k/ (k-1)
RADON-FOURIER TRANSFORMS ON SYMMETRIC
SPACES AND RELATED GROUP REPRESENTATIONS
lowing property:
kaving boundary
BY S. HELGASON
,ing the minimal
Communicated by G. D. Mostow, May 7, 1965
In §2 we announce some results in continuation of [10], connected
with the Radon transform. §1 deals with tools which also apply to
more general questions and § §2-3 contain some applications to group
representations. A more detailed exposition of §2 appears in Proceedings of the U. S.-Japan Seminar in Differential Geometry, Kyoto,
<1},
's > 0 with the
nanifold of class
4 into Rn having
at
.
1
and
and
June, 1965.
1. Radial components of differential operators. Let V be a manifold, v a point in V and V, the tangent space to V at v. Let G be a Lie
transformation group of V. A C- function f on an open subset of V is
called locally invariant if Xf= 0 for each vector field X on V induced
by the action of G.
Suppose now W is a submanifold of V satisfying the following
transversality condition:
(direct sum).
For each w E W, V, = W + (G-w)
(T)
Iff is a function on a subset of V its restriction to W will be denoted
7.
1.1. Let D be a differential operator on V. Then there exists a
unique differential operator A(D) on W such that
LEMMA
:alculus in the large
(Df)- = A(D)f
for each locallyinvariantf.
The operator A(D) is called the radial component of D. Many special
cases have been considered (see e.g. [1, §2], [4, §5], [5, §3], [7, §7],
[8, Chapter IV, §§3-5]).
Suppose now dv (resp. dw) is a positive measure on V (resp. W)
which on any coordinate neighborhood is a nonzero multiple of the
Lebesgue measure. Assume dg is a bi-invariant Haar measure on G.
Given u E Cc (G X W) there exists [7, Theorem 1] a unique
f, G C (G *W) such that
fX
Let
F(g
. w)u(g, w) dgdw = f F(v)f,(v) dv
(F
C(G W)).
CEG
Cc (W) denote the function w-+fu(g, w)dg.
'Work supported in part by the National Science Foundation, NSF GP-2600.
757
758
S. HELGASON
[September
THEOREM1.2. Suppose G leaves dv invariant. Let T be a G-invariant
distribution on G W. Then there exists a unique distribution T on W
such that
u E Cc(G X W).
T(¢u) = T(f),
If D is a G-invariantdifferentialoperatoron V then
(DT)- = A(D)T.
The proof is partly suggested by the special case considered in [7,
§9]. See also [12, §4].
2. The Radon transform and conical distributions. Let G be a connected semisimple Lie group, assumed imbedded in its simply connected complexification. Let K be a maximal compact subgroup of
G and X the symmetric space G/K. Let G=KAN
the dual space c
Plancherel meas
(Harish-Chandrn
creasing functioi
corresponds to
multiplicity) of
ordering given I
eP(kaMN) = exp I
with base K/M
suitable function
restriction to an
determines an o]
tioned one prov
THEOREM 2.1.
be an Iwasawa
decomposition of G (A abelian, N nilpotent) and let M and M', respectively, denote the centralizer and normalizer of A in K. The space
' of all horocycles in X can be identified with G/MN [10, §3]. Let
D(X) and D(z) denote the algebras of G-invariant differential operators on X and X, respectively; let S(A) denote the symmetric algebra over the vector space A and I(A) the set of elements in S(A)
which are invariant under the Weyl group W= M'/M. There are isoof
morphisms r of D(X) onto I(A) [6, p. 260], [9, p. 432] and
D(S) onto S(A) [10, p. 676].
The Radon transformf-
I965]
Cj(fE
C (X)) and its dual q- (E C (Z))
(2)
(3)
for allfE Cc (X)
If all Cartan
are differential (
element in I(A)
in the form
are defined by
f()
=
f(x)dm(x),
(x)==
(d(
(x E X, E )
which is more cc
G is complex a
Graev [2, §5.5].
where dm is the measure on t induced by the canonical Riemannian
structure of X, x is the set of horocycles passing through x and d/ is the
measure on x invariant under the isotropy subgroup of G at x, satisfying IA(x)=1. The easily proved relation
(1)
f(x)(x)d = f f(t)(fdt
(fG Cl*(X>,0
C-_(:())
dx and dS being G-invariant measures on X and 5, respectively, suggests immediately how to extend the integral transforms above to
distributions.
Let @0 and W be the Lie algebras of G and A, respectively, and W9*
Let xoand 0o(
B = K/M can be
xo in X. If E= ka
is normal to t an
bEB let ,(x, b)
and let A (x, b) c
THEOREM 2.2
f(, b)=
Then
RADON-FOURIER
1965]
[September
759
TRANSFORMS
the dual space of W. Let X--c(X) be the function on * giving the
Plancherel measure I c(X)I -2dX for the K-invariant functions on X
(Harish-Chandra [6, p. 612]). Let j be the operator on rapidly decreasing functions on A which under the Fourier transform on A
corresponds to multiplication by c-1. Let p denote the sum (with
multiplicity) of the restricted roots on [ which are positive in the
ordering given by N. Let ep denote the function on Z defined by
eP(kaMN) = exp [p(log a) ] (kCK, aCA). Viewing ' as a fibre bundle
e a G-invariant
bution T on W
E Cc(G X W).
with base K/M,
fibre A [10, p. 675] we define the operator A on
suitable functions ¢ on Z by (ePA) I F=j((ePo) F), where I F denotes
restriction to any fibre F. Similarly, the complex conjugate of c-l
determines an operator . By means of the Plancherel formula men-
)nsidered in [7,
Let G be a con-
tioned one proves (cf. [11, §6]).
.ts simply conct subgroup of
)e an Iwasawa
THEOREM
M and M', re-
2.1. There exist constants c, c' >0 such that
If(x) dx = c'
(2)
il Af() 2d,
n K. The space
V [10, §3]. Let
f = c(A. )-
(3)
fferential oper-
for all f
Fmmetric alge-
nents in S(A)
There are iso-
If all Cartan subgroups of G are conjugate, the operators j and A
are differential operators (c-l is a polynomial). Considering jj is an
element in I(A) we put F] =r-(jj)ED(X).
Then (3) can be written
in the form
432] and P of
-
(+E C
C (X).
))
f = cEl (() -)
f
CC_(X),
which is more convenient for applications [10, §7]. For the case when
G is complex a formula closely related to (3) was given by GelfandGraev [2, §5.5].
Let xoand 40denote the origins in X and Z, respectively. The space
B = K/M can be viewed as the set of Weyl chambers emanating from
xo in X. If =ka-o (kCK, aCA) we say that the Weyl chamber kM
is normal to and that a is the complex distance from xo to . If xCX,
bEB let (x, b) be the horocycle with normal b passing through x,
and let A (x, b) denote the complex distance from x0 to (x, b).
al Riemannian
x and d is the
G at x, satisfy-
C), CC(;))
THEOREM 2.2. For fE
pectively, sug)rms above to
f(x, b) =
tively, and 92*
Then
I
C~(X) definethe Fouriertransformf by
f(x) exp[(-iX + p)(A(x, b))]dx
(x C * , b E B).
760
[September
S. HELGASON
(Xb) exp (i
*xB
f() =
(4)
f
(X)
12d =
,XB
+ p)(A(x, b))] c(X) -2dXdb
be defined by (5).
(i) In view of the analogy between horocycles in X and
hyperplanes in Rn formula (4) corresponds exactly to the Fourier inversion formula in Rn when written in polar coordinate form.
(ii) If f is a K-invariant function on X, Theorem 2.2 reduces to
Harish-Chandra's Plancherel formula [6, p. 612]. Nevertheless,
Theorem 2.2 can be derived from Harish-Chandra's formula.
(iii) A "plane wave" on X is by definition a function on X which is
constant on each member of a family of parallel horocycles. Writing
(4) in the form
tribution on Z, is
THEOREM 2.5.
t given by (5) wi
(7)
DEFINITION.I
(2) conical if thel
we get a continuous decomposition of f into plane waves. On the
other hand, if we write (4) in the form
fx(x)l
for each restricted
induced by the 1
called (1) spheric
f(x) = fb(x) db
f(x) =
Re
(6)
REMARKS.
(4/")
In general b,
However, we ha
THEOREM 2.4.
I(X,b) 12 c(x) -2ddb,
where db is a suitably normalized K-invariant measure on B.
(4')
I9651
The correspon
cal representatio
situation
for
:, J
vector space spa:
natural represent
(X) -2dX
THEOREM 2.6.
we obtain a decomposition of f into simultaneous eigenfunctions of
all DE D(X).
We now define for Z the analogs of the spherical functions on X.
A distribution (resp. C function) on =G/MN is
DEFINITION.
called conical if it is (1) MN-invariant; (2) eigendistribution (resp.
eigenfunction) of each DED(Z).
Let o = MN, **=m*MN, where m* is any element in M' such
that the automorphism a--+m*am*- l of A maps p into -p. By the
Bruhat lemma, Z will consist of finitely many MNA-orbits; exactly
one, namely * = MNA .*, has maximum dimension and given
EZ* there exists a unique element a(t) EA such that EMNa(})
.*
[10, p. 673]. Using Theorem 1.2 we find:
THEOREM2.3. Let T be a conical distribution on S. Then there exists
aPGeC(*) such that T=p on * and a linear function t: I->-Csuch
that
(5)
(5())= V,(/*)exp[u(log a(t))]
(NE
*).
on ~ onto the set
tions of G. The m
functions and ide
where e and e', r
of era and of its t
highest weight of:
COROLLARY 2.'
tion of G. Then 7r
The highest w
acterized by (7).
of the spherical r
3. The case of
of the results of
[September
II c(X)
I965]
RADON-FOURIER
itl
TRANSFORMS
In general I is singular on the lower-dimensional MNA-orbits.
However, we have:
-2dXdb
THEOREM 2.4. Let j4: A---C be a linear function and let 4f
C°(E*)
be defined by (5). Then 4' is locally integrable on Z if and only if
(6)
on B.
cycles in X and
the Fourier in:e form.
2.2 reduces to
Nevertheless,
rmula.
1 on X which is
:ycles. Writing
Re ((a,.
for each restricted root a> 0; here ( , ) denotes the inner product on 2i*
induced by the Killing form of ®. If (6) is satisfied then {, as a dis-
tributionon ~, is a conicaldistribution.
THEOREM 2.5. The conical functions on Z are precisely the functions
a given by (5) where for each restricted root a > O,
(I,
) .
-" "~ is an integer>0.
(7)
(a, a)
DEFINITION. A representation
vaves. On the
(Re = real part)
+ p)) > 0
r of G on a vector space E will be
called (1) spherical if there exists a nonzero vector in E fixed by r(K);
(2) conical if there exists a nonzero vector in E fixed by r(MN).
The correspondence between spherical functions on X and spherical representations is well known. In order to describe the analogous
situation for ~, for an arbitrary function f on ~, let Ek denote the
vector space spanned by the G-translates of 4and let 7r,denote the
natural representation
of G on En.
THEOREM 2.6. The mapping 4--try maps the set of conical functions
enfunctions of
tnctions on X.
on onto the set of finite-dimensional, irreducible conical representations of G. The mapping is one-to-oneif we identify proportional conical
functions and identify equivalent representations. Also
=G/MN is
:ibution (resp.
i,(g o)= (ro(g-1)e, e'),
it in M' such
where e and e', respectively, are contained in the highest weight spaces
of rg and of its contragredient representation. Finally, tu in (5) is the
o -p. By the
)rbits; exactly
highest weight of
on and given
~E MNa()
*
ten there exists
yu: -->C such
iri.
COROLLARY2.7. Let 7rbe a finite-dimensional
irreducible
representa-
tion of G. Then wr
is sphericalif and only if it is conical.
The highest weights of these representations are therefore characterized by (7). Compare Sugiura [13], where the highest weights
of the spherical representations are determined.
3. The case of a complex G. If G is complex, M is a torus and some
(VIE
"*)-
of the results of §2 can be improved. Let
.
be a Cartan subalgebra
762
S. HELGASON
[September
I965
7.
of (Dcontaining Wand H the corresponding analytic subgroup of G.
Now we assume G simply connected.
Let D(G/N) denote the algebra of all G-invariant differential operators on GIN. Let o, v*eG/N be constructed similarly as 0oand *
in §2. Then §1 applies to the submanifold W=H.v* of V=NH.,*
and for each differential operator D on G/N, A(D) is defined and can
be viewed as a differential operator on H.
9. ---10.
form, Bull. Amer. Ma
12. P-D. Methee,
Lorentz, Comment. M
13. M. Sugiura,
Moreover
f(g o) = (rf(g-1)e, e'),
where e and e', respectively, are contained in the highest weight spaces
of crfand of its contragredientrepresentation.
REFERENCES
1. F. A. Berezin, Laplace operatorson semisimple Lie groups, Trudy Moskov.
Mat. Ob. 6 (1957), 371-463; Amer. Math. Soc. Transl. 21 (1962), 239-299.
2. I. M. Gelfand and M. I. Graev, The geometry of homogeneous spaces, group
representations in homogeneous spaces and questions in integral geometry related to them.
I, Trudy Moskov. Mat. ObgW.8 (1959), 321-390;Amer. Math. Soc. Transl. 37 (1964),
351-429.
3. R. Godement, A theory of spherical functions. I, Trans. Amer. Math. Soc. 73
(1952), 496-556.
4. Harish-Chandra, The charactersof semisimple Lie groups, Trans. Amer. Math.
Soc. 83 (1956), 98-163.
5. , Differential operatorson a semisimple Lie algebra,Amer. J. Math. 79
(1957), 87-120.
6.
~,
Sphericalfunctions on a semisimple Lie group. I, II, Amer. J. Math. 80
(1958), 241-310, 553-613.
2
INSTITUTE FOR Ar
I
set of N-invariant holomorphic eigenfunctions of all DED(G/N) (proportional f identified) onto the set of all finite-dimensional2 irreducible
holomorphic representations of G (equivalent representations identified).
Compare the problem indicated in 3, p. 553].
1
on symmetric spaces, I
As a consequence one finds that the N-invariant eigenfunctions
fC C°(G/N) of all DCD(G/N) have a representation analogous to
(5) in terms of the characters of H. Let Ef denote the vector space
spanned by the G-translates of f and let 7rfbe the natural representaTHEOREM 3.2. The mapping f-->rf is a one-to-one mapping of the
, Dualit
(1963), 667-692.
, A duG
11.
In particular, D(G/N) is com-
tion of G on Ef.
, Differe
1962.
THEOREM3.1. The mapping D--A(D) is an isomorphismof D(G/N)
onto the (real) symmetric algebra S().
mutative.
, Invaric
309.
8. S. Helgason, 1
(1959), 239-299.
I
[September
RADON-FOURIER TRANSFORMS
I9651
*
subgroup of G.
7.
763
, Invariant distributions on Lie algebras,Amer. J. Math. 86 (1964), 271-
309.
8. S. Helgason, Differential operators on homogeneous spaces, Acta Math. 102
(1959), 239-299.
9.
~, Differential geometry and symmetric spaces, Academic Press, New York,
lifferential operArlyas 6oand *
of V=NH.v*
1962.
defined and can
10.
~,
Duality and Radon transformfor symmetric spaces, Amer. J. Math. 85
(1963), 667-692.
11. -, A duality in integralgeometry;some generalizationsof the Radon trans.
form, Bull. Amer. Math. Soc. 70 (1964), 435-446.
hism of D(G/N)
12. P-D. Methee, Sur les distributions invariantes dans le groupe des rotations de
D(G/N) is com-
Lorentz, Comment. Math. Helv. 28 (1954), 225-269.
13. M. Sugiura, Representationsof compact groups realized by sphericalfunctions
on symmetric spaces, Proc. Japan Acad. 38 (3) (1962), 111-113.
: eigenfunctions
rn analogous to
he vector space
ural representa-
INSTITUTE FOR ADVANCED STUDY
mapping of the
_D(G/N) (pronal2 irreducible
,ions identified).
,st weight spaces
s, Trudy Moskov.
), 239-299.
zeous spaces, group
etry related to them.
:. Transl. 37 (1964),
ner. Math. Soc. 73
rans. Amer. Math.
Amer. J. Math. 79
Amer. J. Math. 80
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