INOUE, TORU

Inoue, Toru

Osaka University Knowledge Archive

Integral formulas for harmonic functions associated with boundaries of a bounded symmetric domain

Osaka City University Repository

Inoue, T.
Osaka J. Math.
26 (1989), 527-534
INTEGRAL FORMULAS FOR HARMONIC FUNCTIONS
ASSOCIATED WITH BOUNDARIES OF A BOUNDED
SYMMETRIC DOMAIN
TORU INOUE
(Received October 20, 1988)
1. Introduction
In the case of the unit disc, or the upper half-plane in the theory of one
complex variable, the Poisson kernel can be expressed in terms of the Cauchy
kernel in the following simple way; in either case, denoting the Cauchy and the
Poisson kernels by ίf(z, w), £P(z, u) respectively
It is natural, therefore, to extend this definition whenever the Cauchy kernel is
defined. Hua [3] did this for four classical types of bounded symmetric domains
and established some of its basic properties. For generalized half-planes this was
done by Kor&nyi [6] who then used the theory of Cayley transform to determine
the Cauchy and Poisson kernels for all the bounded symmetric domains (See
also [8], [5]).
It is known that the Poisson kernel has another interpretation; it can be
regarded as the Jacobian of an automorphism restricted to the boundary. This
way of viewing the Poisson kernel was shown to work on arbitrary non-compact
Riemannian symmetric spaces by Furstenberg [1]. For any symmetric domain
it turns out that these two possible definitions of the Poisson kernel coincide
(See [6], though it is not explicitly stated).
Now let D be an irreducible bounded symmetric domain in the canonical
Harish-Chandra realization. If r is the rank of D, then the topological bound-
ary dD breaks into r boundaries Bly •••, Bn such that BiZDBi+1(ί<i<r— 1), and
B
r
 is the Silov boundary. As is shown in [4], for each boundary JBt ( l < * < r ) ,
there is a natural measure σ, on Bt and a Cauchy type kernel function ίft (#, w)
such that
whenever / is holomorphic in a neighborhood of D9 the closure of D. For the
528 T. INOUE
Silov boundary B
r
, the function ίf
r
(z, w) is the usual Cauchy(-Szegϋ) kernel of
Z), from which Hua et al. denned the Poisson kernel by (1). Therefore it is
natural to define, for each boundary Bi9 the Poisson type kernel ίPt (#, u) by
putting
| 52 (fM ) ' 2 zeD,u<=Bt.g,(*,u) 2f
if ,-(#, z)
In this note we show that the kernel .£?,•(#, u) represents harmonic functions
/ in D in terms of the boundary values on Bi9 i.e.,
whenever / is harmonic in D and continuous on its closure D. We also show
that the kernel ίPt (#, u) can be regarded as the Jacobian of an automorphism
restricted to the boundary Bi9 i.e., if g is an automorphism of Z),
where o is the origin of D.
2. Preliminaries
We begin by reviewing the general background on bounded symmetric do-
mains (cf. [2], [9]). Every bounded symmetric domain D can be written as
D=G/K, where G is a connected semisimple linear Lie group and K is a max-
imal compact subgroup of G, such that G operates holomorphically on D. In
this note we assume that G is simple, i.e., that D is irreducible. We further
assume that the complexification G
c
 of G is simply connected. Let g, ϊ be the
Lie algebras of G, K and g=f-f-p be the corresponding Cartan decomposition.
We denote the complexifications of g, I, p by g
c
, f
c
, p
c
, respectively. Then p
c
 is
decomposed into the direct sum of two complex subalgebras p+, p~, which are
( ± V— l)-eigenspaces of the complex structure of py respectively, and are abelian
subalgebras of g
c
 normalized by ϊ
c
. Let P±y Kc, be the connected subgroups of
G
c
 corresponding to | ) ± , ϊ
c
, respectively. Then the map p+ X K
c
 X t>"->G
c
, given
by (X+, k, JY"~)-^exρ X + A exp X~, is a holomorphic diffeomorphism onto a
dense open subset P+KCP~ of Gc, which contains G. Therefore every element
g^P+K
c
P~ can be written in a unique way as
(2) g = *+(g)-*o(g)-*-(g), π
o
(g)^KCyπ±(g)^P±.
Furthermore, the map ζ: P+KCP~ —>$>+, given by ζ(g)=log (π+(g)) induces a
holomorphic diffeomorphism of D=G/K onto ζ(G), and ζ(G) is a bounded
domain in p+. Henceforce we assume that D is a bounded symmetric domain
INTEGRAL FORMULAS FOR HARMONIC FUNCTIONS 529
in p+ realized in this manner. In this realization the action of G on D is given
by
and extends smoothly to D.
Let t be a maximal abelian subalgebra of f. Then t
c
, the complexification
of t, is a Cartan subalgebra of g
c
. Let Φ be the root system of g
c
 relative to t
c
.
For each α E Φ , let H
Λ
, E
Λ
 denote the usual basis elements of g
c
. We can choose
a linear order in the dual of the real vector space V— 11 such that p+ is spanned
by the root spaces for noncompact positive roots. We let Φ + be the resulting
set of positive roots.
We choose a maximal set {γ^ •••,%.} of strongly orthogonal noncompact
positive roots as follows. Let y1 be the highest root of Φ and for each j , yj+1 be
the highest positive noncompact root that is strongly orthogonal to each of
{Ύv '">ΎJ}' We write HjyEj for Hyjy EΎj. For each \<i<ry we define the
partial Cayley transform c{ e Gc by
Since ci^P
+K
c
P~y we can define oi=ζ{ci). Let B{ denote the G-orbit of o, .
Then
D—D = U B{ (disjoint union).
Moreover
 JBί Z)JBί+1 ( l < i < r — 1), and Br is the Silov boundary.
Let C^CZBi) be the boundary component of D containing oiy and let JP, —
{ ^ G G ; g C—Ci} and S~{gG:G\ gΌ—o^. Then P f is a maximal parabolic
subgroup of G, and we have a Langlands decomposition P~Mi A
έ
 Nt such
that if we put Li=Mif]Si then Si=LiAiNi (cf. [4]). Further there exists a
semisimple subgroup G, of G such that Ci=GiΌi.
For each l<i<ry we define a C°° function ρ{ on G as follows. Since P{=
Mi A{ Ni is a parabolic subgroup, each g^G can be uniquely written in the form
g=kman (ΛeϋΓ, meM^ Πexp p, a^Ah neiVi); so put p ί(^)=(det(Ad(α)|n.))"1,
where rtt is the Lie algebra of N{. Let dk denote the Haar measure on K such
that \ dk=l. Then we can normalize various left Haar measures in such a way
that J *
(3) ( f{g) dg = \ f(kgiSi) Pi(Sir dk dgi ds(
for any integrable/ on G (cf. [4], p. 89).
Let {aly •••, a} be an enumeration of the set of simple roots for Φ + such
530 T. INOUE
that a
x
 is the unique noncompact simple root for Φ + , and let λ be the linear
form on t
c
 such that
2(λ, cCijlipί^ #i) = 1 and (λ, aj) = 0 for / = 2, •••,/
where ( , ) is the inner product induced by Killing form of g
c
. Then λ is the
differential of a holomorphic character of K
c
.
Let tc= ^ΣRHj. Then the restrictions of t
c
-roots to tc are of the form
1
± 7 / (each with multiplicity one), i—(7y±7*) (j<k, each with the same multi-
1 ^
plicity u>0)y ± — Ύj (each with the same multiplicity 2 Ϊ ; > 0 ) . For each 1 <i<r,
let (as in [4], p. 91)
(4) A = i - u ( i - l ) + « ( r - i ) + ^ + l >
and set ω{ = —pi λ. Note that each j ^ is an integer or a half-integer. If _pf. is
an integer, ω, is also the differential of a holomorphic character of K
c
. For the
moment we assume that this is the case and let τ, be the corresponding character
of KCy i.e., Ti=e
ωi. We define/,-: GxD->Cx (C x =the multiplicative group of
non-zero compex numbers) by
Mg> *) = Ti(πo(g exp z))
where π0 is as in (2). Then we have
/*(&&> *) = / i ( f o f t *)/i(fi) *
Let %: KC->C
X
 be a holomorphic character of JK"C defined by
X(Λ) = det(Ad(Λ)|p+),
and let J C : u x O ~ > C x bea function defined by
JC(z, w) = %(τr
o
(exp(—
where w-*ffi denotes the complex conjugation of g
c
 with respect to g. Then,
up to a constant factor, Jζ(z, w) is the Bergman kernel functoin of D (cf. [7], [4]).
Let n=diτn
c
 D, ni=dhnc Ch and έ/l =dim Λ Bh and set
q, =
Since DxD is simply connected, we can define powers JC(z, w)"' of
with JC(o,oγι= 1. We let
y,(*. w) =
INTEGRAL FORMULAS FOR HARMONIC FUNCTIONS 531
For a fixed #eZ), #*,(#, •) extends smoothly to D. If p{ (in (4)) is an integer,
then it follows from Lemma 6.24 of [4] that
ίfi(z, w) =
 Ti(π0(exp(—W) exp *)) ,
and we have
Up to now we have assumed that the p{ is an integer. We note that, even if
pi is a half-integer, Ji{g,z)2 is a well defined function on GxD, and satisfies
the following properties
(5)
(6) (r{(g *>g"»?=Mi>*?U*>
Let dσ{ be the quasi-invariant measure on B—G/Si defined by
f f 2
for all /e= CC(B{) (continuous functions with compact support). Then Proposi-
tion 4.38 of [4] implies that
C dσi{u)=\ \Ukgi,o,)-2\dkdgi<oo.
JBi JK*Gi
Therefore we can (and do) normalize the Haar measure dg{ on Gt so that
(7) j ^ ώ - , ( t t ) = l .
It then follows from formula (6.15) of [4] that
whenever / is holomorphic in the neighborhood of D .
3. Integral formulas
As in the introduction we define, for each boundary JS, ( l ^ / < r ) , the Poisson
type kernel function £?,(#, u) by putting
Proposition. For g^G,u^Bhwe have
2>i(g-o,»)= \Mg-\un =
532 T. INOUE
Proof. Since ίf^o, w)=l, (5) and (6) imply
and
So the first equality follows from the definition of ζP^g Oy u).
For the second equality, it suffices to show that
t /(*'«) dσ{(u) = ( \Ug-\u)-*\f{u)dσi{u)
JBi JBi
for / e C
c
(Bi). We first note that
(8) \Ji(Si,od-2\=Pi(*i) for *,&S,;
this follows from the argument in the proof of Lemma 6.30 of [4]. Now for
each/eCXB,-), we can take f^C
c
(G) such that
Hence
)Bf(S'u) σΛu)
= \
κχG \Ji{kgi, ox.)"2l fighgroi) dk dgi
=
 \ XGXS ^^kgiSh O ί ' ) " 2 ' f(SkSisi) Pi^d'1 dkdgi dSi (by (5) and (8))
h (by (3))
= [ I Mg'1 kgish o,.)-21 / ( ^ Λ ) PiiSiY1 dk dgi ds(
= ( \Ug-1 kg,, od-'lfikgrod dk dgi
= ( \Ug-\kgi ojγ2\ \Ji{kgi,oi)-*\f(kgroi)dkdgi (by (5))
This proves the Proposition.
Theorem. If f is harmonic on D and continuous on D, then for all
f(z)=\ <?,(*,«)/(«) dσ,{u).
INTEGRAL FORMULAS FOR HARMONIC FUNCTIONS 533
Proof. For each ^Gfl, choose g^G such that z=g o. Then, by the mean
value theorem for harmonic functions (cf. [2]), we have
/(*) = \j{gk w) dk
for weD. The continuity of / on D implies that this formula is valid for all
. Therefore
/(*)*•*(«) (by (7))
~\ (\ f(S'u) d<Γi(u)) dk (by ^ -invariance of dσf )
= \Bf(g u)dσi(u)
= I 3*i(z, u)f(u) dσi(u) (by Proposition).
J Bi
This finished the proof.
REMARK. If z'Φr, the maximal compact subgroup K of G does not act
transitively on the boundary B{. Therefore Proposition implies that the Poisson
type kernel £?,(#, u) is not necessarily harmonic in the variable z.
EXAMPLE. Let p > q and
D= iz^Mfitq(C); l f-***>0} .
Here MPtq(C) refers to all p by q complex matrices, lq is the identity matrix
of size q, z* is the conjugate transpose of z and " > 0 " means "is positive definite".
Then (cf. [9]) D is the bounded symmetric domain of rank q, and for each
?, the j-th boundary B{ is given by
B{ = {ztEMp>q(C)\ l , - ^ * ^ > 0 and rank( l ,-Λr) = q—i} .
On the other hand the Cauchy type kernel function 5P,. (z> w) associated with
the boundary B{ is given by (cf. [4], p. 129)
Therefore the Poisson type kernel function ίPf (#, «) is given by
I det (!.—«**) I
534 T. INOUE
References
[1] H. Furstenberg: A Poisson formula for semi-simple Lie groups, Ann. of Math. 77
(1963), 335-386.
[2] S. Helgason: Differential Geometry and Symmetric Spaces, Academic press,
1962.
[3] L.K. Hua: Harmonic Analysis of Functions of Several Complex Variables in
the Classical Domains, Trans. Math. Monographs, vol. 6, Amer. Math. Soc. 1963.
[4] T. Inoue: Unitary representations and kernel functions associated with boundaries of
a bounded symmetric domain, Hiroshima Math. J. 10 (1980), 75-140.
[5] A. Koranyi: A Poisson formula for homogeneous wedge domains, J. Analyse Math.
14 (1965), 275-284.
[6]
 : The Poisson integral for generalized half-planes and bounded symmetric
domains, Ann. of math. 82 (1965), 332-350.
[7] I. Satake: Algebraic Structures of Symmetric Domains, Pub. Math. Soc. Japan
14, Iwanami, Tokyo and Princeton Univ. Press, 1980.
[8] E.M. Stein, G. Weiss and M. Weiss: Hp-classes of holomorphic functions in tube
domains, Proc. Nat. Acad. Sci. U.S.A. 52 (1964), 1035-1039.
[9] J.A. Wolf: Fine structure of hermitian symmetric spaces, in "Symmetric spaces",
(ed. W. Boothby and G. Weiss), Marcel Dekker, 1972, 271-357.
Department of Mathematics
Fuculty of Science
Yamaguchi University
Yamaguchi 753, Japan


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Integral formulas for harmonic functions associated with boundaries of a bounded symmetric domain

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