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].
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Hideyuki Ishi
Department of MathematicsFaculty of ScienceKyoto UniversitySakyo, Kyoto 606-8502Japan
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