y X is self-adjoint (Prop. 2.4.6 [6]).
− −
Theorem 1 ( Prop. 16 [1] and 13.8.5, 13.8.4 [2]). Let Φ,Ψ be two moderated elements in P2(G,M ).
n
Suppose that Φ Ψ=Ψ Φ. Then Φ,Ψ 0. Let
∗ ∗ h i≥
Φ Φ ... Φ ...
1 2 n
(cid:22) (cid:22) (cid:22) (cid:22)
be an increasing sequence of moderated positive definite functions in L2(G). Suppose that Φ mutually
i
commute. If sup Φi L2 < , then Φ=limΦi exists in P2(G,Mn).
k k ∞
Proof: Φ Ψ=Ψ φ implies that λ(Φ)λ(Ψ) =λ(Ψ)λ(Φ) as bounded operators on L2(G,Cn). Since
∗ ∗
λ(Φ) and λ(Ψ) are both positive, they must be self-adjoint. So λ(Φ)λ(ψ) must be positive and self-
adjoint. In other words, λ(Φ Ψ) is positive on L2(G,Cn). In particular, it is positive with respect to
C (G,Cn). Hence Φ Ψ, as a∗matrix-valued continuous function, is positive definite. Φ Ψ(e) must be
c
∗ ∗
a positive semi-definite matrix. By Lemma (1)
Φ,Ψ =Trace(Φ Ψ)(e) 0.
h i ∗ ≥
Forj i,noticethat Φ 2 = Φ 2+ Φ ,Φ Φ + Φ Φ ,Φ + Φ Φ 2 Φ 2.The sequence
j i i j i j i i j i i
≥ k k k k h − i h − i k − k ≥k k
Φ is an increasing sequence with limit sup Φ . In particular,it is a Cauchy sequence. Notice that
i
{k k} k k
for j i Φ Φ 2 Φ 2 Φ 2. This implies that Φ is a Cauchy sequence in L2(G,M ).
j i j i i n
Let Φ≥be tkhe L−2-limkit o≤f kΦ k. F−orkevekry u C (G,Cn), sinc{e co}nvolution with a compactly supported
i c
{ } ∈
continuous function is bounded on L2(G), we obtain
0 λ(Φ )u,u λ(Φ)u,u .
i
≤h i→h i
It follows that Φ is positive definite. (cid:3)
Theorem 2 (Thm. 17 [1], Theorem 13.8.6 [2]). Suppose that Φ is a moderated element in P2(G,M )
n
such that Φ Θ with Θ a continuous positive definite function. Then there is a moderate element Ψ in
(cid:22)
P2(G,M ) such that Φ = Ψ Ψ in L2. In particular, Φ equals a continuous positive definite function
n
∗
almost everywhere and Ψ 2 Tr(Θ(e)).
k k ≤
In particular, if Φ is continuous, its square root Ψ exists.
The proof is almost the same as the proof of Theorem 13.8.6 in [2]. The original idea of Godement
is to construct an increasing sequence of positive definite moderated elements Ψ in L2(G,M ) that
k n
approaches the square root. In Dixmier’s book, Ψ = λ(Φ) p ( Φ ). Here p (t) is an increasing
k k k k kλ(Φ)k { k }
sequence of nonnegative polynomials on [0,1] such that p (t) √t on [0,1] and p (0) = 0. Es-
k k
→
sentially, we will have Ψ Ψ and Ψ Ψ Φ Θ. By taking the value at e, we have
k k+1 k k
(cid:22) ∗ (cid:22) (cid:22)
Trace(Ψ Ψ (e)) Trace(Θ(e)). By Lemma (1), Ψ is bounded by TrΘ(e). Since Ψ mu-
k k k k
∗ ≤ k k
tually commute, by Theorem 1, the L2-limit of Ψ exists. Put Ψ = lim pΨ . Our assertion then
k k→∞ k
follows. (cid:3)
4. Square Roots: Proof of Theorem A
Now we would like to give a proof of Theorem A. Our proof is somewhat different from the proof of
Theorem 13.8.6 given in [2]. The basic idea is the same, namely, to construct a sequence of moderated
continuous positive definite functions Φ Φ. Let Ψ be the square root of Φ . Then the square root
k k k
→
ofΦ canbe obtainedas the L2-limit of Ψ . The constructionis cannonical. In ourproof, the continuity
k
ON MATRIX VALUED SQUARE INTEGRABLE POSITIVE DEFINITE FUNCTIONS 5
of Φ is given by Theorem 2. We do not use Cor. 13.7.11 in [2] which require several more pages of
k
argument. We also wish to point out in the scalar case λ(Φ ) acts on L2(G) and in our case λ(Φ ) acts
k k
on L2(G,Cn) not on L2(G,M ).
n
Fixanx G. Letρ(x)actonL2(G,Cn)by(ρ(x)u)(g)=u(gx).ρ(x)isaunitaryoperatoronL2(G,Cn).
∈
If Φ L (G,M ), then
loc n
∈
(3) ρ(x)λ(Φ)ρ(x−1)=λ(Φ)
on C (G,Cn).
c
Let Φ P2(G,M ). Then λ(Φ) is a positive symmetric operator densely define on C (G,Cn), by the
n c
∈
definition of positive definiteness of Φ. Let Λ(Φ) be the Friedrichs extension of λ(Φ). Then Λ(Φ) is an
(unbounded)positiveandself-adjointoperator. ByEquation(3),wemusthaveρ(x)Λ(Φ)ρ(x−1)=Λ(Φ).
∞
Let Λ(Φ) = tdP be the spectral decomposition. Here P is an increasing sequence of mutually
0 t t
commutativeRprojectionsonL2(G,Cn). Thenρ(x)Λ(Φ)ρ(x−1)= ∞td[ρ(x)P ρ(x−1)]. Notice here that
0 t
ρ(x)isunitary. Henceρ(x)P ρ(x−1)remainsaprojection. TheunRiquenessofthespectraldecomposition
t
of self-adjoint operators implies that ρ(x)P ρ(x−1)=P . Since P is bounded, we haveρ(x)P =P ρ(x)
t t t t t
for any x G.
∈
Define Φ by letting the j-th column vector to be [Φ ] =P [Φ] . Clearly Φ L2(G,M ).
t t ∗j t ∗j t n
∈
Claim 1: λ(Φ )=P λ(Φ) on C (G,Cn).
t t c
Proof: Let u C (G,Cn). Then
c
∈
[λ(Φ)u](g)= [Φ] (gx−1)[u] (x)dx= (ρ(x−1)[Φ] )(g)[u] (x)dx.
Z ∗j j Z ∗j j
Xj x∈G Xj x∈G
Now fixing x, we have
(P ρ(x−1)[Φ] )(g)=(ρ(x−1)P [Φ] )(g)=(P [Φ] )(gx−1).
t ∗j t ∗j t ∗j
Since P is a bounded operator and [u] (x) are all in L1(G), we have
t j
[P (λ(Φ)u)](g)= P (ρ(x−1)[Φ] )(g)[u] (x)dx
t tZ ∗j j
X
j
= P (ρ(x−1)[Φ] )(g)[u] (x)dx
Z t ∗j j
X
j
= (P [Φ] )(gx−1)[u] (x)dx
(4) XZ t ∗j j
j
= ([Φ ] )(gx−1)[u] (x)dx
Z t ∗j j
X
j
= Φ (gx−1)u(x)dx
Z t
=(λ(Φ )u)(g)
t
Ourclaimis proved. ObservethatP λ(Φ)=P Λ(Φ)onC (G,Cn)andP Λ(Φ)is positive andbounded.
t t c t
Therefore λ(Φ ) = P λ(Φ) is bounded on C (G,Cn) and positive with respect to C (G,Cn). So Φ is
t t c c t
6 HONGYUHE
moderated and positive definite. We must have λ(Φ )=P Λ(φ) on L2(G,Cn). In addition if s t
t t
≥
λ(Φ Φ )=(P P )Λ(Φ)
s t s t
− −
on C (G,Cn) and the right hand side is positive and self adjoint. So Φ Φ . Similarly Φ Φ. Thus
c t s t
(cid:22) (cid:22)
we have obtained an increasing sequence of moderated positive definite functions
Φ Φ ... Φ ... Φ.
1 2 k
(cid:22) (cid:22) (cid:22) (cid:22) (cid:22)
Due to the way [Φ ] are defined, Φ Φ in L2-norm. As a by product, we have
k ∗j k
→
Lemma 4 (Prop 14 [1]). Every Φ P2(G,M ) is a L2-limit of an increasing sequence of mutually
n
∈
commutative moderated elements in P2(G,M ).
n
By Theorem 2, there is a moderated element Ψ P2(G,M ) such that Φ = Ψ Ψ almost
i n i i i
∈ ∗
everywhere. Consequently, λ(Φ ) = λ(Ψ )2. Without loss of generality, suppose that Φ = Ψ Ψ .
i i i i i
∗
Since λ(Φ ) mutually commutes and λ(Φ ) increases, λ(Ψ ) must mutually commute and λ(Ψ ) must
i i i i
also increase. It follows that Ψ commutes with each other and
i
Ψ Ψ ... Ψ ...
1 2 i
(cid:22) (cid:22) (cid:22) (cid:22)
Observe that Ψ 2 = Tr(Ψ Ψ (e)) Tr(Φ(e)). By Theorem 1, Ψ converges in L2(G,M ). Let
i i i i n
k k ∗ ≤ { }
Ψ ΨinL2(G,M ). ThenΦ =Ψ Ψ convergesuniformlytoΨ Ψ. SinceΦ Φ inL2(K,M )
i n i i i i K K n
→ ∗ ∗ | → |
for any compact set K, Φ =Ψ Ψ almost everywhere. So Φ=Ψ Ψ. Theorem A is proved.
K K
| ∗ | ∗
5. Nonnegative Integral: Proof of Theorem B
The main idea of the proof here is essentially due to Godement. See Prop.18 in [1].
Lemma 5. Every Φ in P2(G,M ) is a limit of an increasing sequence of moderated elements in
n
2(G,M ) under the L2 norm.
n
P
Proof: By Lemma 4, it suffices to show that every moderated element Φ in P2(G,M ) is the L2
n
limitofanincreasingsequenceofmoderatedelementsin 2(G,M ). Withoutlossofgenerality,suppose
n
P
that λ(Φ) = 1. Let q (t) be an increasing sequence of nonnegative polynomial functions on [0,1]
k
k k
such that t2 q (t) and q (t) t uniformly on [0,1]. Let Φ = q (Φ). Then λ(Φ ) = q (λ(Φ)) is an
k k k k k k
| →
increasing sequence of positive self-adjoint operators that approaches λ(Φ). In addition, λ(Φ ), λ(Φ)
k
all mutually commute. Since λ(Φ ) λ(Φ), (λ(Φ ))2 (λ(Φ))2. So Φ Φ Φ Φ. This implies
k k k k
(cid:22) (cid:22) ∗ (cid:22) ∗
Tr(Φ Φ (e)) Tr(Φ Φ(e)). So Φ Φ . By Theorem 1, let Ψ be the L2-limit of Φ . For any
k k k k
∗ ≤ ∗ k k ≤ k k
u C (G,M ), λ(Ψ)u=limλ(Φ )u=λ(Φ)u. It follows that Ψ=Φ almost everywhere. So Φ Φ in
c n k k
∈ →
L2-norm.
Observe that Φ Φ is always continuous. So Φ = q (Φ) is always continuous. Φ is obviously moder-
k k k
∗
ated. We have obtained an increasing sequence of moderated elements in 2(G,M ) such that Φ Φ
n k
in L2-norm. (cid:3) P →
Now we shall give a proof of Theorem B. We will first show that
Lemma 6. Let Φ be a moderated element in P2(G,M ) and Φ 2(G,M ). We have
1 n 2 n
∈P
Φ ,Φ 0.
1 2
h i≥
Proof: Suppose that Φ =Ψ Ψ with Ψ P2(G,M ). Then
2 n
∗ ∈
n
Φ ,Φ =Tr(Φ Φ (e))=Tr(Φ Ψ Ψ(e))=Tr(λ(Φ )Ψ Ψ(e))= λ(Φ )[Ψ] ,[Ψ] .
1 2 1 2 1 1 1 ∗i ∗i
h i ∗ ∗ ∗ ∗ h i
X
i=1
ON MATRIX VALUED SQUARE INTEGRABLE POSITIVE DEFINITE FUNCTIONS 7
Notice that λ(Φ ) is a bounded positive self adjoint operator. We have Φ ,Φ 0.
1 1 2
h i≥
Now for Φ,Ψ P2(G,M ). Let Φ be a sequence of moderated element in P2(G,M ) with L2-limit Φ
n α n
∈
and Ψ be a sequence of elements in 2(G,M ) with L2-limit Ψ. Then we have
β n
P
Φ,Ψ = lim Φ ,Ψ 0.
α β
h i α,β→∞h i≥
Theorem B is proved.
6. Zero Integral
If Φ Ψ = 0, we have Φ,Ψ = Tr(Φ Ψ(e)) = 0. Now we would like to show that the converse is
∗ h i ∗
also true.
Theorem 3. Let G be a unimodular locally compact group. Let Φ,Ψ P2(G,M ). If Φ,Ψ =0, then
n
∈ h i
Φ Ψ=0.
∗
Proof: ByLemma4,letΦ beanincreasingsequenceofmoderatedelementsinP2(G,M )suchthat
m n
Φ Φ 1. By Lemma 5, let Ψ be an increasing sequence in 2(G,M ) such that Ψ Ψ 1.
k m− k≤ m p P n k p− k≤ p
Then
0= Φ,Ψ = Φ Φ ,Ψ + Φ ,Ψ Φ ,Ψ Φ ,Ψ 0.
m m m m p
h i h − i h i≥h i≥h i≥
So allthe inequalities here must be equalities. Suppose that Ψ =Θ Θ with Θ P2(G,M ). Then
p p p p n
∗ ∈
0= Φ ,Ψ =Trace(Φ Θ Θ (e))= λ(Φ )[Θ ] ,[Θ ]
m p m p p m p ∗i p ∗i
h i ∗ ∗ h i
X
i
Since λ(Φ ) is a positive operator on L2(G,Cn), λ(Φ )[Θ ] ,[Θ ] = 0. So λ(Φ )[Θ ] = 0 in
m m p ∗i p ∗i m p ∗i
L2(G,Cn). So Φ Θ =0. Hence Φ Ψ =Φ hΘ Θ =0. It folliows that for any fixed g G
m p m p m p p
∗ ∗ ∗ ∗ ∈
[Φ Ψ(g)] [(Φ Φ ) Ψ(g)] + [Φ Ψ(g)]
ij m ij m ij
| ∗ |≤ | − ∗ | | ∗ |
X X X
i,j i,j i,j
[(Φ Φ ) Ψ(g)] + [Φ (Ψ Ψ )(g)]
m ij m p ij
≤ | − ∗ | | ∗ − |
(5) Xi,j Xi,j
n2 Φ Φ Ψ +n2 Φ Ψ Ψ
m m p
≤ k − kk k k kk − k
1 1 1
n2 Ψ +n2 ( Φ + )
≤ mk k p k k m
We see that as m,p , [Φ Ψ(g)] =0. Therefore Φ Ψ(g)=0 for all g.
→∞ i,j| ∗ ij| ∗
P
Corollary 1. Let G be a locally compact unimodular group. Let φ,ψ P2(G). If φψdg = 0, then
∈ G
φ ψ =0. R
∗
7. Applications in Representation Theory
Let G be a unimodular group. We call a unitary representation (π, ) of G L2 if there is a cyclic
H
vector u in such that (π(g)u,u) is L2. A L2 unitary representation has a G-invariant dense subspace
H
with L2-matrix coefficients.
Theorem 4. Let G be a unimodular locally compact group and (π, ) be a unitary representation of
H
G. Suppose that (π , ) and (π , ) are two L2-unitary representations of G such that
1 1 2 2
H H
(π, )=(π π , ˆ ).
1 2 1 2
H ⊗ H ⊗H
8 HONGYUHE
Let u= n u(i) u(i) such that matrix coefficients with respect to u(i) and u(i) are all L2. Then
i=1 1 ⊗ 2 { 1 } { 2 }
P n
(π(g)u,u)dg = (π (g)u(i),u(j))(π (g)u(i),u(j))dg 0.
Z 1 1 1 2 2 2 ≥
G iX,j=1
Observe that Φ defined by [Φ ] = (π (g)u(i),u(j)) is square integrable and positive definite. Sim-
1 1 ij 1 1 1
ilarly, Φ L2(G,M ) defined by [Φ ] = (u(i),π (g)u(j)) is square integrable and positive definite.
This the2or∈em followsneasily from Theo2reijm B. (cid:3)2 2 2
Now we shall apply our result to Howe’s correspondence ( [5]). Let (G(m),G′(n)) be a dual reduc-
tive pair in Sp. Let (G′(n ),G′(n )) be two G′-groupsdiagonally embedded in G′(n) with n +n =n.
1 2 1 2
Then (G(m),G′(n )) is a dual reductive pair in some Sp(i) such that (Sp(1),Sp(2)) are diagonally em-
i
]
bedded in Sp. Let ω be the oscillator representation of Sp(i). Let ω be the oscillator representation of
i
Sp. Thenω canbeidentifiedwithω ω . ThisidentificationpreservesthatactionsofG(m)andG′(n ).
1 2 i
⊗
Nowsupposethatthe matrixcoefficientsofω withrespecttotheSchwartzspaceareL2. Letπ be
1|G(˜m)
an irreducible unitary representation of G˜(m). Suppose that the matrix coefficients for ω∞ π∞
2 |G˜(m)⊗
are all square integrable. Then for any v π∞, u(j) ω∞,u(j) ω∞ with j [1,N], we have
∈ 1 ∈ 1 2 ∈ 2 ∈
(π(g)( u(j) u(j)),( u(k) u(k)))(v,π(g)v)dg
ZG˜(m) X 1 ⊗ 2 X 1 ⊗ 2
(6)
= (π (g)u(j),u(k))(π (g)u(j)),u(k))(v,π(g)v)dg.
Z 1 1 1 2 2 2
Xj,k G˜(m)
By Theorem 4, this integral must be nonnegative.
Corollary 2. Consider a dual reductive pair (G(m),G′(n)) in Sp. Let n=n +n . Let (G(m),G′(n ))
1 2 i
be a dual reductive pair in Sp(i). Let ω be the oscillator representation of Sp(i). Let π be an irre-
i
ducible unitary representation of G˜(m). Suppose that the matrix coefficients with respect to ω∞
1 |G˜(m)
and ω∞ π∞ are square integrable. Let ξ ω∞ ω∞ and u π∞, then
2 |G˜(m)⊗ ∈ 1 ⊗ 2 ∈
(ω(g)ξ,ξ)(π(g)u,u)dg 0.
ZG˜(m) ≥
See [3] and [4] for the importance of this integral in Howe’s correspondence. This Corollaryholds for
both p-adic groups and real groups.
References
[1] R.Godement, “Lesfonctions detypepositifetlathoriedesgroupes,”Trans. Amer. Math. Soc.63,(1948). 1-84
[2] J.Dixmier,C∗-Algebra,North-HollandPublishingCompany, 1969.
[3] H.He, “Theta Correspondence I-SemistableRange: Construction andIrreducibility”,Communications in Contem-
porary Mathematics (Vol2),2000, (255-283).
[4] H.He,“UnitaryrepresentationsandthetacorrespondencefortypeIclassicalgroups,”J. Funct.Anal.199(2003),no.
1,92-121
[5] R.Howe,“TranscendingClassicalInvariantTheory”,Journal of Amer. Math. Soc.(Vol.2),1989(535-552).
[6] R.KadisonandRingrose,Fundamentals of the Theory of Operator Algebras,AcademicPress,1983.
Departmentof Mathematics,Louisiana StateUniversity,Baton Rouge,LA 70803
E-mail address: [email protected]