SOLUTIO'
FUNDAMENTAL SOLUTIONS OF INVARIANT
DIFFERENTIAL OPERATORS ON
SYMMETRIC SPACES
BY S. HELGASON
1
Communicated by I. MI.Singer, June 27, 1963
1. Introduction and notation. Let S be a Riemannian globally
symmetric space, G the largest connected group of isometries of S
in the compact open topology. We assume that S is of the noncompact type, that is, G is semisimple and has no compact normal subgroup
{e}. Let o be any point in S, K the isotropy subgroup of G
at o, and g their respective Lie algebras, and p the orthogonal complement of in g with respect to the Killing form B of g. Let a be any
maximal abelian subspace of p and let A =exp(a). For each X in the
dual space of a (which we identify with a, via B) let gx= {XE g [H, X]
=X(H)X for all HEa}. Let dx=dim(gx). Choose some order on a
and let
p =
X>0
Z dxX,
I>=
X>0
transformation of
sinh
Let D(G) and D(5
differential operat
algebra
over a, S(
decreasing togeth,
note the set of 14
taken with its usi
is a closed subspa
[2, Theorem
timrQ ton G whi
/a
44arIeach D(
"
I
ist
: ) denote
0 xdX
x>0
and let r denote the product of the distinct prime factors in r'.
G=KAN
Then we have the Iwasawa decompositions =f+a+n,
let
H(g)
denote
Given
gG,
exp(n).
where N is the nilpotent group
the unique element in a for which gEK exp H(g)N. Let TV denote the
Weyl group M1'/.Mwhere M and M', respectively, denote the centralizer and normalizer of a in K.
For each XGa consider the spherical function
where aGS(a) an
p. 586], Io(G) C1
seminorms TD,gti
LEMMA 1. Io(G:
Under the rest
isomorphically
Ox()
e(iX-p)(H(xk))dk
=
(x E G)
,2
dk being the normalized Haar measure on K. Let c(X) denote HarishChandra's function on a which occurs in the leading term of the
asymptotic
expansion of Ox [2, p. 283], i.e.,
Ox(expH)
-
E
c(sX)e(i- P)
(H)
EW
where X and H are suitably restricted in a.
Each xEG can be written uniquely in the form x=k exp X(kEK,
XEP). We put XI =(B(X, X))12 and co(X)= {det (sinh ad
X/ad X)p }112where the subscript p indicates restriction to p of the linear
1
Work supported in part by the National Science Foundation NSF GP-149.
778
1, p.
r oftb algebra I
on
tions on A. We c
over on Io(A) by
under convolution
2. Transmutati
called a radial fu
continuous (resp.
respondencef-*f
functions on A.
[2, p. 265] there {
set of regular elen
functionsf. The o
with the radial
(D,=d/dr). It is
SOLUTIONS OF INVARIANT DIFFERENTIAL
OPERATORS
779
transformation of g given by
{VARIANT
sinh ad X/ad X =
ON
E
(adX)2-/(2n + 1)!.
n1O
B of g. Let a be any
). For each X in the
Let D(G) and D(S) denote the set of left invariant (resp. G-invariant)
differential operators on G (resp. S). Let S(a) denote the symmetric
algebra over a, (a) the space of Co functions on a which are rapidly
decreasing together with all their derivatives. Let I(a) and (a) denote the set of W-invariants in S(a) and 8(a), respectively; 8(a) is
taken with its usual locally convex topology [7, p. 90] and then g(a)
is a closed subspace. According to a theorem of Harish-Chandra (cf.
[2, Theorem 1, p. 260], also [3, p. 432]) there exists an isomorphism
r of the algebra D(S) onto I(a). Let I(G) denote the set of C" functions f on G which are bi-invariant under K and for each integer
q>_0 and each DED(G) satisfy
{x X c=lg [H,X]
rD,,(f) = sup (1 + I H )qw(H) (Df) (exp H) I < o.
1963
iemannian globally
of isometries of S
S is of the noncom)mpact normal subropy subgroup of G
:he orthogonal com-
;e some order on a
Hea
Let Io(G) denote the space of functions of the form
.J Xdx
>0
,.a() = f7r()a(X)4x(x)dX
(x E G)
,rime factors in r'.
:f+a+n, G=KAN
-G, let H(g) denote
. Let W denote the
ly, denote the cen-
G)
c(X)denote Harish.ading term of the
where aES(a) and dX is the Euclidean measure on a. Then, by [2,
p. 586], Io(G)CI(G) (it can be shown using [1] that r=7ro). The
seminorms TD, turn Io(G) and I(G) into locally convex spaces.
LEMMA
1. Io(G) is an algebra under convolution on G.
Under the restriction from G to A, Io(G) and I(G) are mapped
isomorphically onto spaces Io(A) and I(A) of W-invariant Ca functions on A. We carry the algebraic and topological structure of Io(G)
over on Io(A) by means of this mapping. The space 4 (a) is an algebra
under convolution on a.
2. Transmutation operators. A function f on the space S=G/K is
called a radialfunction iff(k p) =f(p) for all kCK, pCS. The set of
continuous (resp. CO) radial functions on GIK is in one-to-one correspondence f--j with the set of all continuous (resp. Co) W-invariant
functions on A. Here f(aK) =j(a) for aCA. Let DGD(S); then by
x=k expX(kEK,
)=
{det (sinh
ad
[2, p. 265] there exists a unique differential operator /t'(D)on A' (the
set of regular elements in A) such that (Df)-= ' (D)J for all COradial
tion to P of the linear
functionsf. The operator 8'(D) is called the radial part of D in analogy
with the radial part D,2+ (n - 1)/r D, of the Laplacian on R",
lation NSF GP-149.
(D,=d/dr).
It is known [5] that there exists an isomorphism X
780
[November
S. HELGASON
l
SOLUTIO
i963]
("transmutation operator") of the vector space of even C functions
on R, onto itself, under which the singular operator D2+ (n - 1)/r D,
corresponds to D,. The operators 6'(D) (DED(S)) are singular when
considered as differential operators on A but Theorem 1 shows that
given by (f)=f(c
they have a simultaneous transmutation operator X under which
on G/K induced
they correspond to differential operators on the Euclidean space a
with constant coefficients.
Given a W-invariant function 0 on A, let f denote the corresponding radial function on S. For EI(A), put
(X+)(H) = eP(H)
((exp H)n.o)dn
(H E a)
on S is called a fu:
a fundamental sc
Du=f by putting
THEOREM 2. E,
on the symmetric s
This is a conse
differential opera
tempered fundam
. r7
where dn is a suitably normalized invariant measure on N. As proved
by Harish-Chandra [11, p. 595], X is a continuous mapping of I(A)
into (a).
I
. ___
Aaed n prooj
mula (which is a
simpler proof of'
THEOREM 1. The mapping X is a topological isomorphism of the
algebra Io(A) onto the algebra 9(a). Moreover, if DED(S) then
X6(D) = r(D)x4,
4 E Io(A).
Here r(D) is consideredas a differential operator on a.
The proof is based on the Plancherel formula for functions in
Io(G), proved by Harish-Chandra [2]. It also uses the recent result
of Gindikin and Karpelevi6 [1] according to which the function c(X)
above can be expressed in terms of Gamma functions.
REMARKS. At the end of [2], Harish-Chandra states the following
two conjectures which would imply that lo(A) contains all the Winvariant Cx functions on A with compact support.
(I) There exists a polynomial pES(a) such that c(X)r(X)p(X) _ 1
for all XEa. (Here we have used the fact that r=7ro.)
(II) The mapping X is one-to-one on I(A).
Now (I) can be verified on the basis of the mentioned result of Gindikin and Karpelevi&. Theorem 1 shows that (II) is equivalent to
Io(G) =.I(G). On the other hand, (II) is easily implied by the Plancherel formula for the functions in I(G). This formula is not proved in
[2] but I understand that Harish-Chandra has proved it in recent,
as yet unpublished, work. In the next section we shall therefore as-
sume that Io(G)= I(G).
3. Fundamental solutions. Let C (S) denote the space of Cx functions on S with compact support. Let denote the distribution on S
Here Exp is the u
isomorphism
of
nomials on
and
1. S. G. Gindikin
metric spaces of nonpo
Soviet Math. Dokl. 3
2. Harish-Chandr
Math. 80 (1958), 241
3. S. Helgason, D
York, 1962.
4. L. H6rmander
(1958), 555-568.
5. J. L. Lions, Pq
6. S. Lojasiewicz,
ables rtelles. C. R. Ac
7. L. Schwartz, 2
MASSACHUSETTS I
[November
even C functions
- D + (n- 1)/r D,
are singular when
rem 1 shows that
r X under which
Euclidean space a
-
19631
---
SOLUTIONS OF INVARIANT
----------
--
DIFFERENTIAL
----
--------
As to
OPERATORS
7/51
given by 6(f)=f(o) where fCC (S). Let DED(S). A distribution T
on S is called a fundamental solution of D if DT= S. If fE Cc (S), then
a fundamental solution T of D gives a solution of the equation
Du =f by putting u =f * T where * is the operation on distributions
on GIK induced by the convolution product of distributions on G.
THEOREM 2.
Each invariant differential operatorDED(S) (DO)
:e the correspond-
on the symmetric space S has a fundamental solution.
HE a)
This is a consequence of Theorem 1 and the fact that a nonzero
differential operator on Rn with constant coefficients always has a
tempered fundamental solution [4; 6].
on N. As proved
mapping of I(A)
Added in proof. In the case when G is complex the following formula (which is a simple consequence of Lemma 55 in [2]) gives a
simpler proof of Theorem 2.
1
omorphism of the
9(S) then
(DF) o Exp = -
X(D)(w(F o Exp))
(D E D(S)).
c
Here Exp is the usual Exponential mapping of p onto S, X is a certain
isomorphism of D(S) onto the algebra of Ad(K)-invariant polynomials on and F is any radial function on S.
for functions in
the recent result
the function c(X)
iS.
tes the following
tains all the W-
()(r)p(x
_1
REFERENCES
1. S. G. Gindikin and F. I. Karpelevic, Plancherel measure of Riemannian symmetric spaces of nonpositive curvature, Dokl. Akad. Nauk. SSSR 145 (1962), 252-255=
Soviet Math. Dokl. 3 (1962), 962-965.
2. Harish-Chandra, Sphericalfunctions on a semisimple Lie group. I, II, Amer. J.
Math. 80 (1958), 241-310, 553-613.
3. S. Helgason, Differential geometry and symmetric spaces, Academic Press, New
York, 1962.
4. L. H6rmander, On the division of distributions by polynomials, Ark. Mat. 3
(1958), 555-568.
5. J. L. Lions, quations diffirentielles optrationelles, Springer, Berlin, 1961.
6. S. Lojasiewicz, Division d'une distribution par une fonction analytique de vari-
I result of Gindiis equivalent to
I by the Planchis not proved in
ved it in recent,
,all therefore aspace of C func-
istribution on S
ables reelles. C. R. Acad. Sci. Paris 246 (1958), 683-686.
7. L. Schwartz, Thlorie des distributions. II, Hermann, Paris, 1951.
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