Positive Riesz distributions on

homogeneous cones

By Hideyuki Ishi

Abstract. Riesz distributions are relatively invariant distributions sup-

ported by the closure Ω of a homogeneous cone Ω. In this paper, we

clarify the positivity condition of Riesz distributions by relating it to the

orbit structure of Ω. Moreover each of the positive Riesz distributions

is described explicitly as a measure on an orbit in Ω.

Introduction.

The study of Riesz distributions originates from the paper [9] of Marcel Riesz,

where he introduces a family of convolution operators as a generalization of the

classical Riemann-Liouville operators on the positive reals to the Lorentz cone in Rn.

Riesz’s distributions are the distributions determined by the homogeneous functions

rα(x) := (x21 − x2

2 − · · · − x2n)α/2 (α ∈ C) divided by Γ -factors (see for example the

book [13, (II.3;31)] or the article [2] for details). His aim was to solve the wave

equation, and a systematic use of analytic continuation is made. Among subsequent

works carried out by various mathematicians, Gindikin’s paper [5] was the first to

consider a generalization of Riesz’s distribution on an arbitrary homogeneous cone Ω.

Gindikin replaces the above rα by functions ∆s (s = (s1, s2, . . . , sr) ∈ Cr) on Ω which

are relatively invariant under a split solvable Lie group H acting linearly and simply

transitively on Ω. When Ω is the cone of positive definite real symmetric matrices, ∆s

is the product Ds1−s21 · · ·Dsr−1−sr

r−1 Dsrr of the principal minors D1, . . . , Dr−1, Dr = det.

He also investigates the gamma function ΓΩ associated to the cone Ω playing the role

of the Γ -factors in Riesz’s distribution. Such being the case, we call the distributions

Rs determined by ΓΩ(s)−1∆s dµ the Riesz distributions on Ω, where dµ is the H-

invariant measure on Ω.

1991 Mathematics Subject Classification: Primary 43A85; Secondary 46F10, 22E30.

Key Words and Phrases : Riesz distribution, homogeneous cone, normal j-algebra.

1

2

Our major concern for the Riesz distributions Rs is to determine the set Ξ of

the parameter s for which Rs is a positive measure. The set Ξ plays an important

role in the study of Hilbert spaces of holomorphic functions on Siegel domains as

is done by Vergne and Rossi [14] to deal with the analytic continuation of the

holomorphic discrete series representations of semisimple Lie groups (see also [3],

[16] for this). For general cones, the set Ξ is described in Gindikin’s paper [6]. Since

the intersection of Ξ with the set (α, α, . . . , α) |α ∈ C is seen to be identical with

the Wallach set in the case of symmetric cones (cf. [3]), we shall call Ξ the Gindikin-

Wallach set. However, only a very rough sketch is given in [6], and the complete

details seem to be unpublished until now. Proofs available to us as of now are for the

case of symmetric cones treated in [3, Chapter VII, Section 3], where a fine structure

of Euclidean Jordan algebras is thoroughly utilized. The motivation of the present

work was to improve this situation. But more is done in this paper. In fact, not

only do we have a clearer description of Ξ by relating it to the H-orbit structure of

Ω, but also we give explicitly a positive measure with its support specified for each

of the elements in Ξ. Moreover, a simple algorithm for the specification is supplied.

Let us explain our results in more detail. The basic tool with which we work in

place of Jordan algebras is the structure theory of normal j-algebras developed in

[8]. Based on the fact that the normal j-algebras are in one-to-one correspondence

with the Siegel domains, we shall start with a normal j-algebra g such that the

corresponding Siegel domain is of tube type. Thus g is graded as g = g(0) ⊕ g(1)

and our cone Ω is sitting in V := g(1). Moreover, h := g(0) is the Lie algebra

of the group H acting simply transitively on Ω. The structure theorem of normal

j-algebras (see Theorem 1.1 for details) says that there is a basis α1, . . . , αr for the

roots of g, so that h and V are respectively direct sums of the root spaces:

h = a⊕( ⊕

1≤k<m≤r

g(αm−αk)/2

)

, V =( r⊕

k=1

gαk

) ⊕( ⊕

1≤k<m≤r

g(αm+αk)/2

)

,

see (1.3), (1.7) and (1.8). Let Ek ∈ gαk(k = 1, 2, . . . , r) be the root vectors such

that [jEk, El] = δklEl. Our first theorem is the following.

Theorem A. The 2r elements Eε := ε1E1 + · · · + εrEr (ε = (ε1, . . . , εr) ∈0, 1r) form a complete set of representatives of the H-orbits in Ω:

Ω =⊔

ε∈0,1r

Oε (Oε := H · Eε).

3

In order to state our second theorem, we put for every ε ∈ 0, 1r

pk(ε) :=∑

i<k

εi dim g(αk−αi)/2,

Ξ(ε) := s ∈ Rr | sk = pk(ε)/2 (if εk = 0), sk > pk(ε)/2 (if εk = 1) .

Theorem B. The Riesz distribution Rs is a relatively invariant positive mea-

sure on Oε under the action of H if and only if s ∈ Ξ(ε). Thus the Gindikin-Wallach

set Ξ is the disjoint union

Ξ =⊔

ε∈0,1r

Ξ(ε).

It is easy to verify that our description of Ξ reduces to that of Faraut and Korany

[3, p. 138] for the case of symmetric cones (see Remark given after Theorem 6.2).

To describe the measure that Rs (s ∈ Ξ(ε)) induces on Oε we will pick up a

subgroup H(Oε) of H which is diffeomorphic to Oε (see (3.4) or (3.5) for an explicit

description of H(Oε)). Then transferring the left Haar measure on H(Oε) to Oε

by the orbit map Ψε : t 7→ t · Eε, we have an H(Oε)-invariant measure µε on Oε.

The measure µε turns out to be relatively invariant under H (Proposition 4.1).

Then the measure dRs (s ∈ Ξ(ε)) is a positive number multiple of ∆εε·s dµε with

the density ∆εε·s which is the transfer of a character of H(Oε) to Oε, where ε · s :=

(ε1 ·s1, . . . , εr ·sr). Furthermore, the positive constant is the reciprocal of the gamma

function ΓOεassociated to the orbit Oε (see (4.8) for the definition of ΓOε

).

Theorem C. The measure dRs (s ∈ Ξ(ε)) is given by

dRs = ΓOε(ε · s)−1∆ε

ε·s dµε.(0.1)

Theorem C is essentially proved in Section 5. The function ΓOεitself is a gener-

alization of the function ΓΩ studied by Gindikin [5]. We show in Theorem 4.2 that

it is expressed as a product of the usual gamma functions.

Theorem D. Up to a positive multiple depending on the normalization of the

measure concerned, one has

ΓOε(s) =

∏

εi=1

Γ

(

si −pi(ε)

2

)

.

We now describe the organization of this paper. In Section 1, we summarize the

basic structure of normal j-algebras. Section 2 is devoted to giving various explicit

formulas. This is done by expressing the elements in H by lower triangular matrices

with vector entries (see (2.4)). The idea of such expression is due to Vinberg [15] (see

4

also [5]), where the theory is based on the left symmetric algebras. Our formulation

is inspired by the book [3, Chapter VI, Section 3] of Faraut and Korany in which

the framework of Jordan algebra is developed.

In Section 3, we study the H-orbit structure and an inductive structure of Ω.

The explicit formulas obtained in Section 2 play a fundamental role here. We remark

that the inductive description of homogeneous cones is first given by Vinberg [15]

and further pursued by Rothaus [12] and Dorfmeister [1]. Theorem A above is

proved in Theorem 3.5

The purpose of Section 4 is to introduce and investigate the gamma function

ΓOεassociated to Oε for every ε ∈ 0, 1r mentioned above. The definition of ΓOε

requires a function and a measure on Oε which are relatively invariant under the

action of H. These two relative invariants are defined through the above-mentioned

diffeomorphism Ψε : H(Oε) → Oε. It is essentially from the convergence condition

of the defining integral (4.8) of ΓOεthat the condition for Ξ(ε) is derived.

In Section 5, we consider the analytic continuations of the distributions Rεs

obtained from the relatively invariant measures on Oε defined by the right-hand

side of (0.1). This is done in Theorem 5.1, and comparing the Laplace transforms

of these distributions, we see in Theorem 5.2 that these distributions coincide with

the Riesz distributions in each of their definition domains of the parameter s.

In the last section, Section 6, the Gindikin-Wallach set Ξ is determined. Prop-

erly speaking, we define Ξ as the disjoint union of Ξ(ε)’s and prove that Ξ is exactly

the positivity set for the Riesz distribution. The proof is carried out by induction

on the rank of cones, and in this regard, the inductive description of Ω in Sec-

tion 3 is of substantial importance. The algorithm mentioned above is given after

Proposition 6.1.

Let us fix some general notations used in this paper. Let U be a real vector space

and U∗ its dual vector space. For A ∈ End(U), A∗ denotes the adjoint operator of

A, that is, A∗ξ := ξ A (ξ ∈ U ∗). Moreover, if U is endowed with an inner product,

we denote by S(U) the Schwartz space of rapidly decreasing functions on U .

The present author would like to express his sincere gratitude to Professor

Takaaki Nomura for the encouragement in writing this paper. The author is also

grateful to Professor Masakazu Muro for his interest in this work.

5

§1. Preliminaries.

Let V be a finite-dimensional vector space over R and Ω an open convex cone

in V containing no line. We assume that Ω is homogeneous. This means that the

group GL(Ω) of linear automorphisms of the cone Ω acts on Ω transitively. Let

TΩ := V + iΩ be the corresponding tube domain in the complexification VC of V .

Then TΩ is homogeneous under the affine transformations

x+ iy 7→ a+ t · x + it · y (a ∈ V, t ∈ GL(Ω)).

The tube domain TΩ is also called a Siegel domain of type I, a special case of Siegel

domains of type II (see [8] for definition). Let D be a Siegel domain of type II on

which the group Hol(D) of holomorphic automorphisms of D acts transitively. By

[7] there is a split solvable closed subgroup G of Hol(D) acting simply transitively on

D, and the Lie algebra g of G has a structure of normal j-algebra. Conversely any

normal j-algebra gives rise to a homogeneous Siegel domain of type II as described

in [8, Chapter 2, Section 5], [11, Section 4A]. These observations ensure that we lose

no generality by beginning the present paper with a normal j-algebra and assuming

our homogeneous cone to be defined through the normal j-algebra.

Now let g be a real split solvable Lie algebra, j a linear automorphism on g such

that j2 = −idg, and ω0 a linear form on g. The triple (g, j, ω0) is called a normal

j-algebra if the following three conditions are satisfied:

(i) [Y, Y ′] + j[Y, jY ′] + j[jY, Y ′] − [jY, jY ′] = 0 for all Y, Y ′ ∈ g,

(ii) 〈[jY, jY ′], ω0〉 = 〈[Y, Y ′], ω0〉 for all Y, Y ′ ∈ g,

(iii) 〈[Y, jY ], ω0〉 > 0 for all non-zero Y ∈ g.

By (ii) and (iii), we define an inner product (·|·)ω0 on g as

(Y |Y ′)ω0 := 〈[Y, jY ′], ω0〉 (Y, Y ′ ∈ g).(1.1)

Let a be the orthogonal complement of [g, g] relative to this inner product. It is

known that a is a commutative subalgebra of g and ad(a) is a commutative family

of self-adjoint operators on g. The dimension r := dim a is called the rank of g. For

a linear form α on a, we set

gα := Y ∈ g | [C, Y ] = 〈C, α〉Y for all C ∈ a .

Then [gα, gβ] ⊂ gα+β, and if α 6= β, we have gα ⊥ gβ with respect to the inner prod-

uct (1.1). If α 6= 0 and gα 6= 0, we call α a root and gα the root space corresponding

to α. Concerning normal j-algebras, the following theorem is fundamental.

6

Theorem 1.1 (Pyatetskii-Shapiro [8]). (i) There is a linear basis A1, . . . , Arof a such that if one puts El := −jAl, then [Ak, El] = δklEl (1 ≤ k, l ≤ r).

(ii) Let α1, . . . , αr be the basis of a∗ dual to A1, . . . , Ar. Then the possible roots

are of the following forms:

αk,

(αm − αk)/2,

αk/2

(αm + αk)/2

(1 ≤ k ≤ r),

(1 ≤ k < m ≤ r).

(iii) The root spaces gαk(1 ≤ k ≤ r) are one-dimensional: gαk

= REk.

(iv) If m > k, then jg(αm−αk)/2 = g(αm+αk)/2, and the action of j is given by

jY = −[Y,Ek] (Y ∈ g(αm−αk)/2).(1.2)

Moreover jgαk/2 = gαk/2.

Let us put

g(1) :=( r⊕

k=1

gαk

) ⊕( ⊕

1≤k<m≤r

g(αm+αk)/2

)

, g(1/2) :=

r⊕

k=1

gαk/2,

g(0) := a⊕( ⊕

1≤k<m≤r

g(αm−αk)/2

)

.

(1.3)

Then we have g = g(1) ⊕ g(1/2) ⊕ g(0). We also put

E := E1 + · · ·+ Er, A := A1 + · · · + Ar(1.4)

for simplicity. Then ad(A)Y = µY for Y ∈ g(µ). Furthermore

[g(µ), g(ν)] ⊂ g(µ+ ν) (µ, ν = 0, 1/2, 1),(1.5)

where we understand g(µ) = 0 for µ > 1. It is immediately seen from Theorem 1.1

that jg(0) = g(1) with

jY = −[Y,E] (Y ∈ g(0))(1.6)

and jg(1/2) = g(1/2).

It is known that normal j-algebras corresponding to tube domains are those for

which g(1/2) = 0. Henceforth, let g be a normal j-algebra such that g(1/2) = 0.Put V = g(1) and h = g(0). By (1.5), g is a semidirect product V o h with V a

commutative ideal. Let G be the simply connected Lie group corresponding to the

Lie algebra g. Since g is split solvable, the exponential mapping is a diffeomorphism

from g onto G. Put H := exp h ⊂ G. The adjoint action of H leaves V invariant,

and we write t · x for Ad(t)x (t ∈ H, x ∈ V ). By [11, Theorem 4.15] the H-orbit Ω

7

through E is an open convex cone containing no line. Moreover, H acts on Ω simply

transitively. In order to make the notation clearer, we shall put

Vmk := g(αm+αk)/2 (1 ≤ k ≤ m ≤ r), hmk := g(αm−αk)/2 (1 ≤ k < m ≤ r).

Note that Vkk = REk (1 ≤ k ≤ r). We have by (1.3)

V =( r⊕

k=1

REk

) ⊕( ⊕

1≤k<m≤r

Vmk

)

,(1.7)

h =( r⊕

k=1

RAk

) ⊕( ⊕

1≤k<m≤r

hmk

)

.(1.8)

Before closing this preliminary section, we shall make a brief observation about

the dual cones. The dual cone Ω∗ of Ω is by definition

Ω∗ :=ξ ∈ V ∗ | 〈x, ξ〉 > 0 for all x ∈ Ω \ 0

.(1.9)

Then Ω∗ is also an open convex cone in V ∗ containing no line. It is a fundamental

fact that Ω∗∗ := (Ω∗)∗ equals Ω under the canonical identification of V ∗∗ with V . Let

H∗ := t∗ | t ∈ H ⊂ GL(V ∗). It is known that H∗ acts simply transitively on Ω∗

(see [15, Proposition 9]). The following lemma is a consequence of this observation.

Lemma 1.2. Take ξ0 ∈ Ω∗. One has

Ω = x ∈ V | 〈x, ξ〉 ≥ 0 for all ξ ∈ Ω∗ = x ∈ V | 〈t · x, ξ0〉 ≥ 0 for all t ∈ H .

§2. Explicit description of H-action on V .

In this section, retaining the notation of the previous section, we describe the

action of H on V explicitly by expressing the elements of H as lower triangular

matrices with vector entries. This expression visualizes not only the multiplication

of the elements of H but also the action of H on V , and makes the resultant formulas

easier to understand.

We start by expressing every T ∈ h, according to (1.8), uniquely by

T =

r∑

k=1

tkkAk +∑

m>k

Tmk (tkk ∈ R, Tmk ∈ hmk).(2.1)

For the element T ∈ h in (2.1), we put

Lk :=∑

m>k

Tmk (1 ≤ k ≤ r − 1).(2.2)

8

Given T ∈ h, the symbols Tmk (m > k) as well as Lk will be used to denote the

elements in (2.1) and (2.2) without any comments in this paper. Let Π be the open

subset of h defined by

Π := T ∈ h | tkk > 0 for all k = 1, 2, . . . , r .

Putting Tkk := (2 log tkk)Ak (1 ≤ k ≤ r) for T ∈ Π, we set

γ(T ) := exp T11 · expL1 · expT22 · · · expLr−1 · exp Trr.(2.3)

Then γ(T ) ∈ H, and we shall write it in the following form of lower triangular

matrix:

t11T21 t22...

. . .

Tr1 Tr2 trr

.(2.4)

Proposition 2.1. (i) For T, T ′ ∈ Π, one has

t11T21 t22...

. . .

Tr1 Tr2 trr

t′11T ′

21 t′22...

. . .

T ′r1 T ′

r2 t′rr

=

t′′11T ′′

21 t′′22...

. . .

T ′′r1 T ′′

r2 t′′rr

with

t′′kk = tkkt′kk (1 ≤ k ≤ r),(2.5)

T ′′mk = tmmT

′mk +

∑

k<l<m

[Tml, T′lk] + t′kkTmk (1 ≤ k < m ≤ r).(2.6)

(ii) The map γ is a diffeomorphism from Π onto H.

Proof. (i) Let 1 ≤ i ≤ k ≤ r. We have

expLk · expL′i = exp Ad(expLk)L

′i · expLk = exp ead LkL′

i · expLk.(2.7)

In the same way, we get

exp Tkk · expL′i = exp ead TkkL′

i · exp Tkk,(2.8)

expLk · exp T ′ii =

expT ′ii · expLk (i < k),

expT ′kk · exp e− ad T ′

kkLk (i = k).(2.9)

Using (2.9) and noting that exp a is commutative, we obtain

γ(T ) expT ′11 = exp T11 exp T ′

11 · exp e− ad T ′11L1 · exp T22 · · · expLr−1 exp Trr.

9

Then owing to (2.7) and (2.8), we see that γ(T ) expT ′11 expL′

1 is equal to

exp T11 exp T ′11 · exp e− ad T ′

11L1 exp(ead T22ead L2 · · · ead Lr−1ead Trr

)L′

1 ·· exp T22 · · · expLr−1 exp Trr.

Repetition of this argument yields γ(T )γ(T ′) = γ(T ′′), where

T ′′kk = Tkk + T ′

kk,(2.10)

L′′k = e− ad T ′

kkLk + ead Tk+1, k+1ead Lk+1 · · · ead Lr−1ead TrrL′k.(2.11)

Thus (2.5) follows immediately from (2.10). To show (2.6) we observe that (1.8),

(2.2) and Theorem 1.1 imply ad(Lk)2L′

i = 0, so that if i < k,

ead LkL′i = L′

i + [Lk, L′i] = L′

i + [Lk, T′ki].(2.12)

Similarly,

ead TkkL′i = tkkT

′ki +

∑

l 6=k,l>i

T ′li (if i < k), e− ad T ′

kkLk = t′kkLk.(2.13)

Making use of (2.12) and (2.13), we calculate the right-hand side of (2.11) to arrive

at the right-hand side of (2.6).

(ii) Let H be the image of γ. By (i), H is a subgroup of H. Recall the element

A in (1.4). Then γ(A) equals the unit element of H. Since

d

dh

∣∣∣∣h=0

γ(A + hT ) = T (T ∈ h)

as is easily seen, the map γ is a local diffeomorphism at A, so that the subgroup

H is open. Since H is connected, we have H = H. It remains to show that γ is

injective. Here we need the following simple lemma.

Lemma 2.2. For two subalgebras h1 and h2 of h such that h1 ∩ h2 = 0, the

map exp h1 × exp h2 3 (t1, t2) 7→ t1t2 ∈ H is injective.

Proof. Suppose that t1t2 = t′1t′2 (ti, t

′i ∈ exp hi, i = 1, 2). Then there exists

T ∈ h such that exp T = t1−1t′1 = t2t

′2−1. This implies T ∈ h1 ∩ h2, so that T = 0.

Hence t1 = t′1 and t2 = t′2.

We now prove the injectivity of γ. Let ni (1 ≤ i ≤ r − 1) be the commutative

subalgebras of h given by

ni :=⊕

m>i

hmi.(2.14)

10

We shall apply Lemma 2.2 repeatedly to the pairs of the subalgebras

h(l) :=( l⊕

i=1

RAi

) ⊕( l⊕

i=1

ni

)

, h(l) :=( l⊕

i=1

RAi

) ⊕( l−1⊕

i=1

ni

)

for l = 1, . . . , r − 1. Suppose γ(T ) = γ(T ′) for T, T ′ ∈ Π. Then Lemma 2.2 with

h1 := h(r−1) and h2 := RAr gives trr = t′rr and γ(T ) exp(−Trr) = γ(T ′) exp(−T ′rr) ∈

exp h(r−1). Next, putting h1 := h(r−1) and h2 := nr−1 in Lemma 2.2, we get

Lr−1 = L′r−1. Repeating these arguments, we obtain T = T ′.

We next observe the action of H on V . To do so we express every x ∈ V as

x =

r∑

k=1

xkkEk +∑

m>k

Xmk (xkk ∈ R, Xmk ∈ Vmk).(2.15)

We put Xkk := xkkEk (1 ≤ k ≤ r). We shall use the symbol Xmk (m ≥ k) to denote

the Vmk-component of a given x ∈ V without mentioning it. The action of exp a is

diagonal and easy to describe:

(

expr∑

l=1

(2 log tll)Al

)

· x =∑

m≥k

tmmtkkXmk.(2.16)

In order to see the action of the other elements of H, we introduce the lexicographic

order in the index set Λ := (m, k) | 1 ≤ k ≤ m ≤ r by defining (m, l) (k, i)

if either m > k or m = k, l ≥ i. Since Theorem 1.1 leads us to

ad(h)Vki ⊂⊕

(m,l)(k,i)

Vml ((k, i) ∈ Λ),

we see that the action of H on V is “lower triangular” with respect to the order .

Let us set

V [k] :=⊕

m≥l>k

Vml.(2.17)

Since the summation range can be interpreted as (m, l) (k + 1, k + 1), V [k] is

H-invariant.

We now investigate the action of H on the specific elements more closely. For

this purpose, we first need to introduce a new inner product on g. Let E∗ be the

linear form on g defined by

〈x+ T, E∗〉 :=

r∑

k=1

xkk (x ∈ V, T ∈ h).

11

Lemma 2.3. The bilinear form (Y |Y ′) := (1/2)〈[jY, Y ′], E∗〉 defines an inner

product on g.

Proof. We begin the proof by noting that if ω0 is as in (1.1), then

〈x+ T, ω0〉 = −r∑

k=1

ηkxkk (x ∈ V, T ∈ h),(2.18)

where ηk := −〈Ek, ω0〉 > 0 for all k (see [11, (4.6)]). Now let x, x′ ∈ V and T, T ′ ∈ h

and express them as in (2.15) and (2.1) respectively. By Theorem 1.1, we have

[jXmk, X′mk], [jTmk, T

′mk] ∈ REm. Thus (1.1) and (2.18) tell us that for m > k,

[jXmk, X′mk] =

1

ηm(Xmk|X ′

mk)ω0Em, [jTmk, T′mk] =

1

ηm(Tmk|T ′

mk)ω0Em.

This together with [jEk, Ek] = Ek and [jAk, Ak] = Ek gives

2(x+ T |x′ + T ′) =

r∑

k=1

(

xkkx′kk +

∑

m>k

1

ηm(Xmk|X ′

mk)ω0

)

+

r∑

k=1

(

tkkt′kk +

∑

m>k

1

ηm(Tmk|T ′

mk)ω0

)

.

(2.19)

Therefore the bilinear form (·|·) is symmetric and positive definite.

The inner product (·|·) in Lemma 2.3 will be called the standard inner product

in what follows. By (2.19), the change to the standard inner product does not affect

the orthogonality of the decomposition of V and h in (1.7) and (1.8) respectively.

Corollary 2.4. [Tmk, [T′mk, Ek]] = 2(T ′

mk|Tmk)Em.

Proof. Immediate from [T ′mk, Ek] = −jT ′

mk by (1.2).

We set

Tmk T ′lk := [Tmk, [T

′lk, Ek]] ∈

Vml (m > l),

Vlm (l > m).(2.20)

Since [Tmk, T′lk] = 0 (m 6= l) by Theorem 1.1, Jacobi’s identity gives the commuta-

tivity Tmk T ′lk = T ′

lk Tmk. With these preparations, we are now able to describe

explicitly the action of H on the elements

Eε :=

r∑

k=1

εkEk ∈ V (ε = (ε1, . . . , εr) ∈ 0, 1r ).(2.21)

12

Proposition 2.5. Let x := γ(T ) · Eε (T ∈ Π). Then

xkk = εk(tkk)2 +

∑

i<k

εi‖Tki‖2 (1 ≤ k ≤ r),(2.22)

Xmk = εktkk[Tmk, Ek] +∑

i<k

εiTmi Tki (1 ≤ k < m ≤ r),(2.23)

where ‖ · ‖ stands for the norm defined by the standard inner product on g.

Proof. Let 1 ≤ i ≤ r. If l > i, then Theorem 1.1 implies that expLl ·Ei = Ei

and exp Tll · Ei = Ei. Thus

γ(T ) · Ei = exp T11 · · · expLi−1 · (exp Tii expLi · Ei).

Since Ei lies in the H-invariant subspace V [i−1] (see (2.17)), it holds that

expTii expLi · Ei ∈ V [i−1]. Furthermore, if h ≤ i − 1, Theorem 1.1 shows that

expLh as well as exp Thh acts trivially on V [i−1]. It follows that

γ(T ) · Ei = expTii expLi · Ei.

Noting ad(Li)3Ei = 0 and using (2.16), we have

expTii expLi ·Ei = exp Tii · (Ei + [Li, Ei] + (1/2)[Li, [Li, Ei]])

= (tii)2Ei + tii[Li, Ei] + (1/2)[Li, [Li, Ei]].

(2.24)

Substituting Li =∑

m>i Tmi in the last term , we obtain by Corollary 2.4 and (2.20)

γ(T ) · Ei = (tii)2Ei +

∑

m>i

tii[Tmi, Ei] +∑

k>i

‖Tki‖2Ek +∑

m>k>i

Tmi Tki,

from which the formulas (2.22) and (2.23) follow immediately.

Remark. If one expresses x ∈ V by the symmetric matrix

x11 X21 . . . Xr1

X21 x22 Xr2...

. . .Xrr Xr2 xrr

,

then Proposition 2.5 may be better understood. The elements Eε being expressed

by the diagonal matrices Dε = diag[ε1, . . . , εr], the formulas (2.22) and (2.23) are

the matrix multiplication rule for x = γ(T ) ·Dε · tγ(T ) under an appropriate inter-

pretation.

13

§3. Orbit decomposition of Ω.

From now on, we shall regard H as a subgroup of GL(V ) by the adjoint represen-

tation. Let H denote the closure of H in End(V ). Then H is a semigroup. Recalling

(1.7), we denote by Pml (1 ≤ l ≤ m ≤ r) the orthogonal projection V → Vml. For

k = 1, . . . , r, we define a map ψk : [0,∞) → End(V ) by

ψk(t) := t2Pkk + t(∑

i<k

Pki +∑

m>k

Pmk

)

+∑

m>i

m6=k,i6=k

Pmi.(3.1)

The formula (2.16) says that ψk(t) = exp((2 log t)Ak

)∈ H for t > 0. Thus ψk(t)

belongs to H for any t ≥ 0. Let Π be the closure of Π in h:

Π = T ∈ h | tkk ≥ 0 for all k = 1, 2, . . . , r .

In view of (2.3), we define a map γ : Π → H by

γ(T ) := ψ1(t11) · expL1 · ψ2(t22) · · · expLr−1 · ψr(trr).(3.2)

Evidently γ extends γ, so that we express γ(T ) still by the matrix (2.4). We remark

that the formulas in Propositions 2.1 (i) and 2.5 remain valid for γ(T ) by continuity.

Proposition 3.1. (i) Let Ω be the closure of Ω in V . Then H · E = Ω.

(ii) γ(Π) = H.

Proof. (i) Clearly H · E ⊂ Ω. To prove the converse inclusion we take x ∈ Ω.

Then there exists a sequence x(ν)ν∈N in Ω such that x(ν) → x as ν → ∞. According

to Proposition 2.1 (ii), we take T (ν) ∈ Π for which x(ν) = γ(T (ν)) ·E. By (2.22) with

ε = (1, . . . , 1), we have

x(ν)kk = (t

(ν)kk )2 +

∑

i<k

‖T (ν)ki ‖2 (1 ≤ k ≤ r).(3.3)

Since (3.3) says that all the sequences t(ν)kk ν∈N and T (ν)

ki ν∈N are bounded, we

take a strictly increasing sequence νn of positive integers such that all of the lim-

its tkk := lim t(νn)kk and Tki := limT

(νn)ki exist. Put T :=

∑tkkAkk +

∑Tki. Then

T (νn) → T ∈ Π as n→ ∞. Therefore x = γ(T ) · E ∈ H · E, whence (i) follows.

(ii) For t ∈ H, we take a sequence t(ν)ν∈N in H such that t(ν) → t as ν → ∞.

Put x := t ·E ∈ Ω and x(ν) := t(ν) ·E ∈ Ω. Then x(ν) → x as ν → ∞, and repeating

the argument in (i), we find T ∈ Π for which t = γ(T ). Hence (ii) is proved.

14

We denote by Oε the H-orbit in V through Eε ∈ Ω. Then Oε is contained

in Ω. Note that O(1,...,1) = Ω and Oε ⊂ ∂Ω if ε 6= (1, . . . , 1). For x ∈ V , let

Hx := t ∈ H | t · x = x be the stabilizer at x in H.

Proposition 3.2. (i) Let x := γ(T ) · Eε with T ∈ Π. If x ∈⊕r

k=1 REk, then

εkTmk = 0 for all 1 ≤ k < m ≤ r and x =∑r

k=1 εk(tkk)2Ek.

(ii) If ε 6= ε′, then the two orbits Oε and Oε′ are distinct.

(iii) One has

HEε= γ(T ) ∈ H | if εi = 1, then tii = 1 and Tki = 0 (k > i) .

Proof. (i) We shall prove εkTmk = 0 by induction on k. By (2.23) and (1.2),

we have

0 = Xm1 = ε1t11[Tm1, E1] = −ε1t11jTm1.

Since t11 6= 0, we obtain ε1Tm1 = 0. Assume next that εiTmi = 0 (m > i) for

i = 1, . . . , k − 1. Then, again by (2.23) and (1.2) we have for m > k,

0 = εktkk[Tmk, Ek] +∑

i<k

εiTmi Tki = −εktkkjTmk.

Hence εkTmk = 0. The second assertion follows from this and (2.22).

(ii) Clear from (i) by considering the intersection with⊕r

k=1 REk.

(iii) The assertion follows from (i) and Proposition 2.5.

Let hEεbe the Lie algebra of HEε

. Then Proposition 3.2 (iii) tells us that

hEε=

⊕

εi=0

(

RAi

⊕

ni

)

,

see (2.14) for the definition of ni. Let h(Oε) be the orthogonal complement of hEε

in h. Then

h(Oε) =⊕

εi=1

(

RAi

⊕

ni

)

.

Note that h(Oε) coincides with hE1−ε, where 1 := (1, . . . , 1) ∈ 0, 1r. In particular,

h(Oε) is a subalgebra of h. Put H(Oε) := exp h(Oε). Then H(Oε) = HE1−ε, so that

H(Oε) = γ(T ) ∈ H | if εi = 0, then tii = 1 and Tki = 0 (k > i) .(3.4)

Define a map πε : H → H(Oε) by

t11T21 t22...

. . .Tr1 Tr2 trr

7→

(t11)ε1

ε1T21 (t22)ε2

.... . .

ε1Tr1 ε2Tr2 (trr)εr

.(3.5)

15

Clearly πε is surjective.

Lemma 3.3. (i) For t ∈ H, one has t · Eε = πε(t) · Eε.

(ii) The map Ψε : H(Oε) 3 t 7→ t ·Eε ∈ Oε is bijective.

Proof. (i) If εi = 0, the right-hand sides of (2.22) and (2.23) do not contain

tii nor Tki (k > i). Hence the assertion holds.

(ii) The surjectivity of Ψε follows from (i). Since hEε∩ h(Oε) = 0, the injec-

tivity is clear.

We set for k = 1, 2, . . . , r − 1,

[k] := (0, . . . , 0︸ ︷︷ ︸

k times

, 1, . . . , 1) ∈ 0, 1r.(3.6)

Then E[k] =∑

m>k Em belongs to the H-invariant subspace V [k] defined in (2.17),

and O[k] is a homogeneous cone in V [k] of rank r − k, where the rank of a cone is

the rank of the corresponding normal j-algebra. In fact, O[k] is the homogeneous

cone corresponding to the normal j-subalgebra h(O[k]) ⊕ V [k]. Thus it is natural to

denote the orbit O[k] by Ω[k]. Accordingly, we write H(Ω[k]) instead of H(O[k]). Let

PV [k] be the orthogonal projection V → V [k]. Define E∗[k] ∈ V ∗ by

〈x, E∗[k]〉 :=

∑

m>k

xmm.(3.7)

By (2.22), we have

〈γ(T ) · E[k], E∗[k]〉 =

∑

m>k

(tmm)2 +∑

m>l>k

‖Tml‖2

(T ∈ Π).(3.8)

If γ(T ) · E[k] 6= 0, then the right-hand side of (3.8) is strictly positive owing to

Proposition 2.5. Thus Proposition 3.1 and (1.9) tell us that E∗[k]|V [k] belongs to

(Ω[k])∗, the dual cone of Ω[k].

Lemma 3.4. (i) For t ∈ H(Ω[k]) and x ∈ V , one has t · PV [k](x) = PV [k](t · x).(ii) For 1 ≤ k ≤ r − 1, one has PV [k](Ω) = Ω

[k].

Proof. (i) Let (V [k])⊥ be the orthogonal complement of V [k] in V . Then by

(2.17),

(V [k])⊥ =

k⊕

i=1

r⊕

l=i

Vli.

If i ≤ k < m, then Theorem 1.1 ensures that⊕r

l=i Vli and hence (V [k])⊥ are stable

under the adjoint action of nm. Since h(Ω[k]) =⊕

m>k(RAm

⊕nm), it follows that

16

(V [k])⊥ is stable under H(Ω[k]). The H(Ω[k])-stability of V [k] being evident, the

assertion (i) holds.

(ii) Let x ∈ Ω. According to Proposition 3.1 (i), we take t ∈ H for which

x = t ·E. If t0 ∈ H(Ω[k]), we obtain by (i), (3.7) and (2.22)

〈t0 · PV [k](x), E∗[k]〉 = 〈PV [k](t0 · x), E∗

[k]〉 = 〈t0t ·E, E∗[k]〉 ≥ 0.

Therefore, applying Lemma 1.2 to the cone Ω[k] in the vector space V [k], we obtain

PV [k](x) ∈ Ω[k]

. Hence PV [k](Ω) ⊂ Ω[k]

. The converse inclusion is clear because

Ω[k] ⊂ Ω.

We now arrive at the main result of this section.

Theorem 3.5. The H-orbit decomposition of Ω is given as

Ω =⊔

ε∈0,1r

Oε.

Proof. By Proposition 3.2 (ii) and Lemma 3.3 (ii), it suffices to show

Claim. Given x ∈ Ω, there exist ε ∈ 0, 1r and t ∈ H(Oε) such that x = t ·Eε.

We shall prove the claim by induction on the rank r of cones. The rank one case

is obvious. Assume that the claim holds when the rank is r − 1. In particular,

the claim holds for Ω[1]. Let x ∈ Ω. According to Proposition 3.1, we take T ∈ Π

such that x = γ(T ) · E (though T is not necessarily unique). We assume first that

x11 = 0. Then t11 = (x11)1/2 = 0 by (2.22), so that Xm1 = t11[Tm1, E1] = 0 (m > 1)

by (2.23). Thus x = PV [1](x) ∈ Ω[1]

by Lemma 3.4 (ii). Then the claim holds in this

case by the induction hypothesis. Next, let us consider the case x11 > 0. Put

t1 :=

t11T21 1...

. . .

Tr1 0 1

, t[1] :=

10 t22...

. . .

0 Tr2 trr

.

Then γ(T ) = t1t[1] ∈ H by Proposition 2.1, and we have t11 = (x11)1/2 > 0 by

(2.22), so that t1 ∈ H. On the other hand, we have t[1] · E[1] ∈ Ω[1]

. By the

induction hypothesis, there exist unique ε′ := (0, ε2, . . . , εr) ∈ 0, 1r and t′ =

γ(T ′) ∈ H(Oε′) (T ′ ∈ Π) such that t[1] · E[1] = t′ · Eε′. Put ε := (1, ε2, . . . , εr) and

t := t1t′ ∈ H(Oε). Then by Proposition 2.5 and Lemma 3.3 (i), we get

x = t1 t[1] · (E1 + E[1]) = t1 · E1 + t[1] · E[1]

= t1 · E1 + t′ · Eε′ = t1t′(E1 + Eε′) = t · Eε,

17

which means that the claim holds when the rank is r. Therefore the theorem is

proved.

The second half of this section is devoted to an inductive description of Ω which

will be needed in the proof of Theorem 6.2.

We set

Π(Oε) := T ∈ h(Oε) | tii > 0 for all i such that εi = 1 .(3.9)

By (3.4), the map Π(Oε) 3 T 7→ γ(A1−ε + T ) ∈ H(Oε) is bijective, where

Aε := jEε =

r∑

k=1

εkAk ∈ a.

Proposition 3.6. Let ε, ε′ ∈ 0, 1r. If ε+ ε′ ∈ 0, 1r, the map

Oε ×Oε′ 3 (x, x′) 7→ x + x′ ∈ V(3.10)

gives a diffeomorphism from Oε ×Oε′ onto Oε+ε′.

Proof. Given x ∈ Oε, we take unique elements t ∈ H(Oε) and T ∈ Π(Oε) for

which x = t·Eε and t = γ(A1−ε+T ). Similarly, we take t′ ∈ H(Oε′) and T ′ ∈ Π(Oε′)

for x′ ∈ Oε′ to have x′ = t′ · Eε′. Put

t := γ(A1−(ε+ε′) + T + T ′) ∈ H(Oε+ε′).

Then it is clear that πε(t) = t and πε′(t) = t′ by (3.5). Thus, by Lemma 3.3 (i),

x + x′ = t · Eε + t′ ·Eε′ = t · (Eε + Eε′) = t · Eε+ε′ ∈ Oε+ε′,

showing that the image of the map in (3.10) is contained in Oε+ε′. Since the map

H(Oε) × H(Oε′) 3 (t, t′) 7→ t ∈ H(Oε+ε′) is visibly bijective by (3.5), our map in

(3.10) is bijective from the above discussion.

We define bilinear maps Q[i] : ni × ni → V [i] (i = 1, 2, . . . , r − 1) by

Q[i](L, L′) :=1

2[L, [L′, Ei]].(3.11)

Lemma 3.7. For every i = 1, 2, . . . , r − 1, one has Q[i](L, L) ∈ Ω[i]

for any

L ∈ ni, and if Li =∑

m>i Tmi, then

Q[i](Li, Li) =∑

m>i

‖Tmi‖2Em +∑

m>k>i

Tmi Tki.(3.12)

18

Proof. By (2.24) we have expL · Ei = Ei + [L,Ei] + Q[i](L, L) (L ∈ ni).

Therefore

Q[i](L, L) = PV [i](expL · Ei) ∈ PV [i](Ω) = Ω[i],

where we have used Lemma 3.4 to get the last equality. The second assertion follows

from this and Proposition 2.5.

Let x[1] := PV [1](x) for x ∈ V . By (2.15) we have x = x11E1 +∑r

m=2Xm1 + x[1].

Proposition 3.8. One has

Ω = Ω[1] ∪

x ∈ V∣∣∣ x11 > 0, x[1] −

1

x11

Q[1](∑

jXm1,∑

jXm1

)

∈ Ω[1]

.

Proof. If x ∈ Ω, then Proposition 3.6 for ε = (1, 0, . . . , 0), ε′ = [1] tells us that

there exist unique t11 > 0, L1 ∈ n1, and y ∈ Ω[1] such that x = ψ1(t11) expL1 ·E1+y.

Using (2.24) and (1.2), we get

x = (t11)2E1 − t11jL1 +Q[1](L1, L1) + y.(3.13)

Hence x[1] = Q[1](L1, L1) + y. On the other hand, comparing (3.13) with (2.15), we

obtain x11 = (t11)2 and Xm1 = −t11jTm1. Therefore

L1 =∑

m>1

Tm1 = (x11)−1/2

∑

m>1

jXm1,

so that

Ω[1] 3 y = x[1] −Q[1](L1, L1) = x[1] −1

x11Q[1]

(∑

jXm1,∑

jXm1

)

.

Thus, for a > 0, we see that the set Ω ∩ x ∈ V | x11 = a equals

aE1 +∑

Xm1 + x[1] ∈ V∣∣∣ x[1] −

1

aQ[1]

(∑

jXm1,∑

jXm1

)

∈ Ω[1]

.

We also obtain Ω ∩ x ∈ V | x11 = 0 = Ω[1]

as was shown in the proof of Theo-

rem 3.5. Hence the proposition is proved.

§4. Gamma integrals on H-orbits.

In this section, we shall define and evaluate Γ -type integrals on the H-orbits Oε

after introducing functions and measures on Oε relatively invariant under the action

of H.

19

For any s = (s1, . . . .sr) ∈ Cr, let χs be the character of H given by

χs :

t11T21 t22...

. . .Tr1 Tr2 trr

7→ (t11)2s1(t22)

2s2 · · · (trr)2sr .

Then χs+s′(t) = χs(t)χs′(t). For each ε ∈ 0, 1r, we set

C(ε) := s ∈ Cr | si = 0 for all i such that εi = 0 .(4.1)

When s ∈ C(ε), we have χs(t) = 1 for all t ∈ HEεby Proposition 3.2 (iii). Thus for

every s ∈ C(ε), we can define a function ∆εs on Oε by

∆εs(t · Eε) := χs(t) (t ∈ H).

Clearly we have

∆εs(t · x) = χs(t)∆

εs(x) (t ∈ H, x ∈ Oε).(4.2)

In other words, ∆εs is relatively invariant under the action of H.

Remark. If s ∈ Zr ∩C(ε), one can show that ∆εs is a rational function. We do

not give the details here, however.

We put

nki := dim hki (1 ≤ i < k ≤ r),(4.3)

and define p(ε) = (p1(ε), . . . , pr(ε)) ∈ Zr by

pk(ε) :=∑

i<k

εinki =∑

i<k,εi=1

nki (1 ≤ k ≤ r).(4.4)

For s ∈ Cr and ε ∈ 0, 1r, let

ε · s := (ε1s1, ε2s2, . . . , εrsr).(4.5)

It is clear that ε · s ∈ C(ε).

When ε 6= 0 := (0, . . . , 0), we make use of Π(Oε) in (3.9) as coordinates for Oε

and define a measure µε on Oε by

dµε(t · Eε) = χ−ε·(1+p(ε))/2(t)∏

εi=1

dtii∏

εi=1, k>i

dTki,(4.6)

where t = γ(A1−ε + T ) ∈ H(Oε) (T ∈ Π(Oε)) and dTki stands for the Euclidean

measure on hki normalized by the standard inner product on g. Let µ0 be the Dirac

measure at x = 0. Recalling the H(Oε)-equivariant diffeomorphism Ψε : H(Oε) →Oε in Lemma 3.3, we see from the following proposition that µε is the transfer of

the left Haar measure on H(Oε) by Ψε.

20

Proposition 4.1. The measure µε is relatively invariant under H:

dµε(t0 · x) = χ(1−ε)·p(ε)/2(t0) dµε(x) (t0 ∈ H).(4.7)

In particular, µε is H(Oε)-invariant.

Proof. The case ε = 0 is trivial. Assume that ε 6= 0. Given t0 ∈ H, we take

T 0 ∈ Π for which t0 = γ(T 0). Consider the map H(Oε) 3 t 7→ t′ := πε(t0t) ∈ H(Oε).

In view of (2.5) and (2.6), differentiation of this map gives

dt′ii = t0ii dtii,

dT ′ki = t0kk dTki + ( terms including dtii or dTli (i < l < k)).

Considering the lexicographic order introduced in Section 2, we obtain∏

εi=1

dt′ii∏

εi=1,k>i

dT ′ki =

∏

εi=1

t0ii dtii∏

εi=1,k>i

(t0kk)nki dTki

= χ(ε+p(ε))/2(t0)∏

εi=1

dtii∏

εi=1,k>i

dTki.

Now, if x = t · Eε, then t0 · x = t′ · Eε by Lemma 3.3 (i). Therefore

dµε(t0 · x) = χ−ε·(1+p(ε))/2(t′)

∏

dt′ii∏

dT ′ki

= χ−ε·(1+p(ε))/2(t0t)χ(ε+p(ε))/2(t0)∏

dtii∏

dTki

= χ(1−ε)·p(ε)/2(t0) dµε(x).

Hence (4.7) is proved. For the last assertion, note (1 − ε) · p(ε)/2 ∈ C(1 − ε) and

H(Oε) = HE1−ε.

We now study the integral

ΓOε(s) :=

∫

Oε

e−〈x,E∗〉∆εs(x) dµε(x) (s ∈ C(ε)).(4.8)

Theorem 4.2. The integral (4.8) converges if and only if s ∈ C(ε) satisfies the

following condition:

<si > pi(ε)/2 for all i such that εi = 1.(4.9)

Moreover, when this condition is satisfied, one has

ΓOε(s) = 2−|ε|π|p(ε)|/2

∏

εi=1

Γ

(

si −pi(ε)

2

)

,(4.10)

where |ε| :=∑r

i=1 εi and |p(ε)| :=∑r

i=1 pi(ε).

21

Proof. If ε = 0, the integral (4.8) reduces to 1. Thus (4.9) and (4.10) hold

trivially. Assume now that ε 6= 0. By (2.22), we have for T ∈ Π(Oε)

〈γ(T ) ·E,E∗〉 :=∑

εi=1

(tii)2 +

∑

εi=1, k>i

‖Tki‖2 =∑

εi=1

(tii)

2 + ‖Li‖2.

Since (2.22) and (2.23) lead us to

γ(T ) · E = γ(T ) · Eε = γ(A1−ε + T ) ·Eε,(4.11)

we get from (4.6) by setting t = γ(A1−ε + T ) and dLi =∏

k>i dTki,

ΓOε(s) =

∫

Π(Oε)

exp

(

−∑

εi=1

(tii)

2 + ‖Li‖2)

χs−ε·(1+p(ε))/2(t)∏

εi=1

dtii dLi

=∏

εi=1

∫ ∞

0

e−(tii)2

(tii)2si−1−pi(ε) dtii

∏

εi=1

∫

ni

e−‖Li‖2

dLi

=∏

εi=1

(1

2

∫ ∞

0

e−uusi−1−pi(ε)/2 du

)∏

εi=1

π(dim ni)/2.

Therefore the convergence argument is reduced to the one for the ordinary gamma

functions. Since (2.14), (4.3) and (4.4) imply

∑

εi=1

dim ni =∑

εi=1, k>i

nki =

r∑

k=1

pk(ε) = |p(ε)|,

we obtain (4.10).

We introduce the following subsets in Cr:

D(ε) := s ∈ Cr | si = pi(ε)/2 for all i such that εi = 0 ,(4.12)

ΞC(ε) := s ∈ D(ε) | <si > pi(ε)/2 for all i such that εi = 1 .(4.13)

Clearly we have

D(ε) = (1 − ε) · p(ε)/2 + C(ε).

For every s ∈ Cr, let

s := s− (1 − ε) · p(ε)/2.(4.14)

If s ∈ D(ε), then we have s ∈ C(ε), so that ∆εs is defined. Proposition 4.1 and (4.2)

tell us that if t ∈ H and s ∈ D(ε), then

∆εs(t · x) dµε(t · x) = χs(t)∆

εs(x) dµε(x).(4.15)

22

Proposition 4.3. Let s ∈ ΞC(ε) and t ∈ H. Then s satisfies (4.9) and∫

Oε

e−〈x, t∗·E∗〉∆εs(x) dµε(x) = ΓOε

(s)χ−s(t).

Proof. Observe that <si = <si provided εi = 1. Hence s satisfies (4.9), and

the proposition follows from (4.8) and (4.15).

§5. Riesz distributions.

We begin this section by considering the map Θ(T ) := γ(T ) · E from Π to V .

By Proposition 2.5 with ε = 1, we see that Θ is a polynomial map. Therefore we

consider from now on that the definition domain of Θ is the whole space h. Thus,

if x = Θ(T ) (T ∈ h) with the expressions (2.15) and (2.1), we have

xkk = (tkk)2 +

∑

i<k

‖Tki‖2, Xmk = tkk[Tmk, Ek] +∑

i<k

Tmi Tki.(5.1)

For c = (c1 . . . , cr) ∈ −1, 1r, let gc be the linear automorphism on h defined by

gc

(∑

tkk Ak +∑

Tmk

)

:=∑

cktkk Ak +∑

ckTmk.

Then by (5.1), Θ is gc-invariant:

Θ(gcT ) = Θ(T ) (T ∈ h).(5.2)

Since Θ(Π) = Ω by Proposition 3.1, the equality (5.2) tells us that Θ(h) = Ω. For

every ε ∈ 0, 1r, we set

R(ε) := C(ε) ∩ Rr = s ∈ R

r | si = 0 for all i such that εi = 0 ,(5.3)

R+(ε) := s ∈ R(ε) | si > 0 for all i such that εi = 1 .(5.4)

Theorem 5.1. (i) Let s ∈ ΞC(ε) and s be as in (4.14). Then, for ϕ ∈ S(V ),

the following integral converges:

〈Rεs, ϕ〉 :=

1

ΓOε(s)

∫

Oε

ϕ(x)∆εs(x) dµε(x).(5.5)

(ii) 〈Rεs, ϕ〉 admits an analytic continuation as a holomorphic function of s ∈

D(ε), and defines a tempered distribution.

Proof. (i) Put x = Θ(T ) in the integral (5.5). We have (5.1) and, proceeding

as in the proof of Theorem 4.2, we get

〈Rεs, ϕ〉 =

1

ΓOε(s)

∫

Π(Oε)

ϕ(Θ(T ))∏

εi=1

dLi

∏

εi=1

(tii)2si−pi(ε)−1 dtii.(5.6)

23

Here putting nε :=⊕

εi=1 ni for simplicity, we define Iεϕ ∈ C∞(R(ε)) and Iεϕ ∈C∞(R+(ε)) by

Iεϕ(t11, . . . , trr) :=

∫

nε

ϕ

(

Θ

(∑

εi=1

(tiiAi + Li)

))∏

εi=1

dLi,(5.7)

Iεϕ(u1, . . . , ur) := 2−|ε|Iεϕ((u1)1/2, . . . , (ur)

1/2).(5.8)

Since |tkk|2, ‖Tki‖2 ≤ xkk by (5.1), the integral in (5.7) converges and Iεϕ ∈ S(V ).

By (5.2), we see that Iεϕ is an even function in each variable. Thus, all the deriva-

tives of Iεϕ can be extended to R+(ε) as continuous functions. By (5.6), (5.7) and

(5.8), we have

〈Rεs, ϕ〉 =

1

ΓOε(s)

∫

R+(ε)

Iεϕ(u1, . . . , ur)∏

εi=1

(ui)si−1−pi(ε)/2 dui.(5.9)

If s ∈ ΞC(ε), then the last integral converges, which proves (i).

(ii) Take β = (β1, . . . , βr) ∈ Zr ∩R+(ε). Using the obvious identity

(ui)si−1−pi(ε)/2 =

βi−1∏

l=0

(

si + l +pi(ε)

2

)−1(∂

∂ui

)βi

(ui)si+βi−1−pi(ε)/2,

we have by (4.10) and (5.9),

〈Rεs, ϕ〉 =

2|ε|π−|p(ε)|/2

∏

εi=1 Γ (si + βi − pi(ε)/2)×

×∫

R+(ε)

∏

εi=1

(

− ∂

∂ui

)βi

[Iεϕ](u1, . . . , ur)∏

εi=1

(ui)si+βi−1−pi(ε)/2 dui.

The right-hand side is holomorphic for s ∈ −β + ΞC(ε). Hence 〈Rεs, ϕ〉 can be con-

tinued analytically to −β + ΞC(ε). Since β ∈ Zr ∩ R+(ε) is arbitrary, the assertion

(ii) holds.

We simply write Rs for R1

s , and call it the Riesz distribution on Ω. We note

that s = s if ε = 1 and that since D(1) = Cr, Rs is defined for all s ∈ Cr.

Theorem 5.2. (i) Let s ∈ D(ε) and ξ ∈ Ω∗. Then the Laplace transform of Rεs

is given as 〈Rεs, e

−〈x,ξ〉〉x = χ−s(t), where t ∈ H is taken so that ξ = t∗ · E∗.

(ii) The distribution Rεs coincides with Rs for any ε ∈ 0, 1r and s ∈ D(ε). In

particular, suppRs ⊂ Oε if s ∈ D(ε).

(iii) For s, s′ ∈ Cr, one has Rs ∗ Rs′ = Rs+s′ .

Proof. (i) If s ∈ ΞC(ε), the claim follows from Proposition 4.3. By analytic

continuation, we see that the assertion holds for any s ∈ D(ε).

24

(ii) The assertion is clear from (i) and the injectivity of the Laplace transform.

(iii) The assertion also follows from the injectivity of the Laplace transform.

We obtain the following proposition from (4.15) by analytic continuation.

Proposition 5.3. For s ∈ Cr, ϕ ∈ S(V ) and t ∈ H, one has

〈Rs, ϕ(t−1 · x)〉x = χs(t)〈Rs, ϕ(x)〉x.

We conclude this section by deriving some formulas which will be used in the

next section. Fix k such that 1 ≤ k ≤ r−1. For s = (0, . . . , 0, sk+1, . . . , sr) ∈ D([k]),

put σ := (sk+1, . . . , sr) ∈ Cr−k . Since O[k] = Ω[k] ⊂ V [k], the definition (5.5) of

R[k]s (= Rs) tells us that

〈Rs, ϕ〉 = 〈R(Ω[k])σ, ϕ|V [k]〉 (ϕ ∈ S(V )),(5.10)

where R(Ω[k])σ stands for the Riesz distribution on Ω[k]. Setting

δk = (0, . . . , 0,(k)

1 , 0, . . . , 0) ∈ 0, 1r,(5.11)

we put

Mk(z) := z · δk + p(δk)/2 (z ∈ C).(5.12)

Then it is easy to check that M k(z) ∈ D(δk) (see (4.12)). Since

p(δk) = (0, . . . , 0, nk+1, k, . . . , nrk),(5.13)

we see that |p(δk)| =∑

m>k nmk = dim nk by (2.14). Assume that <z > 0. Then

Mk(z) ∈ ΞC(δk), and we have by (5.6) and (4.10)

〈RMk(z), ϕ〉 =2π−(dim nk)/2

Γ (z)

∫

Πk

ϕ(Θ(T )) dLk (tkk)2z−1 dtkk,(5.14)

where Πk := Π(Oδk) for simplicity. Now if T ∈ Πk, we have by (4.11)

Θ(T ) = γ(T ) · E = γ(A1−δk + tkkAk + Lk) · Ek = ψk(tkk) expLk · Ek.

Thus by (2.24), (1.2) and (3.11), we get

Θ(T ) = (tkk)2Ek − tkkjLk +Q[k](Lk, Lk).

Then, changing the variable tkk =√u, we see that the right-hand side of (5.14) is

π−(dim nk)/2

Γ (z)

∫ ∞

0

∫

nk

ϕ(uEk −

√ujLk +Q[k](Lk, Lk)

)dLk u

z−1 du.

25

Let ρz denote the Riesz distribution on (0,+∞) given by

〈ρz, ψ〉 :=1

Γ (z)

∫ +∞

0

ψ(u)uz−1 du (ψ ∈ S(R)).(5.15)

Then 〈RMk(z), ϕ〉 is rewritten as

π−(dim nk)/2

⟨

ρz,

∫

nk

ϕ(uEk −

√ujLk +Q[k](Lk, Lk)

)dLk

⟩

u

.(5.16)

By analytic continuation, we see that this expression for 〈RMk(z), ϕ〉 is valid for any

z ∈ C. Since ρ0 is the Dirac measure at u = 0, we get by putting z = 0,

〈Rp(δk)/2, ϕ〉 = π−(dim nk)/2

∫

nk

ϕ(Q[k](Lk, Lk)) dLk.(5.17)

Combining (5.17) with (5.10), we obtain for φ ∈ S(V [k]),∫

nk

φ(Q[k](Lk, Lk)) dLk = π(dim nk)/2⟨R(Ω[k])(n

k+1,k/2,...,n

rk/2), φ

⟩.(5.18)

§6. Gindikin-Wallach sets.

We now investigate the positivity set (Gindikin-Wallach set) of the parameter s

of the Riesz distribution Rs and describe it in connection with the H-orbit structure

in Ω.

Let us set

Ξ(ε) := ΞC(ε) ∩ Rr.(6.1)

By (4.13) and (5.4) it is evident that

Ξ(ε) =1

2p(ε) +R+(ε).(6.2)

For s ∈ Ξ(ε) and 1 ≤ k ≤ r, let

s(k) := s− 1

2

k−1∑

i=1

εi p(δi).(6.3)

We note here that (4.4) and (5.13) give

p(ε) =r−1∑

i=1

εi p(δi).(6.4)

Since pk(δi) = 0 for i ≥ k, we have s

(k)k = s

(r)k . Moreover (6.2), (6.3) and (6.4) imply

s(r) ∈ R+(ε), so that

s(k)k > 0 (if εk = 1), s

(k)k = 0 (if εk = 0).

26

This together with (6.3) gives us

s(k) =

s(k−1) − p(δk−1)/2 (if s(k−1)k−1 > 0),

s(k−1) (if s(k−1)k−1 = 0).

(6.5)

Proposition 6.1. For s ∈ Rr, define σ(1), . . . , σ(r) ∈ Rr inductively by σ(1) :=

s and, for 2 ≤ k ≤ r,

σ(k) :=

σ(k−1) − p(δk−1)/2 (if σ(k−1)k−1 > 0),

σ(k−1) (if σ(k−1)k−1 ≤ 0).

(6.6)

Then s ∈ Ξ(ε) if and only if σ(r) ∈ R+(ε).

Proof. If s ∈ Ξ(ε), then the argument preceding Proposition 6.1 says that

σ(r) = s(r) ∈ R+(ε). Suppose conversely that σ(r) ∈ R+(ε). If m ≥ k − 1, we have

pk−1(δm) = 0, so that σ

(m)k−1 = σ

(m+1)k−1 by (6.6). Thus σ

(k−1)k−1 = σ

(r)k−1, and (6.6) is

rewritten as

σ(k−1) =1

2εk−1 p(δ

k−1) + σ(k).

Therefore by (5.13) and (6.2) we arrive at

s = σ(1) =1

2

r−1∑

k=1

εk p(δk) + σ(r) ∈ 1

2p(ε) +R+(ε) = Ξ(ε),

completing the proof of Proposition 6.1.

Let ε, ε′ ∈ 0, 1r. If ε 6= ε′, then clearly R+(ε) ∩ R+(ε′) = ∅, so that we have

Ξ(ε) ∩ Ξ(ε′) = ∅ by Proposition 6.1. Set

Ξ :=⊔

ε∈0,1r

Ξ(ε).

Based on Theorem 6.2 which we are going to prove, we shall call Ξ the Gindikin-

Wallach set for the Riesz distributions Rs. We note here that Proposition 6.1

supplies us a simple algorithm for determining whether s ∈ Ξ or not for a given

s ∈ Rr. Moreover, if s ∈ Ξ, the algorithm tells us the ε ∈ 0, 1r for which s ∈ Ξ(ε).

In fact, given s ∈ Rr, we compute σ(k) (k = 1, . . . , r) by the formula (6.6). If once

σ(k)k < 0 for some k, we then conclude that s /∈ Ξ. If σ

(k)k ≥ 0 for all k = 1, 2, . . . , r,

then putting

εk := 1 (if σ(k)k > 0), εk := 0 (if σ

(k)k = 0),

we have s ∈ Ξ(ε).

Before stating Theorem 6.2, we note that if s ∈ Ξ(ε), then we have s = ε · s by

(4.5), (4.12), (4.13), (4.14) and (6.1).

27

Theorem 6.2. The Riesz distribution Rs is positive if and only if s ∈ Ξ. More-

over, if s ∈ Ξ(ε), then Rs is a measure on the H-orbit Oε and one has

dRs = ΓOε(ε · s)−1∆ε

ε·s dµε.(6.7)

Proof. If s ∈ Ξ(ε), then Theorems 5.1 and 5.2 say that Rs is a positive

measure given by (6.7) on the H-orbit Oε. To show the converse assertion, we

assume that Rs is positive. Since e−〈x, t∗·E∗〉 > 0 for x ∈ Ω and t ∈ H, we have

χ−s(t) = 〈Rs, e−〈x, t∗·E∗〉〉 ≥ 0 by Theorem 5.2. Hence s is necessarily real. We shall

prove s ∈ Ξ by induction on the rank r of the cone Ω.

Let us consider first the case r = 1. In this case Rs coincides with ρs given by

(5.15). Take φν ∈ S(R) (ν = 1, 2, . . . ) such that φν(x) = (x+(1/ν))e−x (x ≥ 0) and

φν ≥ 0. Then 0 ≤ 〈ρs, φν〉 = s+ (1/ν). Letting ν → +∞, we get s ≥ 0, so that the

claim holds in this case. Assume next that the claim holds for cones of rank r−1. In

particular, the claim holds for the cone Ω[1] ⊂ V [1]. For σ = (σ2, . . . , σr) ∈ Cr−1, let

Rσ be the Riesz distribution R(Ω[1])σ on Ω[1]. Let Ξ be the Gindikin-Wallach set for

Rσ. We have Ξ =⊔

ε∈0,1r−1 Ξ(ε) with Ξ(ε) (ε = (ε2, . . . , εr)) having a description

similar to (6.2): Ξ(ε) = p(ε)/2+R+(ε) for the obvious definition of R+(ε) (see (5.4))

and p(ε) = (p2(ε), . . . , pr(ε)) with

pk(ε) :=∑

2≤i<k

εinki.(6.8)

Put s := (s2, . . . , sr) ∈ Rr−1, n := (n21, . . . , nr1) and

M(s1) := M1(s1) = (s1, n21/2, . . . , nr1/2) ∈ D(δ1),

see (5.12). Then s −M(s1) = (0, s− n/2) ∈ D([1]). Since Rs = RM(s1) ∗ Rs−M(s1)

by Theorem 5.2 (iii), we have, thanks to (5.16) and (5.10),

〈Rs, ϕ〉 = π−(dim n1)/2 ×

×⟨

ρs1 ,

∫

n1

⟨Rs−n/2, ϕ(uE1 −

√ujL1 +Q[1](L1, L1) + y)

⟩

ydL1

⟩

u

.(6.9)

for ϕ ∈ S(V ). We consider the functions of the form

ϕ(x) := ϕ1(x11)ϕ2(x[1])(

x = x11E1 +∑

m>1

Xm1 + x[1] ∈ V)

28

with ϕ1 ∈ C∞c (R), ϕ2 ∈ C∞

c (V [1]). By virtue of Proposition 3.8, we see that

suppϕ ∩ Ω is compact, so that Rs can be applied to ϕ. Now by (6.9), we have

〈Rs, ϕ〉 = π−(dim n1)/2〈ρs1, ϕ1〉∫

n1

〈Rs−n/2, ϕ2(Q[1](L1, L1) + y)〉y dL1

= 〈ρs1, ϕ1〉〈Rn/2 ∗ Rs−n/2, ϕ2〉 = 〈ρs1, ϕ1〉〈Rs, ϕ2〉,(6.10)

where we have used (5.18) for the second equality. Take a positive integer N such

that s1 +N > 0. Since 〈ρs1, (u + N)e−u〉u = s1 +N > 0, a suitable choice of non-

negative ϕ1 ∈ C∞c (R) approximating (u + N)e−u on [ 0,+∞) yields 〈ρs1, ϕ1〉 > 0.

If ϕ2 ≥ 0, the positivity of Rs tells us 〈Rs, ϕ2〉 = 〈ρs1 , ϕ1〉−1〈Rs, ϕ〉 ≥ 0 by (6.10).

Thus Rs is positive and the induction hypothesis ensures that s ∈ Ξ(ε) for some

ε = (ε2, . . . , εr) ∈ 0, 1r−1. We next fix a non-negative ϕ2 such that 〈Rs, ϕ2〉 is

strictly positive. Then using (6.10) again, we get 〈ρs1 , ϕ1〉 ≥ 0 for any ϕ1 ≥ 0 in turn.

Therefore ρs1 is positive, so that s1 ≥ 0. If s1 = 0, then putting ε := (0, ε2, . . . , εr),

we get s ∈ Ξ(ε).

It now remains the case s1 > 0. Before proceeding we remark that the map

(0,+∞) × n1 × V [1] → x ∈ V | x11 > 0 given by

(u, L1, y) 7→ uE1 −√ujL1 +Q[1](L1, L1) + y

is a diffeomorphism. In fact, as is suggested by the proof of Proposition 3.8 and can

be checked directly, the inverse map is given by

x 7→(

x11, (x11)−1/2

∑

jXm1, x[1] −1

x11Q[1]

(∑

jXm1,∑

jXm1

))

.

Now let us consider the functions ϕ ∈ C∞c (V ) given by

ϕ(x) :=

φ1(u)φ2(L1)φ3(y) (x11 > 0),

0 (x11 ≤ 0),

with φ1 ∈ C∞c (0,+∞), φ2 ∈ C∞

c (n1), φ3 ∈ C∞c (V[1]). Then we have by (6.9)

〈Rs, ϕ〉 = π−(dim n1)/2〈ρs1, φ1〉〈Rs−n/2, φ3〉∫

n1

φ2(L1) dL1.

Since s1 > 0, the positivity assumption of Rs yields that Rs−n/2 is positive. Hence by

the induction hypothesis we have s−n/2 ∈ Ξ(ε) for some ε = (ε2, . . . , εr) ∈ 0, 1r−1.

Put ε := (1, ε2, . . . , εr). Then by (4.4) and (6.8) we have pk(ε) = pk(ε) + nk1 for

k = 2, . . . , r. Thus s− n/2 ∈ Ξ(ε) is equivalent to

si > pi(ε)/2 (if εi = 1), si = pi(ε)/2 (if εi = 0)

for i = 2, . . . , r. Since we have s1 > 0 = p1(ε)/2, we see that s ∈ Ξ(ε). Hence the

theorem is completely proved.

29

Remark. For every t ≥ 0, we set

ε(t) = 1 (if t > 0), ε(t) = 0 (if t = 0).

Then we have

R+(ε) = u ∈ Rr | ui ≥ 0, ε(ui) = εi for all i = 1, 2, . . . , r .

Making s, u ∈ Rr correspond to each other by s = u+ p(ε)/2, we see by (6.2) that

s ∈ Ξ(ε) ⇐⇒ u ∈ R+(ε).

Moreover if Ω is symmetric and simple, then d := dim hki = dimVki is independent

of k, i, so that (4.3) and (4.4) imply

pk(ε) = d(ε1 + · · · + εk−1).

From these observations follows the description of Ξ for the case of symmetric cones

given in [3, p. 138].

References

[ 1 ] J. Dorfmeister, Inductive construction of homogeneous cones, Trans. Amer.

Math. Soc., 252 (1979), 321–349.

[ 2 ] J. J. Duistermaat, M. Riesz’s families of operators, Nieuw Arch. Wisk., 9

(1991), 93–101.

[ 3 ] J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford Mathemat-

ical Monographs, Clarendon Press, Oxford, 1994.

[ 4 ] I. M. Gel’fand and G. E. Shilov, Generalized functions, I, Academic Press,

New York, 1964.

[ 5 ] S. G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys,

19 (1964), 1–89.

[ 6 ] —, Invariant generalized functions in homogeneous domains, Funct. Anal.

Appl., 9 (1975), 50–52.

[ 7 ] S. Kaneyuki, Homogeneous bounded domains and Siegel domains, Lecture

Notes in Mathematics, 241, Springer-Verlag, Berlin, 1971.

[ 8 ] I. I. Pyatetskii-Shapiro, Automorphic functions and the geometry of classical

domains, Gordon and Breach, New York, 1969.

[ 9 ] M. Riesz, L’integrale de Riemann-Liouville et le probleme de Cauchy, Acta

Math., 81 (1949), 1–223.

30

[ 10 ] H. Rossi, Lectures on representations of groups of holomorphic transforma-

tions of Siegel domains, Lecture Notes, Brandeis Univ., 1972.

[ 11 ] H. Rossi and M. Vergne, Representations of certain solvable Lie groups on

Hilbert spaces of holomorphic functions and the application to the holomorphic

discrete series of a semisimple Lie group, J. Funct. Anal., 13 (1973), 324–389.

[ 12 ] O. S. Rothaus, The construction of homogeneous convex cones, Ann. of

Math., 83 (1966), 358–376; correction, ibid., 87 (1968), 399.

[ 13 ] L. Schwartz, Theorie des distributions, Hermann, Paris, 1966.

[ 14 ] M. Vergne and H. Rossi, Analytic continuation of the holomorphic discrete

series of a semi-simple Lie group, Acta Math., 136 (1976), 1–59.

[ 15 ] E. B. Vinberg, The theory of convex homogeneous cones, Trans. Moscow

Math. Soc., 12 (1963), 340–403.

[ 16 ] N. Wallach, The analytic continuation of the discrete series, I, II, Trans. Amer.

Math. Soc., 251 (1979), 1–17; ibid., 19–37.

Hideyuki Ishi

Department of MathematicsFaculty of ScienceKyoto UniversitySakyo, Kyoto 606-8502Japan