数理解析研究所講究録 1348 巻 2003 年 61-74

61「Lie Theory のひろがりと新たな進展」 研究集会

Fundamental solutions, Cauchy problems and Huygens principle
for invariant differential operators on prehomogeneous vector
spaces of commutative parabolic type
(July 23, 2003)

Masakazu Muro1 (Gifu University)$\mathrm{M}\mathrm{u}r\mathrm{o}^{1}$

Abstract

Huygens principle and propagation of singularity of Cauchy problems for linear invariant differential operators on symmetric or Hermitian matrix spaces are discussed in this paper.
bc
an invariant linear differential operator on a prehomogeneous vector space of commutative parabolic
type. We consider a Cauchy problem ofis hyperbolic with} with the initial plane that
respect to. We construct an explicit fundamental solution of
by using a Laurent expansion
coefficient of the Laurent expansion of the complex power of the determinant functio . A a consequence we obtain the exact support of the fundamental solution and hence we can give a necessary
and sufficient condition that Huygens principle forholds. Next we construct the fundamental
solution for the Cauchy problem and give the singularity spectrum of it explicitly. Then w can obtain an accurate result on the propagation of singularity of the hyperfunction solution to he Cauchy$1_{r}\mathrm{e}\mathrm{t}P(\partial)$$P(\partial$$P(\partial)$$P(\partial)$$\mathrm{n}$$\mathrm{s}$$P(\partial)$$\mathrm{e}$$\mathrm{t}$problem.

Introduction.
The purpose of this paper is to construct explicit fundamental solutions to invariant cl iffcrential operatorsand determine their support and singularity spectrum of them on a kind of vector space witll group action.
These differential operators are hyperbolic with respect to some initial planes. We prove that Huygens
principle holds for these differential operators by the precise investigation of the fundamental solutions.
In addition we can clarify how the singularity of the solutions to Cauchy problem . with respect to theinitial plane propagates by determining the singularity spectrum of the fundamental s.ollltio116.
Let us begin with an explanation of a typical examaple of hyperbolic differential operator. The
most primitive hyperbolic differential operator may be the wave operator,))$/‘$ )f’, which is called “d’Alembertian” . A distinguished phenomenon we observe ind’Alembertian is the Huygens principle. Namely when the dimension of the space-time is even andthe support of the fundamental solution of ’Alembertian concentrates on the boundary of the convex
cone in the time-positive direction. We prove in this paper that similar phenomena are observed for tlie
differential operators we are concerned. Another important problem is the description of propagation of
singularity in the solutions of Cauchy problems. Since ’Alembertian is a strongly hyperbolic differential
operator, the singularity of the solutions of Cauchy problems propagates along bicharacteristic strips ofd’Alembertian (see Kashiwara. Kawai and Kimura [9, Chapter 6, \S 6], Duistermaat [2, \S 5.1]). However
the differential operators in this paper is not strongly hyperbolic and the singularity propagates along
not only bicharacteristic strips but also other varieties. In order to see the propagation of singularity, we
have to determine the singularity spectrum of the fundamental solution. Though H\"orlnallder ’s
Theorem 12.6.2 in Page
gives an upper estimate of the singularity spectrum, it does not give theexact singularity spectrum. We give in this paper the exact singularity spectrum of the fundamental
solutions of Cauchy problems for the hyperbolic differential operators.
’Alembertian is an invariant differential operator under the action of Lorentz group. It is natural to
ask whether the same properties are valid for similar invariant differential operators. Indeed, Garding[3]$1\mathrm{b}$)$\square =\dot{c}\overline{)}.\sim$$t^{\underline{2}}+$a$\underline{\gamma}/\dot{c}$$x^{\frac{..)}{1}}..+$$\partial^{r}.)/\partial x_{\underline{2}}^{2}.+\cdots+\partial^{2}/\partial x^{\frac{\mathrm{r}}{n}}$$\geq 4.$$\mathrm{d}$$\mathrm{d}$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{r}\prime 1$

.,

$[()$

$1‘ \mathit{2}5_{\rfloor}^{1}$$\mathrm{D}$lThe author is supported in part by the grant-in-aid of The Mitsubishi Foundation and Grant-in-A
Research (B) (2) 15340042 of Japan Society for the Promotion of Science.

$\mathrm{i}\mathrm{d}$for Scientific82
constructed solutions for the Cauchy problem of matrix-type differential operator on the symmetric maby using the approach of Riesz [14].
and the complex Hermitian matrix spacetrix space
Gindikin[5] enlarged their calculus to more general type of cones on which Lie groups operate liomogeand proved the Huygens principle for invariant differential operators on them systematically. On
the other hand, they never mentioned about the propagation of singuarity of the Cauchy problem.
In this paper, we present more precise results on these problems by utilizing the author’s results inthe preceding papers in [13], [14]. The results of this paper are the followings.
$\mathrm{S}\mathrm{y}\mathrm{m}_{n}$$(\mathbb{R})$$\mathrm{H}\mathrm{e}\mathrm{r}_{n}$$(\mathbb{C})\neg$$\Gamma \mathrm{l}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{l}\mathrm{y}$on the real
1. To construct the explicit fundamental solutions of invariant differential operator
and the quaternion
symmetric matrix space, the complex Hermitian matrix space
Hermitian matrix space(I I) in terms of Laurent expansion coeffcients of the complex powersof the determinant functions (Theorem 7.1).
$P(\partial)$$\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{C}()$$\mathrm{S}\mathrm{y}_{\mathrm{l}}\mathrm{n}_{n}(\mathbb{R})$$\mathrm{H}\mathrm{e}\mathrm{r}_{\mathrm{n}}$2. To determine the exact support and the singularity spectrum of the fundamental solutions of(Theorem 7.1 and Theorem 8.1).

$\mathrm{P}\{\mathrm{d})$3. To give a necessary and sufficient condition in order that the Huygens principle holds (Corollary 7.2).
4. To give a law of the propagation of the singularity for the Cauchy problems with an initial plane
whichis hyperbolic with respect to (Theorem 9.2).
$P$ $(\partial)$

have been partly obtained in
Hc results on the exact support of the fundamental solutions ofrding[l,some preceding papers. Fo example, Gindikin[5, p. 112, Example 2] and Atiyah, Bott, anddifof
invariantp. 181, Example 8.8] mentioned about the exact support of the fundamental solutions
operators on. However the complete computations of the exact support seems to be(II ) and on Hern (ffii), in this paper. Our method is
carried out for the first time, especially on
based on the author’s results on invariant hyperfunctions ([13], [[14]. We give the complete answer tothe Huygens principle of the differential operators. The results on the exact singularity spectrum ofand the propagation of singularity are derived for the first time in
the fundamental solutions of
this paper. It is well known that the singularity spectrum propagates along the bicharacteristic strip
for a strongly hyperbolic differential operators. However, since these operators are hyperbolic but notstrongly hyperbolic, the singularity specrum of the hyperfunction solutions propagates not only along
the bicharacteristic strips. In fact, we can observe that the singularity spectrum propagates along thevarieties which does not consists of bicharacteristic strips. On the other hand, we can give examplesof hyperbolic but not-stlongly hyperbolic differential operator whose singularity of solutions propagates$\mathrm{P}(\mathrm{d})$$\mathrm{G}[mathring]_{\mathrm{a}}$$\mathrm{r}$$\mathrm{f}\mathrm{e}\iota\cdot \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}$$\mathrm{H}\mathrm{e}1^{\cdot}n(\mathbb{C})$$\mathrm{S}\mathrm{y}\mathrm{m}_{\mathrm{n}}$$P(\partial)$

along the bicharacteristic strip(Corollary 9.3).

Fundamental solutions of hyperbolic equations.

1be an $m$ -dirnensional real vector space with a linear coordinate $x=$ $(x_{1}, \ldots , x_{m})$ . We
with respect to the variable . We define a monomial of ’s
the partial derivative. We define the degrees of multi-index bywith a $:=$
’s, i.e., a linear. A differential operator of constant coefficients on $V$ is a polynomial ofhave
is homogeneous if all the monomials in
combination of monomials of ’s . We say thatthe$\xi=(\xi_{1},\ldots,\xi_{m})$. We denotedegree is called the homogeneous degree ofthe sa ne degree.
. For a differential operator
and $P(\xi)$ is a polynomial on
dual coordinate of the dual vector space
if it satisfies $P(\partial)E(x)=\delta(x)$ .
, we say that a distribution $E(x)$ is a fundamental solution ofdenotes the Dirac’s delta function on $V$ with respect to the coordinate $(x_{1}, \ldots, x_{m})$ .
Here

Let $V.–$denote byby$A^{m}$$\partial_{i}$$\mathrm{V}_{i}$$\partial^{\alpha}.--\partial_{1}^{\alpha_{1}}\cdots$$x_{i}$$\frac{\partial}{\partial x_{1}}$$|\alpha|:=$$(\alpha_{1_{1}}\ldots , \alpha_{m})\in \mathbb{Z}_{\geq}^{m_{0}}$$\partial_{m^{n\iota}}^{\alpha}$$\partial_{i}$$\alpha_{1}+\cdots+\mathit{0}_{m}$$P(\partial)$$\mathrm{P}(\mathrm{d})$$7_{i}$’$P(\partial)$$\mathrm{I}1_{1}\mathrm{e}$$V^{*}$$V^{*}$$P(\partial)$$\mathrm{P}(\mathrm{d})$$\mathrm{S}(\mathrm{x})$be a homogeneous difDefinition 1.1 (homogeneous hyperbolic differential operator). Let
We say that} forferential operator with constant coefficients. Let{ $x\in V|$ x,$\mathrm{P}(\mathrm{d})$$N_{\theta}:=$$\langle$$\theta\rangle=0$$\theta\in V^{*}$

83
is hyperbolic with respect toifin has% 0 and the algebraic equation
only real roots for all. In particular, we say thatis strongly hyperbolic if all the roots of$P$ () $=0$ are distinct for any $\xi\in V^{*}$ satisfying !
fwith a constant .
$P(\partial)$$N_{\theta}$$\langle$$\mathrm{P}(\mathrm{d})$$\mathrm{P}(\mathrm{f}+\tau?\uparrow)=0$$\in V^{*}$$\tau\theta$$\xi$$\tau$$P(\partial)$$c\theta$

$\xi$

$c$

For a homogeneous hyperbolic differential operator, we denote by
the connected component ofcontaining . This becomes a convex cone in(see H\"ormander[b., Page120]). We define the dual cone ofby { $x\in V|\langle x$ ,for allF} and denote it by$(P$ , ! . The dual cone
is a closed convex cone in $V$ .
When a hyperfunction $E(x)$ satisfies the differential equation
for a given differential
operator
we call $E(x)$ a fundamental solution of. The following proposition about the supportof the fundamental solution of hyperbolic differential operators is well known. See, Theorem
12.5.1 in Page 120].
$P(\partial)$$\{\xi\in V^{*}|\mathrm{P}(\mathrm{d})\neq 0\}$$V^{*}$$\Gamma(P, \theta)$$1^{\backslash 0}$$\mathrm{F}(\mathrm{P}, \theta)$$\theta$$\theta\rangle\geq 0$$\theta$$\in$$(\mathrm{P},$$\theta)$$\Gamma^{0}(P, \theta)$$)$$\mathrm{P}(\mathrm{d})\mathrm{E}(\mathrm{x})=\mathrm{S}(\mathrm{x})$$P(\partial)\}$$P(\partial)$$\mathrm{H}^{\cdot}0^{\cdot}\mathrm{r}\prime 1\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}[6$Proposition 1.1 (Unique fundamental solutions of hyperbolic equation). Let
be a fiornOgeneous hyperbolic differential operator with respect to. Then there exists one and only one fundamental solution $E(x)$ ofwith support in the half space $II\#:=$ { $x\in V|$ x,. support of}.
this fundamental solution is contained in the dual cone.
$P(\partial)$$N_{\theta}$$\mathrm{P}(\mathrm{d})$$\langle$$r\mathit{1}’he$$\theta\rangle\geq 0$$\Gamma^{\mathrm{o}}(P, \theta)$

However, the support of $E$ (x) does not always coincide with. We often observe that the
support ofis contained in the boundary set of I
and the dimension of the support is sometimes
very small. In such cases we say thatsatisfies the Huygens principle. In particular, if the dimensionof Supp(E(x)) is strictly less than $m-$ l, we say thatsatisfies the strong Huygens rinc i[J .
Ifis hyperbolic with respect to the initial plane, then it is also hyperbolic with respectto. Therefore there also exists one and only one fundamental solutionofwithsupport in the half space $H_{-\theta}:=$ { $x\in V|\langle x,$ -d)
be a homogeneous} In particular, let
hyperbolic operator of degree on
and let $E(x)$ be the unique fundamental solution supported in. Then the unique fundamental solution $E’(x)$ supported in
is given by
A .– $x)$
since $P(\partial)(-1)^{n\iota+n}E(-x)=(-1)^{rn}P(-\partial)E(-x)=(-1)^{n\iota}\delta(-x)=$ E(x).
$1^{\urcorner}0(P, \theta)$

0 $(P, \theta)$$F_{\lrcorner}(x)$$\mathrm{P}(\mathrm{d})$$\mathrm{P}(\mathrm{d})$$\mathrm{P}(\mathrm{d})$$\cdot$$p$$l\epsilon$$N_{\theta}$$N_{-\theta}=N_{\theta}$$E’(i\iota\cdot)$$\geq 0$$n$$P(\dot{c}\overline{J})$$\mathrm{P}(\mathrm{d})$$.$$\mathbb{P}_{\backslash }^{m}$$H_{\theta}$$\mathrm{E}’(\mathrm{x})=(-1)^{71+rn}$$H_{-\theta}$$($

Singularity spectrum of the fundamental solution.

2

Letbe the sheaf of hyperfunctions on $V$ and letbundle $T^{*}V$ of $V$ . We have a natural isomorphism$\mathrm{B}_{V}$sp :$\mathrm{e}_{V}$$\mathrm{s}\mathrm{p}$be the sheaf of microfunctions on the cotangent:(1)$\mathit{1}\mathit{3}_{V}arrow\sim 7\Gamma_{*}(\mathrm{G}_{V})$and an exact sequence$0arrow A_{V}arrow \mathit{1}\mathit{3}v$(2)$arrow\pi_{*}(\mathrm{G}_{V}|_{TV-V\mathrm{x}\{0\}}.)arrow 0.$

is . sheaf of real analytic
is the projection map from the cotangent bundle $T^{*}V$ to $V$ and$V$$f(x)$a
hyperfunction
isomorphism
the
we
can
on $V$ as a microfunction
By
on
.
(1),
functionsregard$V$
with the microfunction
we often identify the hyperfunction $f(x)$ on
on $T^{*}V.$ In this$T^{*}V$$-V\cross\{()\}$
onthrough the isomorphism (1). In particular, we call the set Supp(spthe singularlty spectrum of the hyperfunction $f(x)$ and denote it by S. $S.(f(x))$ .
H\"ormander [6 , Theorem 12.6.2 in Page 125] gave an estimate of the singularity spectrum (analytic wave
bc a homogeneous
front set) of the fundamental solution of a hyperbolic differential operator. Let
the lowest, we denote by. For a fixedinhyperbolic differential operator with respect to$P(\xi+\eta)$ . Thenis also hyperbolic with respectorder homogeneous part in the Taylor expansion(H\"ormander[7, Theorem 8.7.2]). Then we have the following estimate of the singularity spectrum
toof the fundamental solution $E(x)$ .
Here,$A_{V}$$\pi$$\mathrm{s}\mathrm{p}(f(x))$$\mathrm{t}\mathrm{l}\iota \mathrm{e}$$\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e}\rangle$$(\mathrm{f}(\mathrm{x}))$$\mathrm{s}\mathrm{p}(f(x))$$\mathrm{P}(\mathrm{d})$$N_{\theta}$$\eta-$$N_{\theta}$$\mathrm{s}\mathrm{z}$$0$$V^{*}$$P_{\xi}$$P_{\xi}$

84
Theorem 2.1 (Hormander[6]). Let $E(x)$ be a fundamental solution of a hyperbolic differential operator
with support in. Then we have
$P$ $(\partial)$$1^{\urcorner}0(P, \theta)$$\mathrm{S}.\mathrm{S}.\{\mathrm{E}(\mathrm{x}\mathrm{j})\subset${

EE $T^{*}V|\xi\neq 0$ and$(x,$ $\xi)$$x\in\Gamma^{\mathrm{o}}(P_{\xi},$$\theta)$}Cauchy Problems for hyperbolic equation.

3

In this section we denoteand bythe partial derivative with respect to. We denote.
by,(x2, . . . ,another coordinate and by) the partial derivative with respect$)11x,)$ . Let
to $x’:=$.
be a hyperbolic differential operator with respect to the initial plane. Then we can write{ $x\in-V|$ x,}. Let be the order of$x_{1}=\langle x, \theta\rangle$$J^{l}.\cdot:=$$\ldots$$x_{1}$$\partial’:=$$(x_{2)}$$N_{\theta}:=$$\partial_{1}$$x_{rn})$$(x_{\underline{J}_{7}}$$\partial_{m}$$\mathrm{s}$$P(\partial)$$\langle$$l$$\theta\rangle=0$$P(\partial)$$P(\partial)=p_{0}\partial_{\rceil}^{l}+p_{1}(\partial’)\partial_{1}^{l-1}\mathrm{f}$$\cdots+p_{l-1}(\partial’)\partial_{1}+p\iota$ $(\partial’)$. a lloll-zcro constant andwhere,... ,are differential operators in . The Cauchy
problem forwith respect tois the following problem: for a given initial data of hyperfunctions
with conlpact support ,$.|0(x’)$, $v_{l-1}(x’)$ on, construct a hyperfunction solution to the differential$l$)$p_{1}(\partial’)$$\circ \mathrm{i}_{\neg}$$P(\zeta))\cap$$\partial’$$p_{l}(\partial’)$$\mathit{1}\mathrm{V}_{\theta}$

1$N_{\theta}$$\ldots$$(_{-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}_{011}}^{\backslash }$$P(\partial)\uparrow\iota(x)=0$$u(x)|_{x_{[perp]}=0}=v_{\mathit{0}}(x’)$, $\partial_{1}u(x)|_{x_{1}=0}=v_{1}(x’)$ , . . . ,$\partial_{1}^{l-1}u(x)|_{x_{1}=\mathrm{U}}=v_{l-1}(x’)$(3).

For a hyperbolic differential operator, there exists a unique local hyperfunction solution to the
Cauchy problem (3). To prove this, we have only to construct a fundamental solution to the Cauchy
problem of hyperbolic differential operator.
We define the fundamental solution for the Cauchy problem. Let $F_{0}(x)$ be a hyperfunction solution
to the Cauchy problem$P(\partial)$$\mathrm{P}(\mathrm{d})$$P(\partial)F_{0}(x)=0$$F_{0}(x)|_{x_{1}=0}=\dot{c}^{-})_{1}F_{0}^{1}(x)|_{x_{1}=\mathrm{U}}=\cdots=\partial_{1}^{l-}arrow’ F_{\zeta)}(_{i\mathrm{L}}\cdot)|_{\mathrm{r}.=0}$$CJ_{1}-1\Gamma_{\lceil)}^{\mathrm{f}}\cap l(x\cdot)|_{x_{1}=0}=\delta(x’)$

Then$F_{0}^{\backslash }(x)$

(4)

$=0,$

.

is uniquely determined by virtue of the Holmgren’s uniqueness theorem if it exists. We put$F_{k}(x).-- \frac{1}{p_{0}}(p_{0}\partial_{1}^{k}+p_{1}(\partial’)\partial_{1}^{k^{\wedge}-1}+\cdot\cdot.

(5)+p_{k}(\partial’))\Gamma_{0}^{\prec}(x)$for $k=$ ( . 1. . . . 1–1$]$

Definition 3.1 (fundamental solution to the Cauchy problem. The -tuple of hyperfunctions$l$Fo (x), $F_{1}(x)$ , . . .

’(6)$F_{l-1}(x))\in$ $\mathrm{B}(V)^{l}$is called the fundarnental solution to the ,auchy problem (3). In fact,$C$$\mathrm{E}(\mathrm{x})=\sum_{k=0}^{l-1}f_{N}$,$F_{l-k-1}(x_{1}, x’-y’)v_{k}(y’)dy$’(7)

satisfies (3).
Tllc funda mental solution to the Cauchy problem can be constructed by using the fundamental sO.
lutions supported in the convex cones in the following way. Let
be the Heaviside function in, which is uniquely
supported in
Then it is proved thatis a fundamental solution of$\theta(x_{1})$$\frac{1}{p_{0}}\theta(x_{1})F_{0}(x)$$P(\partial)$$x_{1}$$H_{\theta}$

85
determined, and hence we haveFo (x) is given by

E(x). On the other hand, we can prove that th solution$\mathrm{g}\mathrm{e}(x_{1})F_{0}(x)=$$\mathrm{e}$$\Gamma_{0}\sqrt(x).--p0(E(x)-E’(x))$

where Fo(x) andrespectively. Since$\mathrm{E}’(\mathrm{x})$(8)are fundamental solutions ofwhose supports are contained in$(P, -\theta)$
and Supp $(E’(x))$, we have$\mathrm{P}(\mathrm{d})$$\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(F_{\lrcorner}(x))\subset\Gamma^{0}(P, \theta)$Supp

I0 ( $P,$$(F_{k}(x))\subset 1\urcorner 0$ $(P, \theta)\cup$-$=$$\ldots$$II_{-\theta}$$\{F_{0}(x)_{\rangle}
F_{1}(x),. . . . ,,F_{\mathit{1}-1}(x)\}$F_{\mathit{1}-1}(x)\}$$\{F_{0}(x)\rangle F_{1}(x),’$\mathrm{P}(\mathrm{d})$$P(\partial)$

4

and

d)

$l-$ l.. This me-ans
for . , 1, . . . , $l-1$
means that the support of the filndamental
fundamental solution
to the Cauchy problem forwith respect to the initial plane
is contained in$h$$II_{\theta}$$\subset\Gamma^{0}$$N_{\theta}$$\cup\Gamma^{0}(P, - \mathrm{t}2)$$\Gamma^{\mathrm{o}}(P,\theta)\cup\Gamma^{0}(P,-\theta)$$\Gamma^{\mathrm{o}}(P, \theta)$.

Prehomogeneous vector spaces of commutative parabolic type
and their properties.

The prehomogeneous vector spaces we are considering here are the following ones.

1. real symmetric matrix space Let $V:=$ Syrnn
be the space ofsymmetric matrices over
the real field
and let $G.–$ GLJR) be the general linear group overof degree . Then thegroup
acts on the vector space $V$ by the representation$(\mathbb{R})$$\uparrow l\cross n$$\mathbb{R}$$\mathrm{i}\mathbb{R}$$n$$\mathrm{G}\mathrm{L}_{n}(\mathbb{R})$$\rho(g)$with $x\in V$ and$\mathit{9}\in \mathrm{G}\mathrm{L}_{n}$:$x\mapsto g\cdot \mathrm{r}$$\cdot\iota g$,$(^{9)}’$(9). Then the subgroup{}

GLn(R)$G^{1}:=$ \mathrm{G}\mathrm{L}_{n}(\mathbb{R})|\det(g\cdot$G^{1}:=\{g\in{}^{t}g)=] \}$$|\det(g\cdot$ ${}^{t}g)=1$$g\in$

acts on $V$ naturally. Heremeans the tl.ans\urcorner
posed matrl.X
transposed
matrix ofspace, we define the coordinate ofby $x=$The derivation with respect to the coordinate is defined by$tg$$\mathrm{H}\mathrm{e}\mathrm{l}\cdot \mathrm{e}{}^{t}g\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\triangleright\neg$

$x$

$g$

$\mathrm{v}$

.

(1 )
$()$

In thcIllthe case of synlnletricsymmetric lnat.rix
matrixwith $x_{ij}=x_{ji}$ .

$(x_{ij})_{n\geq
i.j\geqV=\mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathrm{F}_{[perp]})$1}\in V=\mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathrm{F}_{[perp]})$$x=(x_{ij})_{n\geq
i.j\geq 1}\in$\mathrm{S}\mathrm{y}\mathrm{m}_{\mathfrak{n}}(\mathbb{R})$$\partial:=(\partial_{ij})_{n\geq}i,j\geq 1=(\frac{\partial}{\partial_{J_{lj}}})_{n\geq}i$,y$.\geq 1$an$\mathrm{d}$$\partial^{*}=(\partial_{i_{J}}^{*})=(\epsilon_{ij}\frac{\dot{\overline{\mathrm{c}}}^{1}}{\partial x_{ij}})$The dual coordinate is given by. The canonical bilinear form on

.

with

$\epsilon_{ij}.--\{\begin{array}{l}1i=j1/2i\neq j\end{array}$$\xi=(\xi_{ij})_{n\geq i,j\geq 1}\in V^{*}=\mathrm{S}\mathrm{y}\mathrm{m}_{n}$$(\epsilon_{\dot{\mathrm{c}}j}\xi_{ij})$$2$$(x_{\backslash }\xi)\in V\cross V^{*}$

is given by$(\rfloor 1)$(IIE) and we denote$\langle x$,$\xi)$$:=\mathrm{t}\mathrm{r}(x\xi’)$. complex Hermitian matrix space Let $V:=$ Here (C) be the space of $nx$$n$over the complex field and let
be the special linear group overthe group
acts on the vector space $V$ by the representation$\mathbb{C}$$G:=\mathrm{G}\mathrm{L}_{n}(\mathbb{C})$$\xi^{*}=(_{\backslash }\xi_{ij}^{*}.)=$$= \sum_{\iota>j>i>1},x_{\iota j}.\xi_{ij}$Hermitian matrices
of degree . Then$\mathbb{R}$$\eta$$\mathrm{G}\mathrm{L}_{r\iota}(\mathbb{C})$$\rho(g)$with $x\in V$ and$g\in \mathrm{G}\mathrm{L}_{n}(\mathbb{C})$: $x\mapsto g$ .

$x$.

${}^{t}\overline{g}$

,

(12)

. Then the subgroup$(]3)$$G^{1}.--\{g\mathrm{E} \mathrm{G}\mathrm{L}_{n}(\mathbb{C})|\det(g\cdot\overline{g})t= 1 \}$acts on $V$ naturally. Herethe coordinate

$x$

of

${}^{t}\overline{g}$$\mathrm{H}\mathrm{e}\mathrm{r}_{\mathrm{n}}(\mathbb{C})$means the transposed matrix of the complex conjugate of . We define.
$g$by $x=(x_{ij})_{n\geq i,j\geq 1}$$E$$V=$ Hern(C)with$x_{i_{\mathrm{J}}}.=x_{ij}^{1}\mathrm{f}$$\sqrt{-1}x_{j}^{\frac{)}{i}}$

and

66
. In particular,
to the coordinate is defined by$xJ^{\cdot}i=Xij=x_{ij}^{1}-\sqrt{-1}x_{ij}^{2}$$x_{j}^{\frac{9}{i}}=0$

and hence $x_{ii}=x$ ! . The derivation with respect$i$$\partial.--(\partial_{ij})_{n\geq \mathrm{i},j\geq 1}=(\frac{\partial}{\partial x_{ij}^{1}}+\sqrt{-1}\frac{\partial}{\partial x_{ij}^{2}})_{n\geq i,j\geq 1}$) by using

We denote.
Thcl’hc dual coorcln ate is given bywith$\partial_{ji}=\overline{\partial_{i_{7}}‘.}=\frac{\partial}{\partial x_{J}^{1}}.-\sqrt{-1}\frac{\partial}{\partial x_{J}^{-}}$

coordir]$\xi_{1}.\cdot
i=\overline{\xi ij}$$\xi_{1i}.=\overline{\xi_{ij}}$

by x,,$\langle$

$x$

.

$!)$

$\xi\rangle$$\partial^{*}=(\partial_{ij}^{*})=(\epsilon_{ij}\frac{\acute{d}}{\partial
x_{1}},\cdot$$\partial^{*}=(\partial_{ij}^{*})=(\epsilon_{ij}\frac{\acute{d}}{\partial
x_{1}},\cdot)$$\cdot$$\xi^{*}=(\xi_{ij}^{*})=(\epsilon
ij\xi_{ij})$$\xi’=(\xi_{ij}^{*})=(\epsilon_{ij}\xi_{ij})$

defined in (11).

Here (C) with $\xi_{ij}=\xi_{ij}^{1}.+\sqrt{-1}\xi 5$ andV\cross V^{*}$
V^{*}$; $\in V\crossis given. The canonical bilinear form on $(x_{7}\xi)\in$\xi=(\xi_{i_{J}})_{n\geq i\geq}i1\in ii\geq
V^{*}=\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{C})$$\xi=(\xi_{i_{j}})_{n\geq1}\in V^{*}=$We denote$\epsilon_{ij}$$\epsilon
ij$$\xi_{ij}=\xi_{ij}^{1}.+\sqrt{-1}\xi_{ij}^{2}$$(x_{7}$;$:=1}x_{ij}^{1}\xi_{ij}^{1}+x_{ij}^{2}\xi_{ij}^{2}$$i\geq 1ij\xi^{1}ijix^{1}+x^{2}j\xi^{2}ij$$:= \Re(\mathrm{t}\mathrm{r}(x\xi^{*}))=\sum_{n>i>1}x_{ii}^{1}\xi_{ii}^{1}+\sum_{n>j>i\geq\Re(\mathrm{t}\mathrm{r}(x\xi^{*}))=\sum_{n>i>1}x_{ii}^{1}\xi_{ii}^{1}+\sum_{n>j}$$)$.

be the space ofHermitian matrices
3. quaternion Her nitian matrix space Let $V:=$ Hereof degree .
be the general linear group over
over the quaternion field EI and let$V$by the representation(IHI) acts on the vector space
Then thc group$n$$(\mathbb{H})$$\cross n$$\mathbb{H}$$G:=\mathrm{G}\mathrm{L}_{n}(\mathbb{H})$$n$$\mathrm{G}\mathrm{L}_{n}$$\rho(g)$with $x\in V$ and$g\in \mathrm{G}\mathrm{L}_{n}(\mathbb{H})$:$x\mapsto g\cdot x\cdot{}^{t}\overline{g}$

(14)

,

. Then the subgroup{$G^{1}:=$$g\in \mathrm{G}\mathrm{L}_{n}$(IHI) $|\det(g\cdot\overline{g})t=1$ }means the transposed matrix of the quaternion conjugate of . We
acts on $V$ naturally. Herewithdefine the coordinate of Hern (M) by $x=(x_{ij})_{n\geq i,\geq 1}f$ $\in V=$ IIern.
and are the imaginary unts
andand ijk $=-1$ . In particular,of the quaternion and satisfy the relations
and hence $.’.ii$. The derivation with respect to the coordinate is defined by${}^{t}\overline{g}$$g$$\mathrm{j}’ \mathrm{r}_{\mathrm{j}}$$+\mathrm{k}x_{ij}^{4}\in$$\mathrm{i}$$JC_{ij}=x_{ij}^{1}+\mathrm{i}x_{\dot{ij}}^{\tau_{l}}+$$(\mathbb{H})$$x$$x_{\mathrm{j}i}=\overline{x_{\dot{|}j}}=$$\mathrm{t}:" j$$-\mathrm{i}x_{\tilde{i}j}^{J}’-\mathrm{j}x_{ij}^{3}-\mathrm{k}x_{ij}^{4}$$\mathrm{H}\mathrm{e}_{\lrcorner}\mathrm{l}.\mathrm{e}\mathrm{i},\mathrm{j}$$\mathrm{k}$$\mathrm{i}^{2}=\mathrm{j}^{\underline{r_{J}}}=\mathrm{k}^{2}=-1$$=x_{ii}^{1}$$x_{i}^{\frac{9}{\mathrm{t}}}.=x_{i_{l}}^{3}\cdot=x_{\iota i}^{4}=0$$o.–( \partial_{i_{I}})_{n\geq i_{\mathrm{h}}j\geq 1}=(\frac{\partial}{\partial x_{ij}^{1}}+\mathrm{i}\frac{\partial}{\partial x_{ij}^{2}}+\mathrm{j}\frac{\partial}{\partial x_{ij}^{3}}+\mathrm{k}\frac{\partial}{\partial x_{ij}^{4}})n\geq i,j\geq$$1$

definedby using
We denote,”j4yin (11). The dual coordinate is given by $\xi=$ (zij)1 $V^{*}=$ Here (IHI) with wi$(x,\xi)\inV\cross$V’on. The canonical bilinear formand. We denote.
is given by x,
and$-\mathrm{k}_{\partial x}^{\mathrm{d}}\ulcorner$,$\partial^{*}=(\partial_{ij}^{\star})=(\epsilon_{ij}\frac{\partial}{\partial x.J}.)$$- \mathrm{j}\frac{\partial}{\partial x_{1}^{3}}$$c \dagger_{ji}=\overline{\partial_{ij}}=\frac{\partial}{\partial x,J},-\mathrm{i}\frac{\partial}{\partial x^{\underline{\tau}_{J}}}$$\xi$$\mathrm{E}$$\xi_{ji}=\overline{\xi_{ij}}$$\mathrm{k}\xi ij4$$\langle$$\xi\rangle$$\epsilon_{ij}$$\cdot$$\mathrm{t}\mathrm{h}5;+\mathrm{i}\xi 5$

$+$

$+$

$\xi’=(\xi_{ij}^{*})=(\epsilon_{ij}\xi_{ij})$$:=\Re(\mathrm{t}\mathrm{r}(x\xi^{*}))=$$\sum n\geq i\geq 1$$xi_{i} \xi ii1+\sum_{n\geq j>i\geq 1}x_{ij}^{1}\xi_{ij}^{1}+x_{ij}^{2}\xi_{ij}^{\underline{9}}+x_{ij}^{3}\xi yi_{j}$$+x_{ij}^{4}\xi_{ij}^{4}$

We can define the determinant of a symmetric matrix or a complex Hermitian matrix but thedeterminant of a quaternion Hermitian matrix is not well defined since is not commutative. It isdefined in the following way. Note that we can write$\mathbb{H}$$z=a+\mathrm{i}b+\mathrm{j}c+\mathrm{k}d=(a+\mathrm{i}b)+$$\mathrm{j}$(c-4$\mathrm{i}d$)$=\alpha$d. Then we can regard IBIB as the algebra
andwithhomomorphism from IH to M2(C) by$\alpha=a+\mathrm{i}b$$\beta=c+\mathrm{i}$$+\mathrm{j}\beta$$\mathbb{C}\oplus \mathrm{j}\mathbb{C}$. Consider the algebra$\iota$:/Let $X=(z_{i,j})\in$ Here (IH) be anby(16), $X$ is mapped in$z=\alpha+$$n\cross n$

j,\mbox{\boldmath $\theta$}),$arrow[_{-\beta}^{\alpha}$$1 \frac{\beta}{\alpha}$quaternion Hermitian matrix. By the homomorphism(16)$\iota$in$\mathrm{M}_{2,\iota}(\mathbb{C})$$X\mapsto\iota(X\cdot \mathrm{j})=(\iota(z_{i_{h}\mathrm{j}}\cdot \mathrm{j}))$Since$-^{t}$$(\mathrm{t} (X\cdot \mathrm{j}))$$=\iota(X\cdot \mathrm{j})\mathrm{J}$

we see that$\iota(X\cdot \mathrm{j})$

acts

$=$

(17)

is an alternating matrix. Then by putting
Pf(t$(X\cdot \mathrm{j})$

)

(18)

we can define the determinant for the quaternion Hermitian matrix $X$ . Here Pf(A) means thePfaffian of an alternating matrix .
$A$87
We denote) $:=\det(x)$ and we put $S:=\{x\in V|\mathrm{P}\{\mathrm{x})=0\}$ . We call the set $S$ thof $V$ . The subset $V-S$ decomposes into $n+1$ connected components,$\mathrm{P}\{\mathrm{x}$if
if $V=$ Her
if $V=$ Hcrn$\{x\in \mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{R})|\mathrm{s}\mathrm{g}\mathrm{n}(x)=(i, n-i)\}$$V_{i}:=\{${$x\in$

tter$n$$(\mathbb{C})|\mathrm{s}\mathrm{g}\mathrm{n}(x)=(20)$$V=\mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{R})$$2(n-i))$ }$\{x\in \mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{H})|\mathrm{s}\mathrm{g}\mathrm{n}(x)=(4i, 4(n-i))\}$$n(\mathbb{C})$,$(\mathbb{H})$,$\mathrm{e}$$si|\iota gul$ ($t’$

.

set

,

(19)

with $i=0,1$ , . . . ’ . The vector space $V$ decomposes into a finite number of G-Orbits;
$n$

(20)
$V:=\mathrm{u}_{\leq f\lambda}s_{i}^{j}0--\leq^{\leq i}--j<n-i0$

where$\{x\in \mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{R})|\mathrm{s}\mathrm{g}\mathrm{n}(x)=(j, n-i-j)\}$$s_{i}^{j}:=\{${$x$$\in \mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{C})|\mathrm{s}\mathrm{g}\mathrm{n}(x)=$(20)2 (n–i-7))}$\{x\in \mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{H})|\mathrm{s}\mathrm{g}\mathrm{n}(x)=(4j, 4(n-i-/\cdot))\}$

if
if
if

$V=\mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{R})$(.21)$V=\mathrm{I}\mathrm{I}\mathrm{e}\mathrm{r}_{n}(\mathbb{C})$$V=\mathrm{I}\mathrm{I}\mathrm{e}\mathrm{r}_{n}(\mathrm{F}\lrcorner)$with integers $0\leq i\leq n$ and $0\leq j\leq r\iota$ –i. Hcre,
for $x\in$ Symn
is the signature of th.
quadratic form
onand
for $x\in$ Hern(C) (resp.
$(19)$ is the.
signature of the quadratic form
on(resp.
). It is clear thatfrom the definition. All orbits in are-orbits. A-orbit in is called a singular orbit. Th subset$S_{i}:=$$x\inV|$
rank(a:) $=n-i$ } is the set of elements of rank $-i.$ It is easily seen that
{
and
.
.
The strata $\{S_{i}^{J}\}0<i-<n,0<j<n-i$ have the following closure inclusion relation$\mathrm{s}\mathrm{g}\mathrm{n}(x)$$(\mathbb{R})$$\mathrm{e}$$q_{x}(\vec{v}):=t\mathrm{i}$$x\cdot\vec{v}$$\overline{v}\in \mathbb{R}^{n}$$\mathrm{s}\mathrm{g}\mathrm{n}(x)$$q_{x}(\vec{v}).--{}^{t}\overline{\vec{v}}\cdot x\cdot \mathrm{i}$$G^{1}$$S$$x\in \mathrm{I}\mathrm{I}\mathrm{e}’.n$$\vec{v}\in \mathbb{C}^{n}$$\tilde{\gamma\prime}\in \mathbb{H}^{r\iota}$$G^{1}$$V_{1}=S_{1)}^{l}$$S$$\mathrm{e}$$n$$S_{i}=C\mathit{2}i_{0\leq j\leq n-}$$s_{i}^{J}\supset S_{i+1}^{\dot{J}}-1\cup S_{i+1}^{j}$where$S:=\mathrm{u}_{1\leq i\leq n}s_{i}$$iiS^{j}$. is the closure of$s_{i}^{J}$$\mathrm{t}1_{1}\mathrm{e}$stratum$s_{i}^{J}$

,

(22)

. In particular, we have$\overline{V_{0}}=S_{0}^{0}=\mathrm{S}_{0}^{0}\mathrm{u}\mathrm{S}_{1}^{0}\mathrm{u}\cdots$$\mathrm{u}s_{n}^{0}$$\overline{V_{n}}=\overline{S_{0}^{n}}=\mathrm{S}_{0}^{n}\cup$$\cdots \mathrm{u}$(23)$s_{1}^{n-1}\mathrm{u}$$s_{n}^{0}$and$S_{i}^{0}=S_{i}^{0}\mathrm{U}$ $s_{i+1}^{0}\mathrm{u}\cdots \mathrm{u}$$\overline{S_{i}^{n-i}}=S7$-i$\mathrm{u}$$s_{i+1}^{n-i-1}\mathrm{u}$$S_{n}^{0}$

. . .

(24)
$\mathrm{u}s_{\mathrm{n}}^{(\}}$We denote by
the dual vector space of $V$ . We define the inner producton $(x, y)\in V\cross V$ by
FS(tr(x )) where and tr denote the real part and the, respectively. Then we can ident ify$V$ andby the contragredient action and the $G$ -orbits in
The group $G$ operates on
are thesame as the ones in $V$ . The cotangent bundle $T^{*}V$ of $V$ can be identified with $V\cross V^{*}$$V^{*}$$\langle x, y\rangle:=$$\langle x, y\rangle$

$\Re$

$y$

$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$V^{*}$

5$V^{*}$$V^{*}$

Invariant differential operators on prehomogeneous vector spaces.

Proposition 5.1 (hyperbolic operator). Let $V$ be one of the vector spaces. Then$\mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{R})$, Hern (C) on$d$$\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{C})$1. Every non-tr ivial omogeneous-invariant differential operator
withwritten as a constant multiple ofwith some positive integer .
$G^{1}$$h$$\det(\partial^{*})^{k}$$P(\partial)$$CO\mathfrak{l}\downarrow$stc?\iota t

is hyperbolic with respect to the initial plane
2. The differential operator
only
and
t2
a
positiveisif
if
definite matrix or a negative definite matrix.
$\mathrm{P}(\mathrm{d})$$0\}$$\in V^{*}$

coefficients is$k$$\mathit{1}\mathrm{V}_{\theta}:=\{x\in V|$$\langle$x,$\theta\rangle$

$=$

68
7. In the case above,is strongly hyperbolic if and only if
operator of space dimer\iota s ion 2.
$\mathrm{P}(\mathrm{d})$$n$and $k=1$ . This is the wave

$=2$

Let
a
let
(resp.
) be a positive (resp. negative) definite matrix. Thenthe connected component(resp. ( ,) is the set of positive (resp. negative).
definite matrices inOn the other hand, the dual cone
(resp.
( $P$,)) is the set ofsemi-positive (resp. semi-negative) definite matrices in $V$ . Therefore we have$P(\partial).--\det(\partial^{*})^{k}$$\mathrm{n}\mathrm{d}$$\theta_{-}$$\theta_{+}$$1^{\urcorner}(P, \theta_{+})\subset V^{*}$$P$$\Gamma$$\theta_{-})\subset V^{*}$$V^{*}$$\Gamma^{0}(P, \theta_{+})$$\Gamma^{0}$$\theta_{-}$$\Gamma^{\mathrm{o}}(P, \theta_{+})=\overline{V_{n}}=\mathrm{u}s_{i}^{n-i}0\leq i\leq n$(25)$\Gamma^{0}(P, \theta_{-})=\overline{V_{0}}=\mathrm{u}s_{i}^{0}0<i<n$

By Proposition 1.1 and Proposition 5.1, there exist unique fundamental solutions supported in.

$V_{n}$

and$\ovalbox{\tt\small REJECT}$

6Complex powers of relative invariants.

Wethe fundamental solutions of the differential operator
by using the complex powers of $P(x)$ . We definc the complex powers$\overline{V_{0}}$$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{l}.\backslash \cdot \mathrm{t},\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}$$\mathrm{P}(\mathrm{d})$which arc supported in
or
$(i=
0,1,
\ldots,n)$$(x)$ |.j
of $P(x)$$V_{n}$$|\#$

byif $x\in V_{i)}$,0if$|P(x)|^{s}$$|P(x)$ $|_{i}^{\kappa}:=\{$(26)$x\not\in V_{i}$for a complex number .
, the integral$.\mathrm{s}$

$\in$

(fM.

Let

be the space of rapidly decreasing smooth functions on $V$ . For$\mathrm{S}(V)$$\mathrm{f}(\mathrm{x})\in \mathrm{S}(V)$$Z_{i}$$(f. s).– \int_{V}|P(x)$$|_{i}^{s}\mathrm{f}(\mathrm{x})dx$

(27)

,

is convergent if the real part $\Re(s)$ of 6 is sufficiently large and is mevomorphically extended to the wholecomplex plane. Thus we can regard $|P(x)$ ; as a tempered distribution
and hence a hyperfunction
with a meromorphie parameter $s\in$ C.
. call each $|P(x)$ the complex power of $P(x)$ . We consider alinearmbination of the hyperfunctions
–

–

$|$

$\mathrm{W}\backslash ’\mathrm{e}$$|_{i}^{s}$$|P(x)|_{i}^{s}$$\mathrm{c}\mathrm{o}$(28)$P^{[\vec{a},s]}(x):= \sum_{i=0}^{n}a_{i}\cdot|P(x)|_{i}^{s}$withparameter$s\in \mathbb{C}$

Since

and

$s\in \mathbb{C}$$P^{[\vec{\mathrm{G}}_{)}s}$$\vec{a}:=$$(\mathrm{o}_{0\backslash }a_{1)}\ldots, a_{n})\in \mathbb{C}^{n+1}$, and depends on] $(x)$. Thenlinearly.
is meromorphic with respect to$P^{[\vec{a}.s]}(x)$is a hyperfunction with a meromorphic$\vec{a}\in \mathbb{C}^{n+1}$$s\in \mathbb{C}|$, we can expand)] $(x)$$P^{[\tilde{a}.s}$

to a Laurent series.

Let$P^{[\vec{a},s]}$$(x)$$= \sum_{j\in \mathrm{Z}}P_{j}^{[\tilde{a},s_{0}]}(x)(s-s_{0})^{\mathrm{j}}$]$(x)$ is a
at $s=s_{0}$ . Then each Laurent expansion coefficient
be the Laurent expansion of
linear function on.
. Then we haveand let $e_{0}:=(1,0, . . . , 0)$(0, . . . , 0, 1)
In particular, let
$(x)$
hence
and
and
|7
$P^{[\tilde{a},s]}(x)$$P_{j}^{[\tilde{a}s_{0}}\rangle$$\vec{a}\in \mathbb{C}^{n+1}$$\in \mathbb{C}^{n+1}$$e_{n}^{arrow}.--$$P^{[e_{\mathrm{r}\iota}^{arrow},b}\mathrm{i}$$(x)=\mathrm{P}(\mathrm{x})|_{n}^{s}$$P^{[e_{\tilde{0}},s]}(x)=|7$

Supp
Supp

$(P^{[e_{n}^{arrow},s]}(x))\subset\Gamma^{\mathrm{o}}(P, \theta_{+})$$(P^{[e_{\vec{0}},s]}(x))\subset\Gamma^{\mathrm{o}}(P,\mathit{9}i-)$$\in \mathbb{C}^{n+1}$

69
Therefore, every Laurent expansion coefficient has the same property
Supp$(P_{j}^{[e_{n}^{\wedge},s_{0}]}(x))\subset\Gamma^{\mathrm{o}}(P, \theta_{+})$Supp$(P_{J}^{[e_{\vec{\mathrm{O}}},s_{0}]}.(x))\subset\Gamma^{\mathrm{o}}(P, \theta_{-})$

for each $j\in Zl$ and.
We can construct the fundamental solutions satisfying the property in Proposition 1.1 as a constant]” ] $(x)$ . Th
multiple of the Laurent expansion coefficients$(x)$ andexact supports of th
aregiven in the following proposition.
$s_{0}\in \mathbb{C}$$P_{j}^{[e_{n}^{arrow}}$’$s_{0}$$s_{0}$$P_{0}^{[}$$\mathrm{e}$$\mathrm{e}\mathrm{m}$Proposition 6.1. The hyperfunctions

and$P^{[e_{\vec{n}_{1}}s]}(x)$$P^{[e_{\vec{n}}}$:$s$] $(x)$

have the following properties

7. They have poles of orderat $s_{0}=-1,$,
at $s_{0}=-1,$ -2, . . . $-n$
at $s_{0}=-1,$ -2, . . . $’-2n11$$-2n+1$$-\lfloor s_{0}\rfloor$$- \frac{3}{2}$$-_{\underline{7}}^{\underline{n}\pm\underline{1}}$$\ldots,$$\{$$-s_{0}$$\ldots,$$,$$-\lfloor s_{0}/2\rfloor$$\ldots$2.

(a)VVhen(a)
VVhen$V=\mathrm{s}\}^{\prime \mathrm{m}_{n}}(\mathbb{R})$$V=\mathrm{S}\}’\mathrm{m}_{n}(\mathbb{R})$$,$when $V=$ Symn(R),,
when
,
when
.
$V=\mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{R})$(29)$V=\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{C})$$V=\mathrm{I}\mathrm{I}\mathrm{e}\mathrm{r}_{n}(\mathbb{H})$, $u’ e$ have$u\prime e$Supp$(P_{-}^{[e}\mathrm{o}()k+1^{\cdot})-(k+\mathit{1}^{\underline{\tau_{P}}}\mathrm{J}^{(x))}1)/2]=\overline{S_{k}^{0}.}$$arrd$$\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(P_{-\lfloor(k+1)\mathit{1}^{\underline{J}}\mathrm{J}}^{[-(k+1)/2]}..(e_{n_{1}}^{\vee}x))=\overline{S_{k}^{n-k}.}$for$k=1,\mathit{2}$,...

$r\iota$’(b) When $V=$ Hern (c) , we haveSupp$(P_{-k}^{[e_{\tilde{0}_{1}}-k]}.(x))=\overline{S_{k}^{0}}$

arid
Supp

for $k=1,$ 2, . .
(c) VVhen,$n$$(P_{-k}^{[\mathrm{e}_{r\iota}^{-},-k]}(x))=\overline{S_{k-}^{\mathrm{n}-k}.}$.

$V=\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{H})$, we have

Supp$(P3_{(}’,k+-k\mathrm{j}_{)/2}\rfloor(x))$$=\overline{S_{\lfloor(k+[perp])/2\rfloor}^{0}}$

and
Supp

for $k=1$ , , . . . ,$|2$$\ldots$

)

$2.n$

$-1$

.

$\Delta\rfloor$$(P_{-\lfloor(k+1)/2\rfloor}^{[e_{n\prime}^{arrow}-k]}(x))=s_{\lfloor(k^{\wedge}+1)/2\rfloor’}^{n-\lfloor(k+1)}$.

77Construction
Construction of fundamental solutions.
solutions.
Theorem 7.1 (fundamental solution). Fundamental solutions for the diffe
differential operator] $(x)$ .
$(\det(\partial)^{*})^{k}(k=1,2\}\ldots )$ is $giv$
e,n as a Laurent expansion
given
Let . be a positivecoefficient ofinteger.
$P(\mathrm{c})=$$P(cJ\ulcorner)=$$\Gamma \mathrm{f}^{\supset}.ntial$$P^{[\tilde{a},s}$$P^{[\tilde{a},s]}(x)$

$h$
$k$

70
1. When $V=$ Syr\iota ln$(\mathbb{R})$, we put$:=P_{1\mathrm{n}\mathrm{i}\mathrm{n}\{0,-\lfloor(n\gamma^{\mathfrak{l}}1-2k)/2\rfloor\}}^{[e-(n+1-2k)/2]})(x)$,$P_{-\prime k}(x):=P_{\min\{0,-\lfloor(n+1-2k)/2\rfloor\}}^{[e_{0},-(n+1-2k)\mathit{1}^{\underline{9}}\mathrm{J}}(x)$.

$P_{+,k}(x)$When we have$l_{1}ave$,$\det(\partial^{*})^{k}P_{+,k}(x)=c_{+,k}\delta(x)$$\det(\partial^{*})^{k}P_{+,k(X)}=c_{+,k}\delta(x)$

clet

and
. TTheref
Th erefore
and $F_{-,k}(x):=$supports
are unique fun
da
m
ental
solutionswhosearecontained
intheSOlutiOnS
fundamental
half spaces$respecti\iota$$F_{-,k}’(x)$, respectively.
y. The exact supports ofandare given by

non-zero constantscertain non-zeroo$?n.\iota th$$?l\mathit{1}^{\cdot}\iota.th$$c_{+_{\mathrm{I}}}^{-1}{}_{k}P_{-,k}(k)$and$H_{-\theta}$$(\partial^{*})^{k}P_{-,k}(x)=c_{-,k}\delta(x)\rangle$$F_{\mathrm{T}^{\mathrm{I}}},k(x):=c_{+}^{-1},\cdot{}_{k}P_{+,k}(k)$$oreF_{\tau^{\mathrm{I}}},k(x):=c_{+}^{-[perp]},\cdot{}_{k}P_{+,k}(k)$$c_{-_{1}k}$$c\mathrm{r}_{)}k$$c_{+_{)}k}$$\cdot$) $el$Supp$H_{\theta}$$’\tau he$$F_{+\}k}^{1}(x)$if $k=1$ , , , $(n- 1)$ /2J$(n+1)$ ’/” ,if
if $k=1$ , 2,, . . . , $(n- 1)$ /2J
,
if
$2$

$\overline{\frac{s\frac{9}{n}}{S_{0}^{n}}k-\cdot \mathit{2}k=}$$\frac{s\frac{9}{n}}{s_{0}^{n}}k-\cdot \mathit{2}k=\overline{V_{n}}$V)$;=\lfloor$$k=\lfloor(n+1)/2\rfloor$$\lfloor(r\iota\mathrm{t}1)/2\mathrm{J}+$$\lfloor(r\iota+1)f2\rfloor+]$$\rfloor$

$n$

$2$

Supp $(F_{-,k}(.c)).---\{$$\lfloor(n- 1)$$\lfloor$$\ldots$$(F_{+\prime$(F_{+\prime k}^{\urcorner}(x)):=\{$k}^{\urcorner}(x)):=\{$$\mathcal{W}^{\cdot}h$en $V=$ Hern$(\mathbb{C})$,$w\mathrm{c}$

1,,

. (x)

$:=P_{\mathrm{m}\iota \mathrm{n}\{0-(n-k)\dagger(x)}^{[e_{\tilde{n}}-(n-k)]}|..\lrcorner$’’$P_{-_{1}k}(x):=P_{\min\{0-(}^{[e_{\tilde{0}},-\iota-}$When$1\mathcal{L}\cdot\theta$a$\ldots$put$P_{+\prime h}$$Tf\iota\epsilon\cdot n\mathit{1}L\cdot \mathit{6}$$\ldots$$\lfloor(n-1)/2\rfloor$$k= \lfloor(n+1)\oint\lfloor(n+1)\oint 2\rfloor$
2\rfloor)$ $\lfloor(n+1)/2\rfloor+$$\lfloor(n+1)/2\rfloor+1$

2.

,$\lfloor$$\ldots$$\overline{\frac{S_{n}^{0}}{s_{\mathrm{t}\mathrm{J}}^{0}}-\cdot \mathit{2}k=.}\overline{V_{0}}$$\mathrm{i}$$\mathrm{v}\mathrm{Q}k$

)}

.

$(x)$

$h$
$ha\iota)e$) $e$$\iota$$\mathrm{d}\mathrm{c}\mathrm{t}(\partial^{*})^{k}P_{+_{\mathrm{I}}k}.(x)=c_{+_{1}k}\delta(x)_{\backslash }$$\mathrm{d}\mathrm{c}\mathrm{t}(\partial^{*})^{k}P_{+_{\mathrm{I}}k}.(x)=c_{+_{1}k}\delta(x)$$\det(\dot{c})^{*})^{k}P_{-,k}(x)=c_{-_{1}k}\delta(x)$?t\iota s . . and
r$eF_{+.k}(x):=c_{+,k}^{-\iota}P_{+,k}(k)$ and $F_{-,k}(x):=$constants..k.
$(h.)$ are un$f.u?ldc\iota me$ n1c|l solutions
ed.\iota n the half spaces
npports $aoe$
are contair\iota
contained
uniqueique fundamentaluwhose supports,ue/?byby, $.espectittclq$
and $F_{-.k}(x)$) are given’e. ].. The exact supports of $F_{+,k}(x)nnd$

l.\Gam a lth (.ertatn$w\iotatf\iota(.erta$$w\iota$

ont-zero$.r\iota
on- z\epsilon^{\mathrm{J}}roc\cdot or\iota stu\iota$$.r\iota$$c_{-k}$and$H_{-\theta}$$s\mathrm{o}l$$’\cdot esl)\xi^{\supset}.\prime^{\backslash }ti1$$’\Gamma her.ef(J^{\cdot}re F_{+.k}(x):=c_{+}^{-1},{}_{k}P_{+,k}(k)$$(^{n}.-$$c_{+,\mathrm{A}}$$c_{+}^{-1},\cdot{}_{\mathrm{A}}P_{-,k}(h.)\mathrm{C}l’.\underline{t}^{\supset}$$c_{+}^{-1},\cdot{}_{k}P_{-,k}$

$cmd$

,

$’\tau he\mathfrak{l}.\mathrm{e}f(\mathrm{J}^{\cdot}$$\cdot$$t\iota.ons$$.\iota$$s$$F_{+\rangle k}(x)$$l.1$

Supp $(F_{+\prime k}(x)):=\{$.

$\mathrm{F}-\mathrm{t}\mathrm{k}\{\mathrm{X}$$thcl\iota alf6^{\cdot}pacesH_{\mathrm{t}^{(\}}}$$H_{\mathrm{t}^{(}f}$$a\uparrow^{\backslash }egi$

$n-1$
if
.
$=n,$, $n+1$ ,if $=n$,,if $k=1$ , $n+1,.n..-1$
if $=n,$ $n+1$ ,$k=1,2_{7}\ldots\backslash$$\overline{\frac{S_{n}^{k}}{S_{0}^{n}}-k=.}\overline{V_{n}}$$\frac{S_{n}^{k}}{s_{0}^{n}}-k=$$\overline{V_{n}}$$h$$\ldots$

Supp(
Supp

(x) $:=\{$k. (x))$(F_{-})$$F_{-)h}$

$)$

$.if$
$\overline{\frac{S_{n}^{0}}{S_{0}^{0}}-k=}\overline{V_{0}}$$\mathit{2}$$\ldots$

$nt$

–

$l$

$k$

$\ldots$

3. When$V=\mathrm{I}\mathrm{I}\mathrm{e}\mathrm{r}_{n}$(.E), we put
$(x)$

,

$P_{-,k}(x):=P_{\min\{0,-n-\mathrm{L}-k/2\rfloor\}}^{[c_{\tilde{0}},-\underline{\mathrm{o}}_{n+k+1]}}(x)$.

$P_{+,k}(x):=P_{\min\{0_{l}-n-\mathrm{L}-k/2\rfloor \mathrm{t}}^{[\mathrm{e}_{\tilde{n}\prime}2n+k+1]}$

a\iota fWhen
Th we Iiattc$\mathrm{f}^{\supset},nu’ e\mathit{1}_{l}$$c$$\det(\partial^{*})^{h}P_{+,k}(x)=c_{+,k}\delta(x)$,$-,k(X)=c_{-,k}\delta(x)$,$\det(\dot{c}\overline{J}^{*})^{k}\mathrm{P}$$\det(\dot{c}\overline{J}^{*})^{k}P_{-,k}(x)=c_{-\primek}\delta(x)$and $F_{-,k}(x).--$. Therefore
andta in non-zero constants
th cercertainA..
,ntal solutions whose supports are contained in the half spaces$(k)$ arc uniquc
9unique $fundamt^{3}$
fundamental$F_{-,k}(x)$$\Gamma_{+,k}’(x)cmd$
by
given
are
and
, respectively. The exact supports ofand$w\iota tl\iota$$w\iota$$F_{+,k}(x).--c_{+,k}^{-1}P+)\mathrm{A}$$F_{+,k}(x).--c_{+}^{-1},{}_{k}P_{+)}$$\cdot$$c_{+,k}$$c_{-)k}$$(h^{\wedge})$$(h\wedge)$$H_{1}$$H_{1\mathrm{y}}$$c_{+_{1}}^{-1}{}_{k}P_{-,k}$$c_{+_{1}}^{-1}{}_{k}P_{-,k}(k)$$\Gamma_{+,k}’(x)$$H_{-\theta}$$(F_{+\ranglek}(x)):=\{-$k}(x)):=\{$
Supp $(F_{+\rangle$\frac{s_{n}^{-}}{S_{0}^{n}}+\lfloor-\frac{/2\rfloork[2}{V_{n}}\rfloor\lfloor-k=$$\frac{s_{n}^{-}}{s_{0}^{n}}\frac{/2\rfloork[2}{V_{n}}+\lfloor-\rfloor\lfloor-k=$

2
if $k=1,2$ , , $2(n-1)$$(n -1)$
if $k=2(n-1)+1,$ 2 ( $n-$ l ) $+2$ ,if $=1,2,$ , $2(n-1)$
2 $(n-1)+2$ ,if $k=2(n-1)+1,$ $2(n-1)+2$$\ldots$$\iota.f$$k$Supp $(F_{-,k}(x)):=\{$$\overline{\frac{S_{n}^{0}}{S_{0}^{0}}+\mathrm{L}-\mathrm{J}=\frac{kt2}{V_{0}}}$$\overline{\frac{s_{n}^{0}}{s_{0}^{0}}+\lfloor-\rfloor=\frac{kt2}{V_{0}}}$$l.f$$\ldots$$\ldots$$\ldots$

71
Corollary 7.2 (Huygens principle). The hyperbolic operator
if and only if$k=1,2$ ,$\ldots$

A $=1,2$ ,$\ldots$$k=1,2$ ,$\ldots$,)$\lfloor(n$–1 /2\rfloor when$)$$(?\lambda-1)$$\det(\partial^{*})^{k}$$V=\mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{R})$satisfies the$Hv$ygens principle,

when $V=$ Hern(C),, 2 $(n-1)$ when(30)

$V=\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{H})$In particular, it satisfies the strong Huygens principle except for the case that77
and isodd in, the case that $k=(n-1)$ in $V=$ Hern(C) or the case that $k=2(n-1)$ , $2(rl-1)-1$in(IHI).
$k=\lfloor$$($–$1)/2\mathrm{J}$$r\iota$$V=\mathrm{S}\mathrm{y}\mathrm{m}_{n}$ $(\mathbb{R})$$V=\mathrm{H}\mathrm{e}\mathrm{r}_{n}$Remark 7.1. The exact supports of the fundamental solutions have been partly determined in sollle
preceding papers. For example, see Gindikin[5, p. 112, Example 2] and Atiyah, Bott, and.llg[l,p. 181, Example 8.8]. However, in both of the papers, they mentioned only that the support of thefunda mental solution ofcoincides with the set of positive semi-definite matrices of rankinthe case when $V=$ Here (C) while we have determined in this paper the exact support of the fundamental
solutions in the case of
and $V=$ Here (E). Instead of the precise calculation of the support
of fundamental solutions in the specific examples, they gave a theory to handle a wide range of examples.
For example, Gindikin’s theory can also be applied to a certain kind of parabolic differential operators.
$\mathrm{C}_{\mathrm{Y}}[mathring]_{\mathrm{a}}\mathrm{r}\mathrm{d}_{1}$$\det(\partial^{*})^{k}$$\leq t.$

$V=\mathrm{S}\mathrm{y}_{\mathrm{l}}\mathrm{n}_{n}(]\mathrm{R})$Singularity spectrum of fundamental solutions.

8

Definition 8.1 (conormal bundle of a subvariety). Let
be a non-singular subvariety in $V$ . We
define the conormal bundle
tobe$*V|\xi\in$where $(T_{A}^{*}V)_{x}:=\{(x, \xi)$ofthat satisfies (”, $”\rangle=0$ for all ($;\in$ (TA)X). Here $(T^{*}V)_{x}$ and $(TV)_{x}$ are tangent or cotangent(,
vector spaces of $V$ at $x\in V,$ respectively, and $(TA)_{x}$ is the tangent vector space of at
.
$A$

$A$

$T_{A}^{*}V$$T^{*}V1_{x}$$\mathrm{T}_{A}^{\urcorner}*V:=\bigcup_{x\in A}(T_{A}^{*}V)_{x}$$\in \mathit{7}$$\langle$$A$$x\in\Lambda$

Theorem 8.1 (singularity spectru ). Let $F_{+,k}(x)$ and,k (x) be the fundamental solutions ofin
Theoremsingularity
7.1.
The
spectrumthemgiven ? tlic following forrmxl(’s.
aredefinedof$F$$\mathrm{m}$$-$$(\det\partial^{*})^{\mathrm{A}}$

$l?$

1.

$V\mathrm{P}^{\cdot}f\iota e./?V=\mathrm{S}\mathrm{y}\mathrm{m}_{n}$(F. ) ,$ve$$h$c\iota $ve$

$S.S.(F_{+_{\mathrm{I}}k}^{1}(x))=$$\cup$$.i=$$T_{S}^{*},’ V\iota-\cdot$

rnax $\{n-2k,1\}$$(31_{\grave{)}}$. .

$S$ $S$$\vee c\mathit{1}J$. When$V=1\mathrm{I}\mathrm{e}1^{\cdot}n((\mathbb{C})$$( \Gamma_{-)k}^{\prec}(x))=\cup\prime l_{S_{1}^{0}}^{*}Vi=\max\{n-2k,1\}n$, $ufe$ have.

$S.S$ $(F_{+\prime k}(x))$$= \cup T_{S^{n-\iota}}^{*}.Vi=\max\{n-k,1\}$

(32)
$S$

.

$S.(F_{-_{\mathrm{I}}k}(x))= \cdot i=\max\{n-k,1\}\cup T_{S^{0}}^{*}.Vn$

3. When $V=$ Here (IFII), we have
$S.S.(F_{+\}k}(x))= \iota=\max\{n+\lfloor-kf^{I}..\rfloor 11\}\cup T_{S^{n-1}}^{*},V$(33)

S. $S.(F_{-,k}(x))=$$\cup n$$T_{S_{t}^{0}}^{*}V$$i= \max\{n+\lfloor-k\mathrm{s}/2\rfloor,1\}$

72
Remark 8.1. We have $S.S.(F_{+,k}(x))\subset$
applying
rmandcr’s Theorem 2.1.

$)_{i=1}^{n-1}\overline{T_{S^{n-}}^{*},\cdot V}$

and$S.S.(F_{-,k}(x)) \subset\bigcup_{\mathrm{i}=\mathrm{J}}^{n-1}\overline{T_{S_{\mathrm{t}}^{0}}^{*}V}$in all cases by$\mathrm{H}^{\cdot}0$9

The Cauchy problem and the propagation of singularity.

Let $V$ be one of, Here (C) andand let $P$b the differential operator
on $V$ . For an non-zero element
we put $N_{\theta}:=\{x\in V|\langle x, \theta\rangle=0\}$ . Let
and let$V$$(x_{1}, x’)=(x_{1}, x_{2)}\ldots\rangle x_{m})$ be a coordinate of. We denote by
the partial
derivatives with respect
the coordinate $(x_{1}, x’)=(x_{1}, x_{2}, \ldots)$. Hereis the dimension of $V$ .
We denote by $l=kn$ the order of the differential operator. Thencan be$\mathrm{S}\mathrm{y}_{\mathrm{I}1\mathrm{l}_{\mathrm{n}}}$$(\mathbb{R})$$(\partial)=(\det(\partial^{*}))^{k}$$\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{H})$$\mathrm{e}$$\theta\in V^{*}$$x_{1}:=\langle x, \theta\rangle$$(\partial_{1}, \partial’)=(\partial_{1}, \partial_{2}, \ldots, \partial_{m})$$x_{m})$$\mathrm{t}_{1}\mathrm{o}$$rn$$\mathrm{P}(\mathrm{d})=(\det(\partial^{*}))^{k}$$\mathrm{P}(\mathrm{d})$written as$P(\partial)=p_{0}\partial_{1}^{l}+p_{1}(\partial’)\partial_{1}^{l-1}+\cdots+p_{l-1}(\partial’)\partial_{1}+p\iota$$11\acute{\mathrm{c}}$(34)$(\partial’)$consider the Cauchy problem$P(\partial)u(x)=0$$\partial_{j}u(x)|_{x_{1}=}0$(35)$(j=0,1, \ldots, l-1)$ .

$=\cdot v_{j}(x’)$$(v_{0}(x^{J})\ldots., v_{l-1}(x’))\in$ iB
for a given initial dataconsisting of compact supported hyperfunctions on /. The unique solution to the Cauchy problem (35) is given by$(N_{\theta})^{l}$$\mathrm{v}:=$$u(x)= \sum_{j=0}^{l-1}7_{N}$,$F_{l-j-1}^{\urcorner}(x_{1}, x’ -y’)v_{j}(y’)d\iota f$

by using the fundamental solution$FS_{\theta}:=$$(F_{0}$,... ,(36)$F_{l-1})\in \mathit{1}\mathit{3}$ $(V)^{l}$where$F_{\mathrm{t})}^{1}(x).--p_{\mathrm{U}}(\Gamma_{+,k}^{J}-F_{-,k})$$F_{j}(x).-- \frac{1}{I^{)}0}(p_{()}\partial_{1}^{j}+p_{1}(\partial’)\partial_{1}^{j-1}+:.

.

(for $j=1,$+7j(\partial’))F_{0}(x)$$\ldots$: $l-$

1)

The support and the singularity spectrum of the initial data are defined bySupp$(\mathrm{v}(x))=j=0l-1\cup$Supp$(v_{j}(x’))\subset N_{\theta}$

,

(37)
. .

$S$ $S$$(\mathrm{v}(x))$$=j=0l-1\cup S’$.S.

$(v_{\mathrm{J}}\cdot(x’))$$\subset T$’$N_{\theta}$,

and those of the fundamental solution FS$ are defined by
Sup (F6\mbox{\boldmath $\tau$}\mbox{\boldmath $\theta$})$=j=0l-1\cup \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(F_{j}(x))\subset V,$(38)$S.S.(FS_{\theta})= \bigcup_{J^{=0}}^{l-1}$

S. $S.(F_{j}(x))\subset T^{*}V$ .

The support and the singularity spectrum of the fundamental solution
in the following theorem.

$FS_{\theta}$

can be computed explicitly

73
Theorem 9.1 (support and singular spectrum). The exact support and the exact singularity
trum of the fundamental solutions FS$ to the Cauchy problems (35) are given by (39) and (40), respactivcly.

$.9^{\cdot}’ J\mathrm{C}(.-$$(s_{i}^{0}\mathrm{u}n \mathrm{S}_{i}^{n-i})$$i= \max\{n-2k,0\}\mathrm{u}$if$V=\mathrm{S}\mathrm{y}\mathrm{n}1_{n}(\mathrm{J}\mathrm{B})-)$

if$V=\mathrm{H}\mathrm{e}1_{n}^{\cdot}(\mathbb{C})$if$V=\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{H})$

$n$

Supp

$(F1_{arrow}\overline{\mathrm{t}}_{\theta}’)=$$i= \max\{n-k,0\}\mathrm{u}n$$(\mathrm{S}_{i}^{0}\cup S_{i}^{n-i})$($\mathrm{u}$.

$S_{i}^{0}$

LJ$s_{i}^{n-i}$).

$\cdot$$i= \max\{n\{-\lfloor-k/2\rfloor,0\}$$i= \max_{n-1}^{n-1}\{n-2k,1\}\cup\overline{(T_{S^{0}}^{*}.V\cup\prime\Gamma_{S^{n-}}^{\mathrm{r}}.\cdot.V)}$$S.S$ $(FS_{\theta})=$(. {9)$\cup\overline{(T_{S_{\iota}^{0}}^{*}V\cup T_{S^{n-}}^{*}.\cdot V)}\{n-k,1$

if$V=\mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{R})$if$V=\mathrm{H}\mathrm{c}\mathrm{r}_{n}.(\mathbb{C})$if$V=\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{H})$$(\prime \mathrm{t}0)$]$i= \max$$n-1\cup$$\overline{(T_{S^{0}}^{*}‘ V\cup T_{S_{1}^{\tau\iota-}}^{*}.V)}$$i= \max\{n+\lfloor-k/2\rfloor,1\}$

Theorem 9.2 (propagation of singularity). Let $u(x)$ be the unique hyperfunction solution to the
Cauchy problem (35). Then we have:
1. If$x_{0}\in$Supp(u(x), then$x_{0}\in$?

If$(x_{0}, \xi_{0})\in$(a)$x_{0}${$x_{0}=$

S.S.(u (x)) and$x_{0}\not\in N_{\theta}$$\in \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(\mathrm{v}(x))$, then there exists

and$z_{0}\in$$y_{0}\in \mathrm{N}_{\theta}$

Supp$(FS_{\theta}$

)}.

(41)

satisfying the following condrtions$-y0\in$ Supp $(FS_{\theta})$(b) Let$s_{i}^{p}$

(

$p$

$=0$

or

$p=\uparrow\tau-$$(x_{0}-y_{0}, \xi_{0})\in T_{S_{2}^{p}}^{*}V$(c).l0 $+z_{0}|!\mathit{0}$$(y_{0},\overline{\xi_{\mathrm{U}}})\in S.S.(\mathrm{v}(x))$i) be a$G^{1}$-Orbit in$\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(F1\overline{5} |\theta)$that$x_{0}$–$y_{0}$

belongs to.

Then.
. Hern$\overline{\xi_{0}}$means the projection of$\xi_{0}\in V^{*}$

onto$N_{\theta}^{*}$.

Corollary 9.3. Let.
The singularity spectrum of the hyperfunction solution of the
Cauchy problems for
propagates along
and
if and only if $k=1$ and $?t=2$ in$V=$ Syl nn(F) or $k=1$ in $V=$ IIernor in.
$\mathrm{P}(\mathrm{d})=\det(\partial^{*})^{k}$$\mathrm{P}(\mathrm{d})$$T_{S_{n-1}^{0}}^{*}V$$T_{S_{n-1}^{1}}^{*}V$$\cdot$$(\mathbb{C})$$V=\mathrm{H}\mathrm{e}\mathrm{r}_{n}(\mathbb{H})$In particular,andare subvarieties consisting of bicharacteristic strips of. The differential operator
is not strongly hyperbolic except for the case of
2.
Therefore, for $n\geq 3$ in Hern,or in Hernis an example of a non-strongly hyperbolic
differential operator whose singularity spectrum of solution propagates only along bicharacteristic strips.
$T_{S_{n-1}^{0}}^{*}V$$\mathrm{P}(\mathrm{d})=$$T_{S_{\mathfrak{n}-1}^{1}}^{*}V$$\det$ $(\partial^{*})$$\det(\partial^{*})$$(_{\backslash }\mathbb{C})$$\uparrow\downarrow=$$(\mathbb{H})$$\det(\partial^{*})$

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