REPRESENTATIONS WITH SCALAR K-TYPES ANDAPPLICATIONSCHEN-BO ZHU
Abstract. We discuss some results of Shimura on invariant differential operators and extend a folklore theorem about sphericalrepresentations to representations with scalar K-types. We thenapply the result to obtain non-trivial isomorphisms of certain representations arising from local theta correspondence, many of which
are unipotent in the sense of Vogan.

1. Introduction and main results
Let G be a connected noncompact semisimple Lie group with ﬁnite
center, and K be a maximal compact subgroup. Let g0 and k0 be the
Lie algebra of G and its subalgebra corresponding to K. Then we havea Cartan decomposition
g0 = k0 ⊕ p0 ,where p0 is a certain subspace of g0 . We denote by g, k, p the complexiﬁcations of g0 , k0 , p0 , respectively. The universal enveloping algebrasof g (resp., k) will be denoted by U(g) (resp., U(k)).
For a vector space W , denote by S(W ) the symmetric algebra overW , and Sr (W ) the subspace of S(W ) consisting of homogeneous elements of degree r, where r ∈ Z≥0 . Write
S r (W ) =



0≤k≤r

Sk (W ).

Let U r (g) be the subspace of U(g) spanned by elements of the form
Z1 · · · Zs , with Zi ∈ g, s ≤ r. Recall that there is a C-linear bijection ψof S(g) onto U(g), called the symmetrization map. We have ψ(S r (p)) ⊆U r (g).
1991 Mathematics Subject Classification. Primary 22E46, 11F27.
Key words and phrases. Invariant diﬀerential operators, spherical representations, scalar K-types, local theta correspondence, eigendistributions.
The author was supported in part by the NUS-MOE grant R-146-000-017-112.
The article is based on a talk at the National University of Singapore in February
2000.
1

2

CHEN-BO ZHULet V be a Harish-Chandra (g, K)-module. For ρ ∈ K̂, let Vρ bethe ρ-isotypic component of V . Here and after, K̂ denotes the set of
equivalent classes of irreducible (unitary) representations of K.
Our ﬁrst result is the following
Theorem 1.1. Suppose thatψ(S r (g)G ) + U(g)k ⊇ ψ(S r (p)K ),r ∈ Z≥0(true at least for all classical G), and ρ ∈ K̂ is one-dimensional. Foran irreducible Harish-Chandra module V with Vρ = 0, the infinitesimalcharacter of V determines V up to infinitesimal equivalence.
The key ingredient for the proof of this simple criterion is a certain
surjectivity result of Shimura on invariant diﬀerential operators ([Sh]).
For ρ the trivial representation, Helgason showed [He2] that this surjectivity holds precisely when G does not contain any simple factor of thefollowing four exceptional types: e6(−14) , e6(−24) , e7(−25) , e8(−24) . See §2
for the precise statements and other discussions. Therefore excludingthese four exceptional cases, irreducible spherical representations arecompletely determined by their inﬁnitesimal characters.
We next apply the above result in the context of local theta correspondence. As it turns out, theta lifts of one dimensional representations often have one-dimensional K-types, especially at the so-called
point of ﬁrst occurrence. As it is very easy to compute their inﬁnitesimal characters, our results can be used to identify these lifts quickly.
To be more precise, consider the dual pair [H1](O(p, q), Sp(2n, R)) ⊆ Sp(2N, R),
where N = 2(p + q)n. The theta lift from O(p, q) to Sp(2n, R) will bep,q.
denoted by θ→nDenote by 11 the trivial representation of any group. It should beclear from the context which is the group concerned. Recall that acompact orthogonal group has two characters, the trivial and the determinant characters, while for a non-compact orthogonal group O(k, l),there are four characters. They are denoted by 11, det, 11+,− and 11−,+ .
The character 11+,− is characterized by
11+,− |O(k) = trivial,

11+,− |O(l) = determinant.

Likewise for 11−,+ .
We now highlight two of our results from §4.
Theorem 1.2. Suppose that p ≥ q, and consider the dual pairs(O(p, q), Sp(2q, R)),and (O(p − 1, q + 1), Sp(2q, R)).

REPRESENTATIONS WITH SCALAR K-TYPES

3

We havep,qθ→q(11+,− )p−1,q+1
θ→q
(11).

p,q
The representations θ→n(11) (and in fact some larger representationsp,qcontaining θ→n (11) as unique irreducible quotients) are determined in[LZ2]. See §4.

Theorem 1.3. Suppose that p ≥ q, and consider the theta lift of 11−,+
for the dual pair (O(p, q), Sp(2p, R)). We havep,qθ→p(11−,+ )p+1,q−1
θ→p(11)

p−q+2,0
θ→p
(11),


the unitary lowest weight representation of Sp(2p,R) with lowest weight
p−q+2
1
.
Here
1
=
(1,
·
·
·
,
1)
.
p
p
2



p



p,q
Thus θ→p(11−,+ ) is actually not a ”strange fellow” (cf. [A2]).
Here is the organization of this paper. In §2, we review results of
Shimura on invariant diﬀerential operators. In §3, we combine themwith those of Harish-Chandra, Lepowsky-McCollum to arrive at Theorem 1.1. §4 is devoted to applications of Theorem 1.1 to local theta
correspondence.
Acknowledgment: The author would like to thank Roger Howe andJian-shu Li for helpful discussions, and to Jeﬀrey Adams for encouragement.

2. Invariant differential operators (after Shimura)
Recall that G is a connected noncompact semisimple Lie group withﬁnite center.
Given an irreducible ﬁnite dimensional representation ρ : K →GL(U), denote by C ∞ (G; U) the space of U-valued C ∞ functions on
G. SetC ∞ (ρ) = {f ∈ C ∞ (G; U)|f (gk −1) = ρ(k)f (g), ∀g ∈ G, k ∈ K}.
Denote by D(G) the ring of left-invariant diﬀerential operators on G.
Then D(G) (respectively, U(g)) acts naturally on C ∞ (G; U) throughright translation (respectively, the derived action of right translation).
Let
andD(ρ) = {A ∈ D(G)|A · C ∞ (ρ) ⊆ C ∞ (ρ)},D(ρ) = {B ∈ U(g)|B · C ∞ (ρ) ⊆ C ∞ (ρ)}.
Clearly there is a natural surjective map
π : D(ρ) → D(ρ).

4CHEN-BO ZHUObserve that for g ∈ G, k ∈ K, B ∈ g, f ∈ C ∞ (ρ), we havedf (g · exp(tAd(k)B))|t=0dtdd= f (gk · exp(tB)k −1 )|t=0 = ρ(k) f (gk · exp(tB))|t=0
dtdt= ρ(k)(Bf )(gk).

[(Ad(k)B)f ](g) =The above equation remains true for B ∈ U(g). In particular for B ∈U(g)K (the K-invariants of U(g)), we have (Bf )(g) = ρ(k)(Bf )(gk),namely B ∈ D(ρ). We thus have
U(g)K ⊆ D(ρ).
Observe that for f ∈ C ∞ (ρ) and X ∈ k0 , we have Xf = −dρ(X)f .
We may extend −dρ uniquely to a C-linear antihomomorphism of U(k)into End(U), which will be denoted by ρU . Then we have
Bf = ρU (B)f,

B ∈ U(k), f ∈ C ∞ (ρ).

LetRρ = ker(ρU ) ⊆ U(k)be the kernel of ρU .
Proposition 2.1. (Shimura, [Sh])(a) B ∈ U(g) annihilates C ∞ (ρ) if and only if B ∈ U(g)Rρ .
(b) D(ρ) = U(g)K + U(g)Rρ .
(c) The natural map
π : D(ρ) → D(ρ)
induces an isomorphism of U(g)K /U(g)K ∩ U(g)Rρ onto D(ρ).
Proposition 2.2. (Shimura, [Sh]) If ρ is one dimensional, then(a) D(ρ) is commutative.
(b) The bijection ψ : S(g) → U(g) induces a C-linear bijection of
S(p)K onto D(ρ).
(c) ψ(S r (p)K ) and U r (g)K have the same image in D(ρ), for r ∈ Z≥0 .
Remark 2.3. (a) U(g)K /U(g)K ∩ U(g)Rρ is in fact anti-isomorphic toa subalgebra of U(a) ⊗ U(k)/Rρ , where a ⊆ p is an abelian subalgebraof g. See [Le, Theorem 1.3].
(b) Suppose that ρ is ﬁnite dimensional and irreducible. For i Ei ⊗Bi ∈ End(U) ⊗ S(p), deﬁne Ψ( i Ei ⊗ Bi ) to be the operator on
C ∞ (G, U) given by


Ψ(

i

Ei ⊗ Bi ) =iEi ψ(Bi ).

REPRESENTATIONS WITH SCALAR K-TYPES

5

Then Ψ induces a bijection of (End(U) ⊗ S(p))K onto D(ρ). See [Sh,
Proposition 2.2].
Let Z(g) be the center of U(g). ClearlyZ(g) ⊆ U(g)K ⊆ D(ρ).
Theorem 2.4. (Shimura, [Sh]) Suppose thatψ(S r (g)G ) + U(g)k ⊇ ψ(S r (p)K ),r ∈ Z≥0(2.5)(true at least for all classical G), then the natural map
π|Z(g) : Z(g) → D(ρ)
is surjective for ρ one-dimensional.
Remark 2.6. For ρ = 11, the trivial representation, the we have D(ρ) =D(G/K), the algebra of G-invariant diﬀerential operators on the symmetric space G/K. Here are some historical remarks on the surjectivityofZ(g) → D(G/K).
(a) Berezin stated (incorrectly) the surjectivity (1957).
(b) The mistake was pointed out by Helgason (1964), and he provedthat the surjectivity is true for classical groups, but not true for somereal forms of e6 , e7 , e8 [He1].
(c) Shimura later pointed out a gap in Helgason’s proof. A correct
proof was given by Helgason [He2, 1989] who showed that Z(g) →D(G/K) is surjective if and only if G does not contain any simple
factor of the following four exceptional types: e6(−14) , e6(−24) , e7(−25) ,e8(−24) .
Remark 2.7. If G is simple, then K has a non-trivial one-dimensional
representation if and only if the pair (G, K) is of Hermitian symmetric
type. Suppose this is the case and suppose that G is exceptional,
namely G is of type e6(−14) , e7(−25) , then the condition (2.5) will not
hold. This is because its validity will imply in particular the surjectivityof Z(g) → D(G/K), which is false by the result of Helgason. Thus the
additional cases covered by Theorem 2.4 (apart from ρ = 11) are reallyfor G classical and of Hermitian symmetric type.
3. Admissible representations with scalar K-types
Let (π, V ) be an irreducible admissible representation of G. Denoteby V ∞ the space of smooth vectors of V , and Vρ∞ the ρ-isotypic component of V ∞ , where ρ ∈ K̂. The Harish-Chandra module of (π, V ) is

6CHEN-BO ZHUthen the algebraic direct sum
VK =



Vρ∞ .

ρ∈K̂

For each ρ ∈ K̂, the space Vρ∞ is naturally a U(g)K × K module.
Theorem 3.1. (a) (Harish-Chandra, [Ha]) If Vρ∞ = 0, then the U(g)K
module structure of Vρ∞ determines π up to infinitesimal equivalence.
(b) (Lepowsky-McCollum, [LM]) Denote Iρ = U(g)K ∩U(g)Rρ . ThenIρ is a two-sided ideal in U(g)K , and the action of U(g)K on Vρ∞ factors through U(g)K /Iρ . This establishes a one-to-one correspondence
between irreducible admissible representations V of G with Vρ∞ = 0 andirreducible U(g)K /Iρ modules.
We are now ready to prove the following
Theorem 3.2. Suppose thatψ(S r (g)G ) + U(g)k ⊇ ψ(S r (p)K ),r ∈ Z≥0(true at least for all classical G), and ρ ∈ K̂ is one-dimensional. Foran irreducible Harish-Chandra module V with Vρ = 0, the infinitesimalcharacter of V determines V up to infinitesimal equivalence.
Proof. By Proposition 2.1, we haveK
D(ρ) ∼= U(g) /Iρ .

From Theorem 2.4, we know that the natural map
Z(g) → U(g)K /Iρ
is surjective under the current hypothesis. Therefore for B ∈ U(g)K ,there exists ZB ∈ Z(g) such thatB − ZB ∈ Iρ .
Clearly given B ∈ U(g)K , such ZB is unique modulo Z(g) ∩ Iρ .
Let χ be the inﬁnitesimal character of V . Then we have
B · v = χ(ZB )v,

B ∈ U(g)K , v ∈ Vρ .

By the Theorem of Harish-Chandra (Theorem 3.1 (a)), we conclude
that the inﬁnitesimal character χ of V determines V up to inﬁnitesimalequivalence.

REPRESENTATIONS WITH SCALAR K-TYPES

7

4. Applications to local theta correspondenceFix a reductive dual pair [H1](G , G) ⊆ Sp(2N, R).

Let Sp(2N,R) be the metaplectic cover of Sp(2N, R), and for any sub be the inverse image of L in Sp(2N,R).
group L of Sp(2N, R), let LDenote by K (resp. K) a maximal compact subgroup of G (resp. G).
R). ReWe also ﬁx a (smooth) oscillator representation Ω of Sp(2N,
call [H2] that an irreducible admissible representation π of G and an are said to correspond to
irreducible admissible representation π of G
each other under the local theta correspondence (or the Howe duality
 × G-intertwining
mapcorrespondence) if there exists a non-zero GΩ → π  ⊗ π.
In that case we writeθ(π  ) = π,orθ(π) = π  ,or in a symmetric way π  ↔ π.
A basic property of the local theta correspondence is that it has
 and K types, which
subordinated to it a correspondence of certain K
we shall describe.
Recall that the Fock model of Ω is realized in the space F of poly - andnomials in N variables. Using this, we may associate to each KK-typeoccuring in F a degree, which will be called the Howe degree.
 and K harmonics ([H2]). This is a
Let H be the space of joint K
 × K-invariant
K
subspace of F .
Theorem 4.1. (Howe, [H2]) There is a one to one correspondence of - and K-types
Kin H with the following properties. Suppose that π  and and G respectively,
π are irreducible admissible representations of Gand π ↔ π in the correspondence for the dual pair (G , G). Let σ  be a -type occuring in π  , and suppose that σ  is of minimal Howe degreeK -types of π  . Let σ be the K-typewhich corresponds
among all the Kto σ in H. Then σ is a K-type of minimal Howe degree in π.
In what follows, we shall only examine the dual pair(O(p, q), Sp(2n, R)) ⊆ Sp(2N, R),
where N = 2(p + q)n. The theta lift from O(p, q) to Sp(2n, R) will bep,q, as in the Introduction.
denoted by θ→n8CHEN-BO ZHU

We have K  = O(p)×O(q) and K = U(n). The following descriptionin H is well-known. See for exof correspondence of K  - and K-types
ample [A1]. We shall write this correspondence as σ  ↔ σ. Recall that
irreducible representations of O(p) are indexed by (a1 , ..., ak , 0, ..., 0; *),where (a1 , ..., ak , 0, ..., 0) are the usual highest weights of irreducible
 irreducible reprerepresentations of SO(p), and * = ±1. For U(n),sentations are indexed by their highest weights, with their components
integers or half integers. Denote as before
1n = (1, · · · , 1) .



n



Fact 4.2. Correspondence of joint harmonics for (O(p, q), Sp(2n, R))σ  = (a1 , ..., ak , 0, ..., 0; *) ⊗ (b1 , ..., bl , 0, ..., 0; η) ↔
p−q
σ=1n + (a1 , ..., ak , 1, ..., 1 , 0, ..., 0, −1, ..., −1, −bl , ..., −b1 ),  
21−(p−2k)
21−η
(q−2l)
2

where 2k ≤ p, 2l ≤ q and k + 1−
(p − 2k) + l + 1−η
(q − 2l) ≤ n.
2
2
Denote m = p + q. Recall also the correspondence of inﬁnitesimalcharacters under the local theta correspondence [Pr]. In particular wehavep,q(χ) hasFact 4.3. For any character χ of O(p, q), the theta lift θ→nm
the infinitesimal character with Harish-Chandra parameter ( 2 − 1, m2 −2, · · · , m2 − n).

4.1. The trivial character 11. The trivial character 11 occurs in thelocal theta correspondence of (O(p, q), Sp(2n, R)) for any n. See [Zh]for example.
m,00,m(11) (resp. θ→n(11)) is the unitary lowest (resp. highNote that θ→nest) weight representation of Sp(2n, R) with lowest weight m2 1n (resp.
highest weight − m2 1n ).
p,qp,qThe representations θ→n(11) are very well understood. In fact θ→n(11)
is the unique irreducible quotient of a certain natural Sp(2n, R)-moduleΩp,q→n (11), called the Howe (maximal) quotient corresponding to thetrivial representation of O(p, q), and complete structures of Ωp,q→n (11)p,q(11)) are known [LZ2]. In particular, Ωp,q
(1
1)
is K
(and therefore θ→n→n
of Ωp,qmultiplicity-free, and all K-types→n (11) and of their irreducible
constituents are explicitly described. One may also write down theirLanglands parameters (see [PT] for the unitary case).
The following proposition summaries some basic and relevant propp,q(11). For σ ∈ C and α = 0, 1, 2, 3, let I(σ; α) be theerties of θ→nREPRESENTATIONS WITH SCALAR K-TYPES

9
degenerate principal series of Sp(2n,R) (normalized) induced from theασcharacter χ0 ⊗ | · | of the Siegel parabolic subgroup. Here χ0 is acertain character of order 4 (roughly the square root of the sign ofdeterminant).

Theorem 4.1.1. (see [LZ2])p,q(a) θ→n(11) is isomorphic to the unique irreducible constituent ofp−q
1n , where σ = m−(n+1)
andI(σ; α) containing the scalar K-type22
α ≡ p − q (mod 4).
p,q(b) θ→n(11) is finite dimensional if and only if p, q ≡ n + 1 (mod 2),p,q(11) is isomorphic to F( m2 −n−1)1n ,and p, q ≥ n + 1. In this case, θ→nthe finite dimensional irreducible representation with the highest weight( m2 − n − 1)1n .
p,q(11) is unitary if and only if either pq = 0 or both p, q ≤ n+ 1.
(c) θ→n
The following proposition may be read oﬀ from the results of [LZ2].
We give a proof in the spirit of this article.
Proposition 4.1.2. Let p, q ≤ n + 1, then we haven+1−q,n+1−p
θ→n
(11)

p,q
θ→n
(11).

n+1−q,n+1−pp,qProof. Both θ→n(11) and θ→n(11) contain the scalar K-typep−q
1n (Fact 4.2). The inﬁnitesimal character of the latter has Harish2
Chandra parameter ( m2 − 1, m2 − 2, · · · , m2 − n), while that of the former
has Harish-Chandra parameter (n − m2 , ..., 1 − m2 ). Since they are con-jugate via a Weyl group element, the assertion clearly follows from
Theorem 3.2.
4.2. The characters 11+,− and 11−,+ . Without any loss of generality,
we may assume that p ≥ q. From the correspondence of joint harmonics(Fact 4.2), we see that 11+,− occurs in the local theta correspondence
for the dual pair (O(p, q), Sp(2n, R)) only if q ≤ n. At n = q, we know
from the existance of certain tempered 11+,− -eigendistribution [HZ] that11+,− does occur. In view of the persistence principle of Kudla [K], we
may conclude that
Proposition 4.2.1. The character 11+,− occurs in the local theta correspondence for the dual pair (O(p, q), Sp(2n, R)) if and only if q ≤ n.
At n = q, it will be refered to as the point of ﬁrst occurrence forp,q(11+,− ) contains the scalar K-type11+,− . From Fact 4.2, we see that θ→q
p−q−2
1q . From Fact 4.3 and Theorem 3.2, we immediately have
2
Theorem 4.2.2. Suppose that p ≥ q, and consider the dual pairs(O(p, q), Sp(2q, R)),and (O(p − 1, q + 1), Sp(2q, R)).

10

CHEN-BO ZHU

We havep,q(11+,− )θ→q

p−1,q+1
θ→q(11).

We may then combine Theorem 4.1.1 with Theoem 4.2.2 to obtainthe followingp,qCorollary 4.2.3. (a) θ→q(11+,− ) is finite dimensional if and only ifp,qp ≡ q (mod 2), and p > q. In this case, θ→q(11+,− ) is isomorphic toF( p−q −1)1q .
2p,q(11+,− ) is unitary if and only if p = q, q + 1, q + 2, and(b) θ→qp,qθ→q(11+,− )0,2θ→q (11),θ0,1 (11),
 →q



11,

p = q,
p = q + 1,
p = q + 2.

Remark 4.2.4. The assertion in Corollary 4.2.3 (a) is related to certain interesting phenomenon about solutions of a system of generalized
wave equations, called the ”Generalized Huygens’ Principle”. See [HZ].
Now we examine the character 11−,+ . Similarly for 11−,+ to occur inthe local theta correspondence for the dual pair (O(p, q), Sp(2n, R)),
we must have p ≤ n. The next proposition implies that it does occur
at n = p.
Proposition 4.2.5. There exists a non-zero tempered 11−,+ -eigendistribution of O(p, q) on the matrix space Mp+q,p (R). Hence 11+,− occurs forthe dual pair (O(p, q), Sp(2n, R)) if and only if p ≤ n.
Proof. We consider a larger matrix space M2p+2,p (R). According
to [HZ], there exists a non-zero tempered 11−,+ -eigendistribution Ψ ofO(p, p+2) on M2p+2,p (R). In the notation of [HZ], Ψ = dνp is the diﬀerence of two O + (p, p+2) invariant measures supported on the O(p, p+2)
nullcone in M2p+2,p (R), where O + (p, p+2) is a subgroup of O(p, p+2) ofindex 2. For the case under consideration, the distribution Ψ is invariR) (through the smooth oscillator representation Ω ofant under Sp(2p,
Sp(2p(2p + 2), R) on the Schwarz space S(M2p+2,p (R))). C.f. Corollary
4.2.3 (b). But we shall not use this fact.
a, whereWrite a typical element v ∈ M2p+2,p (R) as v =ba = (aij )(p+q)×p ∈ Mp+q,p (R), and b = (bij )(p−q+2)×p ∈ Mp−q+2,p (R).
Corresponding to this, we have the tensor product decomposition
S(M2p+2,p (R))

S(Mp+q,p (R)) ⊗ S(Mp−q+2,p (R)).

REPRESENTATIONS WITH SCALAR K-TYPES

11

Since Ψ = 0, we see that there exists a φ ∈ S(Mp−q+2,p (R)) suchthat
Ψ|S(Mp+q,p(R))⊗φ = 0.
This restriction will then deﬁne a non-zero tempered distribution on
Mp+q,p (R), which is clearly a 11−,+ -eigendistribution of O(p, q).
Remark 4.2.6. It is not diﬃcult to see that we may takeφ(b) = exp(−tr(bt b)/2),b ∈ Mp−q+2,p (R)in the above proof.
Theorem 4.2.7. Suppose that p ≥ q, and consider the theta lift of
11−,+ for the dual pair (O(p, q), Sp(2p, R)). We havep,qθ→p(11−,+ )p+1,q−1
θ→p(11)

p−q+2,0
θ→p
(11),


R) with lowest weightthe unitary lowest weight representation of Sp(2p,
p−q+2
1p .
2

Proof. From the correspondence of joint harmonics (Fact 4.2),
p−q+2
p,q

(11−,+ ) contains the scalar K-type
1p . Since
we see that θ→p2
p+1,q−1θ→p(11) contains the same scalar K-type, and since the two representations clearly have the same inﬁnitesimal character, the ﬁrst isomorphism follows from Theorem 3.2. The second isomorphism is a
special case of Proposition 4.1.2.
4.3. Remarks about the determinant character. The determinant character det of O(p, q) occurs in the local theta correspondence
for the dual pair (O(p, q), Sp(2n, R)) if and only if p + q ≤ n. As in thep,qp,q multiplicity-free, andcase of θ→n(11), the representation θ→n(det) is K
can be explicitly described. See [LZ2].
all its K-types
The point of ﬁrst occurance for det is thus at n = p + q, which is in
the so-called stable range. Assume this is the case. From Fact 4.2, wep,q
know that θ→n(det) contains the K-typep−q
1n + (1, ..., 1, −1, ..., −1).
   


2
p

q

p,q

(det) contains a scalar K-type
if and only if pq = 0.
It turns out that θ→nWhen this happens, clearly we have
n,0
θ→n
(det)

n+2,0
θ→n
(11),

0,n
θ→n
(det)

0,n+2
θ→n
(11).

In all other cases, namely p + q = n and p, q > 0, the representation
 ,q 
p,q(det) is isomorphic to certain submodule of Ωp→n(11), where p +q  =θ→np ,q (11) is then + 2. See [LZ2], Theorems 4.15 and 4.17. Note that θ→n,qunique irreducible quotient of Ωp→n
(11).

12

CHEN-BO ZHUReferences[A1] J. Adams, The theta correspondence over R, notes for Workshop at the University of Maryland, 1994.
[A2] J. Adams, One-dimensional representations, private notes.
[Ha] Harish-Chandra, Representations of semisimple Lie groups II, Trans. Amer.
Math. Soc., 76 (1954), 26-65.
[He1] S. Helgason, Fundamental solutions of invariant diﬀerential operators on asymmetric space, Amer. J. Math., 86 (1964), 565-601.
[He2] S. Helgason, Invariant diﬀerential operators on symmetric spaces, Amer. J.
Math., 114 (1992), 789-811.
[H1] R. Howe, θ-series and invariant theory, in “Automorphic forms, representations and L-functions,” Proc. Sympos. Pure Math., Vol. 33, Part 1, pp.
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Department of Mathematics, National University of Singapore, 2
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E-mail address: matzhucb@nus.edu.sg