Journal of Lie Theory Volume 14 (2004) 111–140 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

Algorithmic Construction of Hyperfunction Solutions

to Invariant Differential Equations on the Space of Real Symmetric Matrices

Masakazu Muro^{∗}

Communicated by J. Hilgert

Abstract. This is the second paper on invariant hyperfunction solutions of invariant linear differential equations on the vector space of n × n real symmetric matrices. In the preceding paper [22], we proved that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter. Fundamental properties of the complex power have been investigated in [19]. In this paper, we give algorithms to determine the space of invariant hyperfunction solutions and apply the algorithms to some examples. These algorithms enable us to compute in a fully constructive way all the invariant hyperfunction solutions for all the invariant differential operators in terms of Laurent expansion coefficients of the complex power of the determinant function. Keywords: invariant hyperfunctions, symmetric matrix spaces, linear differential equations, 2000 Mathematics Subject Classification: Prim.: 22E45, 58J15; Sec.: 35A27

Introduction.

Let V := Sym_{n}(ℝ) be the space of n × n symmetric matrices over the real field ℝ and let SL_{n}(ℝ) be the special linear group over ℝ of degree n. Then the group G := SL_{n}(ℝ) acts on the vector space V by the representation

ρ(g) : x ⟼ g · x := gx^{t}g,  (1)

with x ∈ V and g ∈ G. Let D(V ) be the algebra of linear differential operators on V with polynomial coefficients and let ℬ(V ) be the space of hyperfunctions on V . We denote by D(V )^{G} and ℬ(V )^{G} the subspaces of G-invariant linear differential operators and of G-invariant hyperfunctions on V , respectively. For a given invariant differential operator P (x, ∂) ∈ D(V )^{G} and an invariant hyperfunction v(x) ∈ ℬ(V )^{G}, we consider the linear differential equation

P (x, ∂)u(x) = v(x)  (2)

∗ Supported in part by The Mitsubishi Foundation and by the Grant-in-Aid for Scientific Research (C)(2)11640161 and (C)(2)13640163 , Ministry of Education, Culture, Sports, Science and Technology, Japan

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag where the unknown function u(x) is in ℬ(V )^{G}.

This paper is the sequel of the author’s paper [22]. In the preceding paper [22], under some suitable conditions, the author proved that every invariant hyperfunction solution to (2) is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the complex parameter. On the other hand, the author [19] has made clear which Laurent expansion coefficients appear from the Laurent expansion of the complex power of the determinant function. In this paper, we give algorithms to construct all invariant hyperfunction solutions to (2) in terms of the Laurent expansion coefficients by using the results of [22] and [19]. The aim of this paper is not only to give solution spaces in an abstract form but also to give algorithms to construct all the solutions for given differential equations of the form (2) using the Laurent expansion coefficients of the complex power of the determinant function.

Our algorithms are described as procedures with a differential equation of the form (2) as input and its solution space or its solution as output. However, they fully depend on the roots of b_{P} -functions of P (x, ∂) ∈ D(V )^{G} and the exact orders of the poles of the complex power functions. The author has given an algorithm to compute the b_{P} -function of a given invariant differential operator in [20]. On the other hand, the exact orders of poles of the complex power functions are completely determined by the author in [19]. By combining the results in [22] and [19], our algorithms works well in practical computations of invariant hyperfunction solutions.

We explain the organization of this paper. In §1, we introduce some notions and notations and review the results obtained in [22]. We state Theorem 1.3, Corollary 1.7, Theorem 1.8 and Theorem 1.9, which are main theoretical results of the paper [22]. They guarantee that every G-invariant hyperfunction solution for P (x, ∂)u(x) = 0 or P (x, ∂)u(x) = v(x) can be written as a finite sum of the Laurent expansion coefficients of | det(x)|^{s} and that the solution space is determined by the b_{P} -function of P (x, ∂) (see Definition 1.1). In §2, we give a method to determine the order of pole of P^{[a⃗,s]}(x) as an application of the author’s results in [19], and introduce “standard basis”. The standard basis will be used in the algorithms in the later sections. Then, we give the algorithms to construct G-invariant hyperfunction solutions in §3 and §4 to given G-invariant differential equations. Some examples of typical G-invariant differential equations and their invariant solutions are given in §5.

The main results of this paper are the algorithms in §3 and §4 and the examples in §5. They are not abstract existence theorems but practical algorithms to construct all the invariant solutions. For example, we prove in Proposition 5.2 that every SL_{n}(ℝ)-invariant hyperfunction solution to the differential equation det(x)u(x) = 0 on V = Sym_{n}(ℝ) is a linear sum of SL_{n}(ℝ)-invariant measures on the SL_{n}(ℝ)-orbits in the set S := {x ∈ Sym_{n}(ℝ) | det(x) = 0}. This is proved by carrying out the procedure of Algorithm 3.2. At the same time, our algorithm guarantees that there are no other solutions except for the invariant measures. This is a natural extension of the fact that the hyperfunction solution to the differential equation xu(x) = 0 on the real line is only a constant multiple of the delta function u(x) = c · δ(x). This may be rather a well-known example, but our algorithm can be applied to all the invariant differential equations if the b_{P} -function of the invariant differential operator can be computed.

Methods to construct explicit solutions to linear partial differential equations which are invariant under the action of Lie groups has been studied for a long time. However, they are supposed to be a special type of differential operators. For example, Riesz [27] considered the Cauchy problem for d’Alembertian, which is an invariant differential equation by the Lorentz group action. Methée [13], [15] dealt with the invariant distribution solutions for d’Alembertian, which was applied to the study of quantum field theory by Bogoliubov et al [4]. Gårding [7] studied the Cauchy problem for the differential operator given of determinant type, which is considered as an invariant differential operator under the group action of the special linear group. These studies can be regarded as investigation on groupinvariant linear differential equations. Modern method that exploit micro-local analysis actively enable us to construct explicit hyperfunction solutions algorithmically for a wider range of invariant differential operators. The author considered invariant differential operators on the real symmetric matrix space to show a sample of the algorithm. We can calculate all the invariant hyperfunction solutions to all the invariant linear differential equations under some additional conditions — the homogeneity of differential operators and the the condition on b_{P} -function — on the real symmetric matrix space.

The author thinks that the method employed in this paper is rather orthodox. Namely, we are only analyzing the distributions defined by the complex powers of polynomials of special type, which had been developed by Riesz [27] and Gårding [7] before the concept of Schwartz’s distributions or Sato’s hyperfunctions was produced. Old analysis is sufficiently great but seems to be insufficient now since we obtain more precise results like the outcomes of this paper — which could not be achieved by old technique. We can analyze a wider range of invariant differential equations, especially on prehomogeneous vector spaces, and they will be analyzed in the future papers. Refer to Sato and Shintani [28] for typical prehomogeneous vector spaces and see Kashiwara, Kawai, and Kimura [12] for micro-local analysis.

Our work has close connections with the analysis on homogeneous cones and the invariant distributions on them. A systematic monograph [6] has been written by Faraut and Korányi. Other recent references on invariant distributions are Blind[3] and Ricci and Stein[26]. On the other hand, we can find many interesting works about invariant differential equations on some kinds of spaces where Lie groups are acting. For example, see the recent works Ames[2] and Bouaziz and Kamoun[5]. Concerning invariant differential operators, Nomura’s works [24] and [25] are of value for reference.

Notations: In this paper, for a square matrix x, we denote by ^{t}x, tr(x) and det(x) the transpose of x, the trace of x and the determinant of x, respectively. The complex numbers, the real numbers and the integers are denoted by ℂ, ℝ and ℤ, respectively. ℤ_{≥0} denotes the non-negative integers and ℤ_{>0} denotes the positive integers.

Acknowledgement: The author expresses his sense of gratitude to the referee for the careful reading of the manuscript and kindhearted advice.

1. Fundamental definitions and theorems.

In this section we review the fundamental definitions and theorems given in the preceding paper [22]. For the precise definitions and the complete proofs of theorems, refer to the paper [22].

Let V be a finite dimensional real vector space of dimension m with linear coordinate (x_{1}, …, x_{m}). Then a polynomial with complex coefficients on V is given as a complex finite linear combination of monomials x^{α} := x_{1}^{α_{1}} ⋯ x_{m}^{α_{m}} with α := (α_{1}, …, α_{m}) ∈ ℤ_{≥0}^{m}. We denote by ∂_{i} the partial derivative \frac{∂}{∂x_{i}}. We define a monomial of \frac{∂}{∂x_{i}} ’s by ∂^{β} := ∂_{1}^{β_{1}} ⋯ ∂_{m}^{β_{m}} with β := (β_{1}, …, β_{m}) ∈ ℤ_{≥0}^{m}. We define the degrees of multi-index by |α| := α_{1} + ⋯ + α_{m} and |β| := β_{1} + ⋯ + β_{m}. For a given monomial differential operator a_{αβ}x^{α}∂^{β}, we call |α| − |β| (resp. |β|) homogeneous degree (resp. order) of the monomial differential operator a_{αβ}x^{α}∂^{β}. A homogeneous differential operator of homogeneous degree k in D(V ) is a differential operator given as a finite linear combination of monomial differential operators of homogeneous degree k. We denote by D_{k}(V ) the ℂ-vector space of homogeneous differential operators of homogeneous degree k. We call a differential operator invariant under the action of all g ∈ G a G-invariant differential operator on V . We denote D(V )^{G} the totality of G-invariant differential operators on V . We can easily check that D(V )^{G} is a subalgebra of D(V ) and that D(V )^{G} = ⨁_{k∈ℤ} D_{k}(V )^{G} := ⨁_{k∈ℤ} D_{k}(V ) ∩ D(V )^{G} is a natural gradation induced from the gradation D(V ) = ⨁_{k∈ℤ} D_{k}(V ).

Let ℬ(V ) be the space of hyperfunctions on V and let ℬ(V )^{G} be the space of G-invariant hyperfunctions on V . We say that v(x) ∈ ℬ(V ) is quasihomogeneous provided that there exist a complex number λ ∈ ℂ and a nonnegative integer k ∈ ℤ_{≥0} satisfying

\underbrace{F_{r,λ} ∘ F_{r,λ} ∘ ⋯ ∘ F_{r,λ}}_{(k+1)-times}(v) = 0  (3)

for all r ∈ ℝ_{>0} with F_{r,λ}(v) := v(r · x) − r^{λ}v(x). We call λ ∈ ℂ the homogeneous degree (or simply the degree) of v(x) and call k ∈ ℤ_{≥0} the quasi-degree of v(x). It is easily checked that (3) is equivalent to

(ϑ − λ)^{k+1}v(x) = 0  (4)

with ϑ := ∑_{i=1}^{m} x_{i}∂_{i}. In particular, when a quasi-homogeneous function v(x) is of quasi-degree k and not k − 1, we say that v(x) is quasi-homogeneous of proper quasi-degree k.

We use the following notations in this paper.

1. QH(λ) := {u(x) ∈ ℬ(V ) | u(x) is quasi-homogeneous of degree λ ∈ ℂ}.

2. QH(λ)^{G} := QH(λ) ∩ ℬ(V )^{G}.

3. QH := ⨁_{λ∈ℂ} QH(λ).

4. QH^{G} := ⨁_{λ∈ℂ} QH(λ)^{G}.

Proposition 1.1. Let P (x, ∂) ∈ D(V ) (resp. ∈ D(V )^{G}) be a non-zero homogeneous differential operator of homogeneous degree μ. If f(x) ∈ ℬ(V ) (resp. ∈ ℬ(V )^{G}) is quasi-homogeneous of degree λ ∈ ℂ, then P (x, ∂)f(x) ∈ ℬ(V ) (resp. ∈ ℬ(V )^{G}) is quasi-homogeneous of degree λ + μ ∈ ℂ.

The proof of Proposition 1.1 is found in [22, Proposition1.1].

¿From now on, let V := Sym_{n}(ℝ) be the space of n×n symmetric matrices over the real field ℝ and let G := SL_{n}(ℝ) be the special linear group over ℝ of degree n. Then the group G acts on the vector space V by the representation

ρ(g) : x ⟼ g · x := gx^{t}g,

with x ∈ V and g ∈ G. The vector space V decomposes into a finite number of GL_{n}(ℝ)-orbits:

V := ⨆_{\begin{matrix}0≤i≤n \\ 0≤j≤n−i \end{matrix}} S_{i}^{j}  (5)

where

S_{i}^{j} := {x ∈ Sym_{n}(ℝ) | sgn(x) = (j, n − i − j)}  (6)

with integers 0 ≤ i ≤ n and 0 ≤ j ≤ n − i. Here, sgn(x) for x ∈ Sym_{n}(ℝ) is the signature of the quadratic form q_{x}(v⃗) := ^{t}v⃗ · x · v⃗ on v⃗ ∈ ℝ^{n}. The subset S_{i} := {x ∈ V | rank(x) = n − i} is the set of elements of rank n − i.

We set P (x) := det(x) and S := {x ∈ V | det(x) = 0}. We call the set S the singular set and call the G-orbits in S the singular orbits. The subset V − S decomposes into (n + 1) connected components,

V_{i} := {x ∈ Sym_{n}(ℝ) | sgn(x) = (i, n − i)}  (7)

with i = 0, 1, …, n.

We define the complex power function of P (x) by

|P (x)|_{i}^{s} := \{ \begin{matrix}|P (x)|^{s} & if x ∈ V_{i}, \\ 0 & if x ∉ V_{i}, \end{matrix}  (8)

for a complex number s ∈ ℂ. We can regard |P (x)|_{i}^{s} as a tempered distribution — and hence a hyperfunction — with a meromorphic parameter s ∈ ℂ. We consider a linear combination of the hyperfunctions |P (x)|_{i}^{s}:

P^{[a⃗,s]}(x) := ∑_{i=0}^{n} a_{i} · |P (x)|_{i}^{s}  (9)

with s ∈ ℂ and a⃗ := (a_{0}, a_{1}, …, a_{n}) ∈ ℂ^{n+1}. Then P^{[a⃗,s]}(x) is a hyperfunction with a meromorphic parameter s ∈ ℂ and it depends on a⃗ ∈ ℂ^{n+1} linearly.

A homogeneous differential operator of degree k ∈ ℤ is given by

P (x, ∂) = ∑_{\begin{matrix}α,β∈ℤ_{≥0}^{m} \\ |α|−|β|=k \end{matrix}} a_{αβ}x^{α}∂^{β}

with m := n(n + 1)/2. The notations here are written as

x = (x_{ij})_{n≥j≥i≥1}, ∂ = (∂_{ij}) = (\frac{∂}{∂x_{ij}} )_{n≥j≥i≥1} ,

x^{α} = ∏_{n≥j≥i≥1} x_{ij}^{α_{ij}}, ∂^{β} = ∏_{n≥j≥i≥1} ∂_{ij}^{β_{ij}}

with

α = (α_{ij}) ∈ ℤ_{≥0}^{m}, |α| = ∑_{n≥j≥i≥1} α_{ij},

β = (β_{ij}) ∈ ℤ_{≥0}^{m}, |β| = ∑_{n≥j≥i≥1} β_{ij}.

We define ∂^{∗} by

∂^{∗} = (∂_{ij}^{∗}) = (ϵ_{ij} \frac{∂}{∂x_{ij}} ) , with ϵ_{ij} := \begin{cases}1 & if i = j, \\ 1/2 & if i ≠ j. \end{cases}  (10)

For an invariant differential operator P (x, ∂) ∈ D(V )^{G}, we have the following proposition.

Proposition 1.2. Let P (x, ∂) ∈ D(V )^{G} be a homogeneous differential operator.

1. The homogeneous degree of P (x, ∂) is in (n · ℤ). If the homogeneous degree of P (x, ∂) is nk, then P (g · x,^{t}g^{−1} · ∂) = det(g)^{2k}P (x, ∂).

2. If the homogeneous degree of P (x, ∂) is nk with k ∈ ℤ and x ∈ Sym_{n}(ℝ) is positive definite, then we have

P (x, ∂)(det x)^{s} = b_{P} (s)(det x)^{s+k}  (11)

where b_{P} (s) is a polynomial in s ∈ ℂ. We have also

P (x, ∂)P^{[a⃗,s]}(x) = b_{P} (s) det(x)^{k}P^{[a⃗,s]}(x)
= b_{P} (s)sgn(det(x))^{k}P^{[a⃗,s+k]}(x)
= b_{P} (s)P^{[a^{#k},s+k]}(x)

(12)

for all x ∈ V − S. Here we put

a^{#k} := ((−1)^{nk}a_{0}, (−1)^{(n−1)k}a_{1}, …, a_{n}) ∈ ℂ^{n+1}.  (13)

3. If the homogeneous degree of P (x, ∂) is nk with k < 0, then b_{P} (s) is divisible by b\underline{−k}(s − 1) where b\underline{−k}(s − 1) := b(s − 1)b(s − 2) ⋯ b(s − (−k)) with b(s) := ∏_{i=1}^{n}(s + \frac{i+1}{2}).

The proof of Proposition 1.2 is given by [22, Proposition 3.1].

Now we give the definition of b_{P} -function for a given G-invariant differential operator P (x, ∂).

Definition 1.1. (b_{P} -function) Let P (x, ∂) ∈ D(V )^{G} be a homogeneous differential operator of homogeneous degree k. We call b_{P} (s) in (11) the b_{P} -function of P (x, ∂).

Then we have the following theorem, whose proof is given in [22, Theorem 4.1].

Theorem 1.3. Let P (x, ∂) ∈ D(V )^{G} be a non-zero homogeneous differential operator with homogeneous degree kn. We suppose that

degree of b_{P} (s) = order of P (x, ∂).  (14)

The space of G-invariant hyperfunction solutions to P (x, ∂)u(x) = 0 is finite dimensional. The solutions u(x) are given as finite linear combinations of quasihomogeneous G-invariant hyperfunctions.

The following proposition is [22, Proposition 5.1]. Its proof is easy.

Proposition 1.4. Let ∂^{∗} be the symmetric matrix of differential operators defined by (10).

1. We have

\begin{aligned}(det(∂^{∗}))P^{[a⃗,s+1]}(x) & = b(s) · P^{[a^{#},s]}(x) \\ & = b(s) · (det(x)) · P^{[a⃗,s−1]}(x) \end{aligned}  (15)

with a^{#} = a^{#1} := ((−1)^{n}a_{0}, …, −a_{n−1}, a_{n}) and

b(s) = c · (s + 1)(s + \frac{3}{2}) ⋯ (s + \frac{n + 1}{2}),  (16)

where c is a constant.

2. P^{[a⃗,s]}(x) is holomorphic with respect to s ∈ ℂ except for poles at s = −(k + 1)/2 with k ∈ ℤ_{≥0}. The possible highest order of the pole of P^{[a⃗,s]}(x) at s = −(k + 1)/2 is

\begin{cases}⌊\frac{k+1}{2}⌋ & if k = 1, 2…. , n − 1, \\ ⌊\frac{n}{2}⌋ & if k = n, n + 1…. , and k + n is odd, \\ ⌊\frac{n+1}{2}⌋ & if k = n, n + 1…. , and k + n is even. \end{cases}  (17)

We give here two definitions, [22, Definition 5.1] and [22, Definition 5.2], which concern the order of poles and the Laurent expansion coefficients of P^{[a⃗,s]}(x).

Definition 1.2. (Possible highest order) Let λ ∈ ℂ be a fixed complex number.

1. We denote by P HO(λ) the possible highest order of the pole of P^{[a⃗,s]}(x) at s = λ. Namely we define

P HO(λ) := \begin{cases}⌊\frac{k+1}{2}⌋ & if λ = −\frac{k+1}{2} (k = 1, 2…. , n − 1), \\ ⌊\frac{n}{2}⌋ & if λ = −\frac{k+1}{2} (k = n, n + 1…. , and k + n is odd), \\ ⌊\frac{n+1}{2}⌋ & if λ = −\frac{k+1}{2} (k = n, n + 1…. , and k + n is even), \\ 0 & otherwise. \end{cases}

(18)

2. Let q ∈ ℤ. We define a vector subspace A(λ, q) of ℂ^{n+1} by

A(λ, q) := {a⃗ ∈ ℂ^{n+1} | P^{[a⃗,s]}(x) has a pole of order ≤ q at s = λ}. (19)

Then we have A(λ, q − 1) ⊂ A(λ, q) by definition. We define \overline{A(λ, q)} by

\overline{A(λ, q)} := A(λ, q)/A(λ, q − 1).  (20)

It is easily verified that \overline{A(λ, q)} = \{0\} if q > P HO(λ) or q < 0. We have

⨁_{q∈ℤ} \overline{A(λ, q)} = ⨁_{0≤q≤P HO(λ)} \overline{A(λ, q)} ≃ ℂ^{n+1}.  (21)

In particular, a⃗ = 0 if a⃗ ∈ A(λ, q) for some q < 0 since A(λ, q) = {0} for q < 0. However, when q < 0, a pole of order q means a zero of order −q.

Definition 1.3. (Laurent expansion coefficients) Let λ ∈ ℂ be a fixed complex number.

1. We define o(a⃗, λ) ∈ ℤ by

o(a⃗, λ) := order of pole of P^{[a⃗,s]}(x) at s = λ.  (22)

Then o(a⃗, λ) ∈ ℤ_{≥0}. We have p = o(a⃗, λ) if and only if a⃗ ∈ A(λ, p) and [a⃗] ∈ \overline{A(λ, p)} is not zero.

2. Let a⃗ ∈ ℂ^{n+1} and let r = o(a⃗, λ) ∈ ℤ_{≥0}. Then we have the Laurent expansion of P^{[a⃗,s]}(x) at s = λ,

P^{[a⃗,s]}(x) = ∑_{w=−r}^{∞} P_{w}^{[a⃗,λ]}(x) (s − λ)^{w} .  (23)

It is easily checked that the Laurent expansion coefficient P_{w}^{[a⃗,λ]}(x) is linear with respect to a⃗ ∈ ℂ^{n+1}.

Next, we give a definition of a standard basis of ℂ^{n+1}, following [22, Definition 5.3].

Definition 1.4. (Standard basis) Let

SB := {a⃗_{0}, a⃗_{1}, …, a⃗_{n}}  (24)

be a basis of ℂ^{n+1}. We say that SB is a standard basis of ℂ^{n+1} at s = λ if the following property holds: there exists an increasing integer sequence

0 < k(0) < k(1) < ⋯ < k(P HO(λ)) = n  (25)

such that SB_{q} := {a⃗_{0}, a⃗_{1}, …, a⃗_{k(q)}} is a basis of A(λ, q) for each q in 0 ≤ q ≤ P HO(λ). It is easily seen that the representatives of SB_{q}−SB_{q−1} form a basis of the quotient vector space \overline{A(λ, q)} := A(λ, q)/A(λ, q − 1).

Proposition 1.5. Let SB := {a⃗_{0}, a⃗_{1}, …, a⃗_{n}} be a standard basis of ℂ^{n+1} at s = λ and let r_{j} := o(a⃗_{j}, λ) ∈ ℤ_{≥0}. Then the Laurent expansion coefficients at s = λ

{P_{−r_{j}+i}^{[a⃗_{j},λ]}(x) | i ∈ ℤ_{≥0} and j = 0, 1, 2, …, n}  (26)

are linearly independent.

The proof of Proposition 1.5 is given in [22, Proposition 5.3].

Theorem 1.6. Let r := o(a⃗, λ) ∈ ℤ_{≥0} be the order of the pole of P^{[a⃗,s]}(x) at s = λ defined by (22).

1. Then the Laurent expansion coefficient of P^{[a⃗,s]}(x) at s = λ defined by (23) P_{w}^{[a⃗,λ]}(x) is a quasi-homogeneous hyperfunction of degree nλ of quasi-degree r + w. Conversely, let v(x) ∈ QH(nλ)^{G}, the space of G-invariant quasihomogeneous hyperfunctions. Then v(x) is written as a linear combination of Laurent expansion coefficients of |P (x)|_{i}^{s} at s = λ.

2. Let

LC(λ, w) := vector space generated by {P_{w}^{[a⃗,λ]}(x) | a⃗ ∈ ℂ^{n+1}}.  (27)

Then we have the direct sum decomposition:

QH(nλ)^{G} = ⨁_{\begin{matrix}w∈ℤ, \\ w≥−P HO(λ) \end{matrix}} LC(λ, w).  (28)

The proof of Theorem 1.6 is given in [22, Theorem 5.3]. By combining Theorem 1.3 and Theorem 1.6, we have the following corollary.

Corollary 1.7. Let P (x, ∂) ∈ D(V )^{G} be a non-zero homogeneous differential operator with homogeneous degree kn satisfying the condition (14). Then the Ginvariant hyperfunction solutions u(x) to the differential equation P (x, ∂)u(x) = 0 are written as finite linear combinations of Laurent expansion coefficients of |P (x)|_{i}^{s} at a finite number of points.

Then we have the following two theorems, which are the main results of [22].

Theorem 1.8. Let P (x, ∂) ∈ D(V )^{G} be a non-zero homogeneous differential operator with homogeneous degree kn and let v(x) be a quasi-homogeneous Ginvariant hyperfunction of homogeneous degree nλ. We suppose that

b_{P} (s) ≢ 0.  (29)

Then

1. We can construct a G-invariant hyperfunction solution u(x) ∈ ℬ(V )^{G} to the differential equation P (x, ∂)u(x) = v(x) which is given as a sum of Laurent expansion coefficients of |P (x)|_{i}^{s} at s = λ − k and hence is quasihomogeneous of degree n(λ − k).

2. Any G-invariant hyperfunction solution u(x) is given as finite linear combinations of quasi-homogeneous G-invariant hyperfunctions, and hence it is written as a finite linear combinations of Laurent expansion coefficients of |P (x)|_{i}^{s} at a finite number of points in ℂ.

The proof of Theorem 1.8 is given in [22, Theorem 6.1].

Theorem 1.9. Let P (x, ∂) ∈ D(V )^{G} be a non-zero homogeneous differential operator of homogeneous degree kn satisfying the condition (14). Then we can construct the G-invariant quasi-homogeneous hyperfunction solution of homogeneous degree nλ to the differential equation P (x, ∂)u(x) = 0 as a finite linear combination of Laurent expansion coefficients of |P (x)|_{i}^{s} (i = 0, …, n) at s = λ. It is determined by the homogeneous degree kn and b_{P} (s) and does not depend on P (x, ∂) itself.

The proof of Theorem 1.9 is given in [22, Theorem 6.2].

2. Orders of poles of complex powers of determinant functions.

In the preceding section, we have reviewed that the solutions of (2) can be constructed in terms of the Laurent expansion coefficients of the complex powers of the determinant functions P^{[a⃗,s]}(x) (Theorem 1.8 and Theorem 1.9). These are main results of the paper [22]. In order to apply these constructions of solutions in concrete examples, we have to see the exact pole order of P^{[a⃗,s]}(x) especially at s = λ ∈ \frac{1}{2}ℤ.

In this section, we shall give a condition to determine the exact order of pole of P^{[a⃗,s]}(x) at s = λ ∈ \frac{1}{2}ℤ for a given vector a⃗ ∈ ℂ^{n+1}. This is a direct application of the author’s result in [19].

In order to determine the exact pole of P^{[a⃗,s]}(x) at s = s_{0}, the author introduced in [19] the coefficient vectors

d^{(k)}[s_{0}] := (d_{0}^{(k)}[s_{0}], d_{1}^{(k)}[s_{0}], …, d_{n−k}^{(k)}[s_{0}]) ∈ ((ℂ^{n+1})^{∗})^{n−k+1}  (30)

with k = 0, 1, …, n. Here, (ℂ^{n+1})^{∗} denotes the dual vector space of ℂ^{n+1}. Each element of d^{(k)}[s_{0}] is a linear form on a⃗ ∈ ℂ^{n+1} depending on s_{0} ∈ ℂ, i.e., a linear map from ℂ^{n+1} to ℂ

d_{i}^{(k)}[s_{0}] : ℂ^{n+1} ∋ a⃗ ⟼ ⟨d_{i}^{(k)}[s_{0}], a⃗⟩ ∈ ℂ.  (31)

We set

⟨d^{(k)}[s_{0}], a⃗⟩ := (⟨d_{0}^{(k)}[s_{0}], a⃗⟩, ⟨d_{1}^{(k)}[s_{0}], a⃗⟩, …, ⟨d_{n−k}^{(k)}[s_{0}], a⃗⟩) ∈ ℂ^{n−k+1}.  (32)

The precise definition of d^{(k)}[s_{0}] is the following.

Definition 2.1. (Coefficient vectors d^{(k)}[s_{0}]) Let s_{0} be a half-integer, i.e., s_{0} = q/2 with q ∈ ℤ. We define the coefficient vectors d^{(k)}[s_{0}] (k = 0, 1, …, n) by induction in the following way.

1. First, we set

d^{(0)}[s_{0}] := (d_{0}^{(0)}[s_{0}], d_{1}^{(0)}[s_{0}], …, d_{n}^{(0)}[s_{0}]) ∈ ((ℂ^{n+1})^{∗})^{n+1}  (33)

where each d_{i}^{(0)}[s_{0}] satisfies ⟨d_{i}^{(0)}[s_{0}], a⃗⟩ = a_{i} for all a⃗ ∈ ℂ^{n+1}.

2. Next, we define d^{(1)}[s_{0}] and d^{(2)}[s_{0}] by

d^{(1)}[s_{0}] := (d_{0}^{(1)}[s_{0}], d_{1}^{(1)}[s_{0}], …, d_{n−1}^{(1)}[s_{0}]) ∈ ((ℂ^{n+1})^{∗})^{n},  (34)

with d_{j}^{(1)}[s_{0}] := d_{j}^{(0)}[s_{0}] + ϵ[s_{0}]d_{j+1}^{(0)}[s_{0}], and

d^{(2)}[s_{0}] := (d_{0}^{(2)}[s_{0}], d_{1}^{(2)}[s_{0}], …, d_{n−2}^{(2)}[s_{0}]) ∈ ((ℂ^{n+1})^{∗})^{n−1},  (35)

with d_{j}^{(2)}[s_{0}] := d_{j}^{(0)}[s_{0}] + d_{j+2}^{(0)}[s_{0}]. Here,

ϵ[s_{0}] := \begin{cases}1 & if s_{0} is a strict half-integer, \\ (−1)^{s_{0}+1} & if s_{0} is an integer. \end{cases}  (36)

A strict half-integer means a rational number given by q/2 with an odd integer q.

3. Lastly, by induction on k, we define the coefficient vectors d^{(k)}[s_{0}] for k = 3, 4, …, n by

d^{(2l+1)}[s_{0}] := (d_{0}^{(2l+1)}[s_{0}], d_{1}^{(2l+1)}[s_{0}], …, d_{n−2l−1}^{(2l+1)}[s_{0}]) ∈ ((ℂ^{n+1})^{∗})^{n−2l}, (37)

with d_{j}^{(2l+1)}[s_{0}] := d_{j}^{(2l−1)}[s_{0}] − d_{j+2}^{(2l−1)}[s_{0}], and

d^{(2l)}[s_{0}] := (d_{0}^{(2l)}[s_{0}], d_{1}^{(2l)}[s_{0}], …, d_{n−2l}^{(2l)}[s_{0}]) ∈ ((ℂ^{n+1})^{∗})^{n−2l+1},  (38)

with d_{j}^{(2l)}[s_{0}] := d_{j}^{(2l−2)}[s_{0}] + d_{j+2}^{(2l−2)}[s_{0}].

By using d^{(k)}[s_{0}] in Definition 2.1, the author proved in [19] an algorithm to compute the exact order of poles of P^{[a⃗,s]}(x). Refer to Theorem A1 in Appendix. In this section, we shall characterize the vector space

A(λ, q) := {a⃗ ∈ ℂ^{n+1} | P^{[a⃗,s]}(x) has a pole of order ≤ q at s = λ}

in terms of the coefficient vectors d^{(k)}[λ].

Definition 2.2. We define the vector subspaces D_{half}^{(l)} , D_{even}^{(l)} and D_{odd}^{(l)} in ℂ^{n+1}.

1. Note that d^{(2l+2)}[λ] does not depend on the choice of λ if it is a half-integer. We define

D_{half}^{(l)} := {a⃗ ∈ ℂ^{n+1} | ⟨d^{(2l+2)}[λ], a⃗⟩ = 0 for any strict half-integer λ}.

2. Note that d^{(2l+1)}[λ] does not depend on the choice of λ if λ is an odd (resp. even) integer. We define

D_{odd}^{(l)} := {a⃗ ∈ ℂ^{n+1} | ⟨d^{(2l+1)}[λ], a⃗⟩ = 0 for any odd integer λ}.

(resp. D_{even}^{(l)} := {a⃗ ∈ ℂ^{n+1} | ⟨d^{(2l+1)}[λ], a⃗⟩ = 0 for any even integer λ}.)

Theorem 2.1. The subspaces D_{half}^{(l)} , D_{even}^{(l)} and D_{odd}^{(l)} in ℂ^{n+1} have the following properties.

1. We define

a^{#} = a^{#1} := ((−1)^{n}a_{0}, (−1)^{n−1}a_{1}, …, a_{n}) ∈ ℂ^{n+1}

for a⃗ = (a_{0}, a_{1}, …, a_{n}) ∈ ℂ^{n+1}. Then we have

a⃗ ∈ D_{odd}^{(l)} ⟺ a^{#} ∈ D_{even}^{(l)}

and

a⃗ ∈ D_{half}^{(l)} ⟺ a^{#} ∈ D_{half}^{(l)}.

2. Let l be an integer 0 ≤ l < P HO(λ). Then we have

a⃗ ∈ A(λ, l) ⟺ \begin{cases}a⃗ ∈ D_{half}^{(l)} & if λ is a strict half-integer, \\ a⃗ ∈ D_{odd}^{(l)} & if λ is an odd integer, \\ a⃗ ∈ D_{even}^{(l)} & if λ is an even integer. \end{cases} (39)

In addition, we have A(λ, P HO(λ)) = ℂ^{n+1}.

Proof. The second statement is nothing but the definition of D_{half}^{(l)} , D_{even}^{(l)} and D_{odd}^{(l)} by Theorem A1 in Appendix.

We shall prove the first statement. Let odd be an odd integer and let even be an even integer. We have only to prove that

⟨d^{(2l+1)}[odd], a⃗⟩ = (−1)^{n}⟨d^{(2l+1)}[even], a^{#}⟩.  (40)

for each l ∈ ℤ_{≥0}. We prove it by induction on l. When l = 0, we have

⟨d^{(1)}[odd], a⃗⟩ = (a_{0} + a_{1}, a_{1} + a_{2}, …, a_{n−1} + a_{n})
= (−1)^{n}(a_{0}^{#} − a_{1}^{#}, a_{1}^{#} − a_{2}^{#}, …, a_{n−1}^{#} − a_{n}^{#})
= (−1)^{n}⟨d^{(1)}[even], a^{#}⟩

since a^{#} = (a_{0}^{#}, a_{1}^{#}, …, a_{n}^{#}) = ((−1)^{n}a_{0}, (−1)^{n−1}a_{1}, …, a_{n}). We see that

⟨d^{(2l+1)}[odd], a⃗⟩ = (−1)^{n}⟨d^{(2l+1)}[even], a^{#}⟩

if

⟨d^{(2l−1)}[odd], a⃗⟩ = (−1)^{n}⟨d^{(2l−1)}[even], a^{#}⟩

by (37). Thus (40) is valid for all l ∈ ℤ_{≥0}. By (40), we have

\begin{aligned}a⃗ ∈ D_{odd}^{(l)} & ⟺ ⟨d^{(2l+1)}[odd], a⃗⟩ = 0 \\ & ⟺ ⟨d^{(2l+1)}[even], a^{#}⟩ = 0 ⟺ a^{#} ∈ D_{even}^{(l)}. \end{aligned}

Next let half be a strict half-integer. We have only to prove that

⟨d^{(2l+2)}[half], a⃗⟩ = ⟨d^{(2l+2)}[half], a^{#}⟩^{#}  (41)

for each l ∈ ℤ_{≥0}. We prove it by induction on l. When l = 0, we have

⟨d^{(2)}[half], a⃗⟩ = (a_{0} + a_{2}, a_{1} + a_{3}, …, a_{n−2} + a_{n})
= ((−1)^{n−2}(a_{0}^{#} + a_{2}^{#}), (−1)^{n−3}(a_{1}^{#} + a_{3}^{#}), …, (a_{n−2}^{#} + a_{n}^{#}))
= ⟨d^{(1)}[half], a^{#}⟩^{#}

since a^{#} = (a_{0}^{#}, a_{1}^{#}, …, a_{n}^{#}) = ((−1)^{n}a_{0}, (−1)^{n−1}a_{1}, …, a_{n}). We see that

⟨d^{(2l+2)}[half], a⃗⟩ = ⟨d^{(2l+2)}[half], a^{#}⟩^{#}

if

⟨d^{(2l)}[half], a⃗⟩ = ⟨d^{(2l)}[half], a^{#}⟩^{#}

by (38). Thus (41) is valid for all l ∈ ℤ_{≥0}. By (41), we have

a⃗ ∈ D_{half}^{(l)} ⟺ ⟨d^{(2l+2)}[half], a⃗⟩ = 0
⟺ ⟨d^{(2l+2)}[half], a^{#}⟩^{#} = 0
⟺ ⟨d^{(2l+2)}[half], a^{#}⟩ = 0 ⟺ a^{#} ∈ D_{half}^{(l)}.

When λ ∉ \frac{1}{2}ℤ, any basis is a standard basis since all P^{[a⃗,s]}(x) is holomorphic at s = λ. When λ ∈ \frac{1}{2}ℤ, we can easily choose one standard basis for a given λ by utilizing Theorem 2.1. However, it is sufficient only to consider the following three kinds of standard basis, SB^{half}, SB^{even} and SB^{odd}.

Definition 2.3. For λ ∈ \frac{1}{2}ℤ, we define the bases SB^{half}, SB^{even} and SB^{odd} of ℂ^{n+1} by

SB^{half} := {a_{0}^{half}, a_{1}^{half}, …, a_{n}^{half}} if λ is a strict half-integer,
SB^{even} := {a_{0}^{even}, a_{1}^{even}, …, a_{n}^{even}} if λ is an even integer,
SB^{odd} := {a_{0}^{odd}, a_{1}^{odd}, …, a_{n}^{odd}} if λ is an odd integer,

(42)

satisfying that there exists an increasing integer sequence

0 < l(0) < l(1) < ⋯ < l(p) = n  (43)

with

p := \begin{cases}⌊\frac{n}{2}⌋ & if i + n is odd, \\ ⌊\frac{n+1}{2}⌋ & if i + n is even, \end{cases}

such that

SB_{q}^{half} := {a_{0}^{half}, a_{1}^{half}, …, a_{l(q)}^{half}} is a basis of D_{half}^{(q)},
SB_{q}^{even} := {a_{0}^{even}, a_{1}^{even}, …, a_{l(q)}^{even}} is a basis of D_{even}^{(q)},
SB_{q}^{odd} := {a_{0}^{odd}, a_{1}^{odd}, …, a_{l(q)}^{odd}} is a basis of D_{odd}^{(q)},

for q = 0, 1, …, p, respectively. In particular, we take SB^{even} and SB^{odd} such that

a_{j}^{odd} = a_{j}^{even#} (j = 0, 1, …, n)  (44)

where a^{#} := ((−1)^{n}a_{0}, (−1)^{n−1}a_{1}, …, a_{n}) for a⃗ := (a_{0}, a_{1}, …, a_{n}) ∈ ℂ^{n+1}. This is possible by Theorem 2.1.

Proposition 2.2. The bases (42) are standard bases for λ ∈ \frac{1}{2}ℤ in the sense of Definition 1.4. When λ ∉ \frac{1}{2}ℤ, every basis is a standard basis since every P^{[a⃗,s]}(x) does not have a pole.

Proof. This is just the definition of the standard basis.

3. Algorithms for constructing solutions — homogeneous equations.

In this section we give algorithms to compute all the hyperfunction solutions to P (x, ∂)u(x) = 0 for a homogeneous G-invariant differential operator P (x, ∂).

Algorithm 3.1. (The case of homogeneous degree zero) For a given non-zero G-invariant differential operator P (x, ∂) ∈ D(V )_{0}^{G} of homogeneous degree 0 satisfying the condition (14) in Theorem 1.3, an algorithm to compute a basis of the G-invariant differential equation P (x, ∂)u(x) = 0 is given in the following.

Input A non-zero G-invariant differential operator P (x, ∂) ∈ D(V )_{0}^{G} satisfying the condition (14).

Output A basis of the G-invariant hyperfunction solutions to the differential equation P (x, ∂)u(x) = 0.

Procedure

1. Compute the b_{P} -function for P (x, ∂). It is denoted by b_{P} (s) = (s − λ_{1})^{p_{1}} ⋯ (s − λ_{l})^{p_{l}}.

2. For each λ_{i} (i = 1, …, l), take one standard basis at s = λ_{i} SB^{λ_{i}} = {a⃗_{0}(λ_{i}), ⋯, a⃗_{n}(λ_{i})}, which is defined in Definition 1.4.

3. Compute the Laurent expansion coefficients P_{k}^{[a⃗_{j}(λ_{i}),λ_{i}]}(x) for each a⃗_{j}(λ_{i}) (i = 1, …, l; j = 0, …, n) and k ∈ ℤ in −o_{ij} ≤ k ≤ −o_{ij} + p_{i} − 1 with o_{ij} := o(a⃗_{j}(λ_{i}), λ_{i}). Then we have the generators of the vector space.

L_{ij} := the vector space generated by

{P_{k}^{[a⃗_{j}(λ_{i}),λ_{i}]}(x) | k = −o_{ij}, …, −o_{ij} + p_{i} − 1}. (45)

4. Then

⨁_{\begin{matrix}i=1,...,l \\ j=0,...,n \end{matrix}} L_{ij}  (46)

forms a basis of the G-invariant hyperfunction solution space to the equation P (x, ∂)u(x) = 0.

Proof. Note that, by Theorem 1.3 and Corollary 1.7, every G-invariant hyperfunction solution to P (x, ∂)u(x) = 0 is written as a finite combination of Laurent expansion coefficients of |P (x)|_{i}^{s} (i = 0, …, n). Suppose that u(x) is written as u(x) = u_{1}(x) + ⋯ + u_{l}(x) where each u_{i}(x) is quasi-homogeneous of degree s_{i} and s_{1}, …, s_{l} are pairwise different. If P (x, ∂)u(x) = 0, then P (x, ∂)u_{i}(x) = 0 for all i = 1, …, l since the homogeneous degrees of P (x, ∂)u_{i}(x) (i = 1, …, l) are pairwise different and hence linearly independent. Then, for each complex number λ ∈ ℂ, we have only to see which u(x) is annihilated by P (x, ∂) when u(x) is a finite combination of Laurent expansion coefficients of |P (x)|_{i}^{s} (i = 0, …, n) at s = λ.

Let SB := {a⃗_{0}, a⃗_{1}, …, a⃗_{n}} be a standard basis of ℂ^{n+1} at s = λ with an increasing sequence

0 < k(0) < k(1) < ⋯ < k(P HO(λ)) = n  (47)

such that SB_{q} := {a⃗_{0}, a⃗_{1}, …, a⃗_{k(q)}} is a basis of A(λ, q) for each q in 0 ≤ q ≤ P HO(λ). Then every u(x) given as a finite combination of Laurent expansion coefficients of |P (x)|_{i}^{s} (i = 0, …, n) at s = λ is expressed as

u(x) = ∑_{\begin{matrix}f,g∈ℤ \\ 0≤g≤n \end{matrix}} c_{f,g}P_{f}^{[a⃗_{g},λ]}(x)  (48)

with c_{f,g} ∈ ℂ. Let b_{P} (s) = ∑_{i=0}^{q} b_{i}(s−λ)^{p+i} be an expansion of b_{P} (s) with respect to (s − λ). The number p is the multiplicity of b_{P} (s) at s = λ. Then what we have to prove is that

c_{f,g} = 0 except for − o(a⃗_{g}, λ) ≤ f ≤ −o(a⃗_{g}, λ) + p − 1
if and only if P (x, ∂)u(x) = 0

(49)

since P_{f}^{[a⃗_{g},λ]}(x) = 0 if f < −o(a⃗_{g}, λ) by the definition. Indeed, the basis of L_{ij} in (45) is the basis of the remainder terms in the expression (48) with the condition (49) when λ = λ_{i} and p = k_{i}. In particular, if λ is not a root of b_{P} (s) = 0, i.e., p = 0, then there is no G-invariant solution to P (x, ∂)u(x) = 0.

The rest of the proof is devoted to showing (49). Let the Laurent expansion of P^{[a⃗,s]}(x) at s = λ be denoted by P^{[a⃗,s]}(x) = ∑_{w∈ℤ} P_{w}^{[a⃗,λ]}(x)(s − λ)^{w}. Then we have

P (x, ∂)P^{[a⃗,s]}(x) = ∑_{w∈ℤ}P (x, ∂)P_{w}^{[a⃗,λ]}(x)(s − λ)^{w} = b_{P} (s)P^{[a⃗,s]}(x)

= (∑_{i=0}^{q} b_{i}(s − λ)^{p+i})(∑_{j∈ℤ}P_{j}^{[a⃗,λ]}(x)(s − λ)^{j})

= ∑_{w∈ℤ} ∑_{i+j+p=w} b_{i}P_{j}^{[a⃗,λ]}(x)(s − λ)^{w}

and hence

P (x, ∂)P_{w}^{[a⃗,λ]}(x) = ∑_{i+j+p=w} b_{i}P_{j}^{[a⃗,λ]}(x).  (50)

Here b_{i} = 0 except for i in 0 ≤ i ≤ q and P_{j}^{[a⃗,λ]}(x) = 0 for sufficiently small j.

Then since u(x) = ∑\begin{matrix}f,g∈ℤ \\ 0≤g≤n \end{matrix} c_{f,g}P_{f}^{[a⃗_{g},λ]}(x), we have

P (x, ∂)u(x) = ∑_{f,g∈ℤ}c_{f,g}P (x, ∂)P_{f}^{[a⃗_{g},λ]}(x) = ∑_{f,g∈ℤ}c_{f,g} ∑_{i=0}^{q} b_{i}P_{f−p−i}^{[a⃗_{g},λ]}(x)

= ∑_{f,g∈ℤ}c_{f,g} ∑_{j∈ℤ}b_{f−p−j}P_{j}^{[a⃗_{g},λ]}(x) = ∑_{j∈ℤ} ∑_{f,g∈ℤ}c_{f,g}b_{f−p−j}P_{j}^{[a⃗_{g},λ]}(x)

= ∑_{j∈ℤ}P_{j}^{[∑_{f,g∈ℤ} c_{f,g}b_{f−p−j}a⃗_{g},λ]}(x) = 0

where g runs in 0 ≤ g ≤ n. Therefore

∑_{f,g∈ℤ}c_{f,g}b_{f−p−j}a⃗_{g} = ∑_{g=0}^{n} ∑_{f∈ℤ}c_{f,g}b_{f−p−j}a⃗_{g} ∈ A(λ, −j − 1)

for all j ∈ ℤ by Theorem 1.6. This means that,

for each g = 0, 1, …, n,
∑_{f∈ℤ}c_{f,g}b_{f−p−j} = 0 for all j ∈ ℤ satisfying g ≥ k(−j)

(51)

since a⃗_{g} ∉ A(λ, −j − 1) if g ≥ k(−j) by definition. Here k(−j) is the number defined by (47) if 0 ≤ −j ≤ P HO(λ) and k(−j) = 0 (resp. k(−j) = n + 1) if −j < 0 (resp. −j > P HO(λ)). Since g ≥ k(−j) is equivalent to o(a⃗_{g}, λ) ≥ −j by definition and ∑_{f∈ℤ} c_{f,g}b_{f−p−j} = ∑_{f=p+j}^{p+j+q} c_{f,g}b_{f−p−j} = ∑_{s=0}^{q} c_{p+j+s,g}b_{s} = 0, the condition (51) is rewritten as the condition:

for each g = 0, 1, …, n,

∑_{s=0}^{q} c_{p+j+s,g}b_{s} = 0 for all j ∈ ℤ satisfying j ≥ −o(a⃗_{g}, λ). (52)

Note that coefficients b_{0} and b_{q} are not zero. Then the condition (52) is equivalent to the condition:

for each g = 0, 1, …, n,
c_{p+j,g} = 0 for all j ∈ ℤ satisfying j ≥ −o(a⃗_{g}, λ).

(53)

This is just equivalent to the condition (49), which we have to prove.

Next we consider P (x, ∂) of non-zero homogeneous degree.

Algorithm 3.2. (The case of non-zero homogeneous degree) For a given nonzero G-invariant differential operator P (x, ∂) ∈ D(V )^{G} of homogeneous degree q_{1}n ≠ 0 satisfying the condition (14) in Theorem 1.3, an algorithm to compute a basis of the G-invariant differential equation P (x, ∂)u(x) = 0 is given in the following.

1. The case of the homogeneous degree q_{1} < 0 Input A non-zero G-invariant differential operator P (x, ∂) ∈ D(V )^{G} of ho-

mogeneous degree q_{1}n < 0 satisfying the condition (14). Output A basis of the G-invariant hyperfunction solutions to the differential

equation P (x, ∂)u(x) = 0. Procedure

(a) Compute the b_{P} -function for P (x, ∂). It is denoted by b_{P} (s) = (s − λ_{1})^{p_{1}} ⋯ (s − λ_{l})^{p_{l}}.

(b) For each λ_{i} (i = 1, …, l), take one standard basis

SB^{λ_{i}} = {a⃗_{0}(λ_{i}), ⋯, a⃗_{n}(λ_{i})}

at s = λ_{i}, which is the standard basis defined by (42) when λ_{i} ∈ \frac{1}{2}ℤ and the one defined in Definition 1.4 otherwise.

(c) Compute the Laurent expansion coefficients P_{k}^{[a⃗_{j}(λ_{i}),λ_{i}]}(x) for each a⃗_{j}(λ_{i}) (i = 1, …, l,j = 0, …, n) and k ∈ ℤ in −o_{ij} ≤ k ≤ −o_{ij}^{#q_{1}} + p_{i} − 1 with o_{ij} := o(a⃗_{j}(λ_{i}), λ_{i}) and o_{ij}^{#q_{1}} := o(a⃗_{j}(λ_{i})^{#q_{1}}, λ_{i} + q_{1}). Then we have the generators of the vector space L_{ij} by (54).

L_{ij} := the vector space generated by

{P_{k}^{[a⃗_{j}(λ_{i}),λ_{i}]}(x) | k = −o_{ij}, …, −o_{ij}^{#q_{1}} + p_{i} − 1}. (54)

We set L_{ij} := {0} if −o_{ij} > −o_{ij}^{#q_{1}} + p_{i} − 1.

(d) Then

⨁_{\begin{matrix}i=1,...,l \\ j=0,...,n \end{matrix}} L_{ij}  (55)

forms a basis of the solution space.

2. The case of the homogeneous degree q_{1} > 0.

Input A non-zero G-invariant differential operator P (x, ∂) ∈ D(V )^{G} of ho-

mogeneous degree q_{1}n > 0 satisfying the condition (14). Output A basis of the G-invariant hyperfunction solutions to the differential

equation P (x, ∂)u(x) = 0. Procedure

(a) Compute the b_{P} -function b_{P} (s) and consider the set R := R_{1} ∪ R_{2} with

R_{1} := \{λ_{i} := −\frac{i + 1}{2} | i = 1, 2, …, n + 2q_{1} − 2\},

R_{2} := {λ ∈ ℂ | b_{P} (λ) = 0}.

Let q_{2} be the number of elements of the set R_{2} − R_{1}. We denote by

λ_{n+2q_{1}−1}, λ_{n+2q_{1}}, …, λ_{n+2q_{1}+q_{2}−2}

the elements of R_{2} − R_{1}. Then we can write the elements of R by

R = {λ_{1}, λ_{2}, …, λ_{n+2q_{1}+q_{2}−2}}.

(b) We define the multiplicity k_{i} of λ_{i} by

p_{i} := \begin{cases}multiplicity of s − λ_{i} in b_{P} (s) & if b_{P} (λ_{i}) = 0, \\ 0 & if b_{P} (λ_{i}) ≠ 0. \end{cases}  (56)

(c) For each λ_{i} (i = 1, …, n + 2q_{1} + q_{2} − 2), take one standard basis SB^{λ_{i}} = {a⃗_{0}(λ_{i}), ⋯, a⃗_{n}(λ_{i})} at s = λ_{i}, which is the standard basis defined by (42) when λ_{i} ∈ \frac{1}{2}ℤ and the one defined in Definition 1.4 otherwise.

(d) Compute the Laurent expansion coefficients P_{k}^{[a⃗_{j}(λ_{i}),λ_{i}]}(x) for each a⃗_{j}(λ_{i}) (i = 1, …, n + 2q_{1} + q_{2} − 2,j = 0, …, n) and k ∈ ℤ in −o_{ij} ≤ k ≤ −o_{ij}^{#q_{1}} + p_{i} − 1 with o_{ij} := o(a⃗_{j}(λ_{i}), λ_{i}) and o_{ij}^{#q_{1}} := o(a⃗_{j}(λ_{i}), λ_{i} + q_{1}). Then we have the generators of the vector space L_{ij} by (57).

L_{ij} := the vector space generated by

{P_{k}^{[a⃗_{j}(λ_{i}),λ_{i}]}(x) | k = −o_{ij}, …, −o_{ij}^{#q_{1}} + p_{i} − 1}. (57)

We set L_{ij} := {0} if −o_{ij} > −o_{ij}^{#q_{1}} + p_{i} − 1.

(e) Then

⨁_{\begin{matrix}i=1,...,n+2q_{1}+q_{2}−2 \\ j=0,...,n \end{matrix}} L_{ij}  (58)

forms a basis of the solution space.

Proof. Note that, for each complex number λ ∈ ℂ, we have only to decide which u(x) is annihilated by P (x, ∂) when u(x) is given as a finite combination of Laurent expansion coefficients of |P (x)|_{i}^{s} (i = 0, …, n) at s = λ. This is because of the same reason as in the proof of Algorithm 3.1.

Let SB := {a⃗_{0}, a⃗_{1}, …, a⃗_{n}} be a standard basis of ℂ^{n+1} at s = λ with an increasing sequence

0 < k(0) < k(1) < ⋯ < k(P HO(λ)) = n  (59)

such that SB_{q} := {a⃗_{0}, a⃗_{1}, …, a⃗_{k(q)}} is a basis of A(λ, q) for each q in 0 ≤ q ≤ P HO(λ). In particular, we suppose that it is the standard basis defined by (42) when λ_{i} ∈ \frac{1}{2}ℤ or the one from Definition 1.4 otherwise. Then, by the property (44), we see easily that SB^{#q_{1}} := {a_{0}^{#q_{1}}, a_{1}^{#q_{1}}, …, a_{n}^{#q_{1}}} is a standard basis of ℂ^{n+1} at s = λ + q_{1} with an increasing sequence

0 < k^{#q_{1}}(0) < k^{#q_{1}}(1) < ⋯ < k^{#q_{1}}(P HO(λ + q_{1})) = n  (60)

such that SB_{q}^{#q_{1}} := {a_{0}^{#q_{1}}, a_{1}^{#q_{1}}, …, a_{k^{#q_{1}}(q)}^{#q_{1}}} is a basis of A(λ + q_{1}, q) for each q in 0 ≤ q ≤ P HO(λ + q_{1}). Here, we see from the definition that

\begin{matrix}P HO(λ + q_{1}) ≥ P HO(λ) & if q_{1} < 0, \\ P HO(λ + q_{1}) ≤ P HO(λ) & if q_{1} > 0, \end{matrix}

and that k(q) = k^{#q_{1}}(q) for q < P HO(λ) if q_{1} < 0 or for q < P HO(λ + q_{1}) if q_{1} > 0.

Every u(x) given as a finite combination of Laurent expansion coefficients of |P (x)|_{i}^{s} (i = 0, …, n) at s = λ is expressed as

u(x) = ∑_{\begin{matrix}f,g∈ℤ \\ 0≤g≤n \end{matrix}} c_{f,g}P_{f}^{[a⃗_{g},λ]}(x)  (61)

with c_{f,g} ∈ ℂ. Let b_{P} (s) = ∑_{i=0}^{q} b_{i}(s−λ)^{p+i} be an expansion of b_{P} (s) with respect to (s − λ). The number p is the multiplicity of b_{P} (s) at s = λ. Now our goal is to prove the statement:

c_{f,g} = 0 except for − o(a⃗_{g}, λ) ≤ f ≤ −o(a_{g}^{#q_{1}}, λ + q_{1}) + p − 1
if and only if P (x, ∂)u(x) = 0

(62)

since P_{f}^{[a⃗_{g},λ]}(x) = 0 if f < −o(a⃗_{g}, λ) from the definition.

First we consider the situation that λ is not a root of b_{P} (s) = 0, i.e., p = 0. When q_{1} < 0 (Algorithm 3.2-1), there is no non-zero G-invariant homogeneous solutions of homogeneous degree nλ to P (x, ∂)u(x) = 0. When q_{1} > 0 (Algorithm 3.2-2), there is no non-zero G-invariant homogeneous solutions of homogeneous degree nλ to P (x, ∂)u(x) = 0 except when λ ∈ R_{1}. Then we have only to consider the cases that λ is a root of b_{P} (λ) = 0 when q_{1} < 0 (Algorithm 3.2-1), and the cases λ is a root of b_{P} (λ) = 0 or λ ∈ R_{1} when q_{1} > 0 (Algorithm 3.2-2). This is the reason why we restrict the λ’s to the finite sets of numbers in the first step of the procedures in the algorithms. We can easily see that the basis of L_{ij} in (57) is just the basis of the terms in the expression (61) with the condition (62) when λ = λ_{i} and p = k_{i}.

The rest of the proof is devoted to showing (62). The Laurent expansion of P^{[a⃗,s]}(x) at s = λ is denoted by P^{[a⃗,s]}(x) = ∑_{w∈ℤ} P_{w}^{[a⃗,λ]}(x)(s − λ)^{w}. Then we have P (x, ∂)P^{[a⃗,s]}(x) = ∑_{w∈ℤ} ∑_{i+j+p=w} b_{i}P_{j}^{[a^{#q_{1}},λ+q_{1}]}(x)(s − λ)^{w} and hence we have

P (x, ∂)P_{w}^{[a⃗,λ]}(x) = ∑_{i+j+p=w} b_{i}P_{j}^{[a^{#q_{1}},λ+q_{1}]}(x).  (63)

Here b_{i} = 0 except for i in 0 ≤ i ≤ q and P_{j}^{[a^{#q_{1}},λ+q_{1}]}(x) = 0 for sufficiently small j. Then, for the function u(x) = ∑\begin{matrix}f,g∈ℤ \\ 0≤g≤n \end{matrix} c_{f,g}P_{f}^{[a⃗_{g},λ]}(x), we see that

P (x, ∂)u(x) = ∑_{j∈ℤ} P_{j}^{[∑_{f,g∈ℤ} c_{f,g}b_{f−p−j}a_{g}^{#q_{1}},λ+q_{1}]}(x) = 0 where g runs in 0 ≤ g ≤ n, i.e., a⃗_{g} = 0 except for 0 ≤ g ≤ n. Thus we have ∑_{f,g∈ℤ} c_{f,g}b_{f−p−j}a⃗_{g} = ∑_{g=0}^{n} ∑_{f∈ℤ} c_{f,g}b_{f−p−j}a_{g}^{#q_{1}} ∈ A(λ + q_{1}, −j − 1) for all j ∈ ℤ by Theorem 1.6. Therefore we see that

for each g = 0, 1, …, n,

∑_{f∈ℤ}c_{f,g}b_{f−p−j} = 0 for all j ∈ ℤ satisfying g ≥ k^{#q_{1}}(−j) (64)

since a_{g}^{#q_{1}} ∉ A(λ + q_{1}, −j − 1) if g ≥ k^{#q_{1}}(−j) by definition. Here k^{#q_{1}}(−j) is the number defined by (60) if 0 ≤ −j ≤ P HO(λ + q_{1}) and k^{#q_{1}}(−j) = 0 (resp. k^{#q_{1}}(−j) = n + 1) if −j < 0 (resp. −j > P HO(λ + q_{1})). Since g ≥ k^{#q_{1}}(−j) is equivalent to o(a_{g}^{#q_{1}}, λ+q_{1}) ≥ −j by definition and ∑_{f∈ℤ} c_{f,g}b_{f−p−j} = ∑_{f=p+j}^{p+j+q} c_{f,g}b_{f−p−j} = ∑_{s=0}^{q} c_{p+j+s,g}b_{s} = 0, the condition (64) is rewritten as the condition:

for each g = 0, 1, …, n,

∑_{s=0}^{q} c_{p+j+s,g}b_{s} = 0 for all j ∈ ℤ satisfying j ≥ −o(a_{g}^{#q_{1}}, λ + q_{1}). (65)

Note that coefficients b_{0} and b_{q} are not zero. Then the condition (65) is equivalent to the condition:

for each g = 0, 1, …, n,

c_{p+j,g} = 0 for all j ∈ ℤ satisfying j ≥ −o(a_{g}^{#q_{1}}, λ + q_{1}).

(66)

This is equivalent to the condition (62), which we wanted to prove.

4. Algorithms for constructing solutions — inhomogeneous equations.

For the inhomogeneous differential equation P (x, ∂)u(x) = v(x), the algorithm for the computation of invariant hyperfunction solutions is carried out under rather a weak condition like (29).

Algorithm 4.1. (the case of inhomogeneous equation) For a given non-zero Ginvariant homogeneous differential operator P (x, ∂) ∈ D(V )^{G} of homogeneous degree kn ∈ ℤ satisfying the condition (29) in Theorem 1.8, an algorithm to compute a G-invariant hyperfunction solution of the G-invariant differential equation P (x, ∂)u(x) = v(x) is given in the following.

Input A non-zero G-invariant homogeneous differential operator P (x, ∂) ∈ D(V )^{G} of homogeneous degree kn ∈ ℤ satisfying the condition (29) and a non-zero quasi-homogeneous G-invariant hyperfunction v(x).

Output A non-zero G-invariant hyperfunction solution u(x) to the differential equation P (x, ∂)u(x) = v(x).

Procedure

1. Write v(x) as a sum of Laurent expansion coefficients. Namely, v(x) is given by v(x) = ∑_{l=1}^{q} P_{w_{l}}^{[a⃗_{l},λ]}(x) with a⃗_{1}, …, a⃗_{k} ∈ ℂ^{n+1} and w_{1}, …, w_{k} ∈ ℤ.

2. Compute the b_{P} -function b_{P} (s) and factorize it as b_{P} (s) = (s−λ+k)^{p}b̃(s) with b̃(λ − k) ≠ 0 and expand b̃(s)^{−1} into a Taylor series b̃(s)^{−1} = ∑_{i=0}^{∞} b_{i}(s − λ + k)^{i} at s = λ − k.

3. Then

u(x) = ∑_{l=1}^{q} ∑_{i+j=w_{l}+p} b_{i}P_{j}^{[a_{l}^{#−k},λ−k]}(x)  (67)

is an G-invariant hyperfunction solution to P (x, ∂)u(x) = v(x).

Proof. The proof can be found in the proof of Theorem 1.8. See [22, Theorem 6.1].

5. Explicit computations of examples

In this section we shall give some examples. Some homogeneous differential equations generated by det(x) and det(∂^{∗}) are treated here.

5.1. The equations det(∂^{∗}) det(x)u(x) = 0 and det(x) det(∂^{∗})u(x) = 0. We consider two examples of differential equation of homogeneous degree 0. Let us consider the case of P (x, ∂) = det(∂^{∗}) det(x) and P (x, ∂) = det(x) det(∂^{∗}). The homogeneous degrees of P (x, ∂) are 0 and the b_{P} -functions are b_{P} (s) = (s + 1)(s + \frac{3}{2}) ⋯ (s + \frac{n+1}{2}) and b_{P} (s) = (s)(s + \frac{1}{2}) ⋯ (s + \frac{n−1}{2}), respectively.

Proposition 5.1. The invariant hyperfunction solution to the differential equations det(∂^{∗}) det(x)u(x) = 0 and det(x) det(∂^{∗})u(x) = 0 are calculated in the following.

1. The G-invariant hyperfunction solution space to the differential equation det(∂^{∗}) det(x)u(x) = 0 is generated by

⋃_{i=1}^{n} ⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋} \{P_{−q}^{[a⃗,− \frac{i+1}{2} ]}(x) | a⃗ ∈ A(−\frac{i+1}{2}, q)\} .  (68)

Here, A(−\frac{i+1}{2}, q) is the vector subspace of ℂ^{n+1} defined by (19) in Definition 1.2 and explicitly computed in (39) of Theorem 2.1. Similarly, the Ginvariant hyperfunction solution space to the equation det(x) det(∂^{∗})u(x) = 0 is generated by

⋃_{i=−1}^{n−2} ⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋} \{P_{−q}^{[a⃗,− \frac{i+1}{2} ]}(x) | a⃗ ∈ A(−\frac{i+1}{2}, q)\} .  (69)

2. In particular, for i = −1, 0, 1, 2, …, n,

⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋} \{P_{−q}^{[a⃗,− \frac{i+1}{2} ]}(x) | a⃗ ∈ A(−\frac{i+1}{2}, q)\}  (70)

spans an (n + 1)-dimensional vector space generated by all the relatively invariant hyperfunctions corresponding to the character det(g)^{−i−1}. The dimensions of G-invariant hyperfunction solutions to det(∂^{∗}) det(x)u(x) = 0 and det(x) det(∂^{∗})u(x) = 0 are n(n + 1).

Proof. 1. We compute the solution space following Algorithm 3.1. For the differential operator P (x, ∂) = det(∂^{∗}) det(x), the b_{P} -function is b_{P} (s) = (s+1)(s+ \frac{3}{2}) ⋯ (s+ \frac{n+1}{2}). In the first step of the procedure of Algorithm 3.1, we have l = n and the roots of b_{P} (s) = 0 are λ_{i} = \frac{i+1}{2} with multiplicity p_{i} = 1 (i = 1, …, n). Since they are all half-integers, we can take a standard basis at s = λ_{i} SB^{λ_{i}} = {a⃗_{0}(λ_{i}), …, a⃗_{n}(λ_{i})} as defined in Definition 2.3. Let SB_{q}^{λ_{i}} be a subset of SB^{λ_{i}} such that SB_{q}^{λ_{i}} forms a basis of A(λ_{i}, q) for each q

in 0 ≤ q ≤ P HO(λ_{i}) = ⌊\frac{i+1}{2}⌋. Then we have SB^{λ_{i}} = ⨆_{q=0}^{⌊ \frac{i+1}{2} ⌋} SB_{q}^{λ_{i}} − SB_{q−1}^{λ_{i}}. and the set SB_{q}^{λ_{i}} − SB_{q−1}^{λ_{i}} forms a basis of \overline{A(λ_{i}, q)} := A(λ_{i}, q)/A(λ_{i}, q − 1), where A(λ_{i}, −1) = {0} and SB_{−1}^{λ_{i}} := Ø. For each a⃗_{j}(λ_{i}) ∈ SB_{q}^{λ_{i}} − SB_{q−1}^{λ_{i}}, we have o_{ij} := o(a⃗_{j}(λ_{i}), λ_{i}) = q and hence

−o_{ij} ≤ k ≤ −o_{ij} + p_{i} − 1 ⇒ −q ≤ k ≤ −q + p_{i} − 1 ⇒ k = −q

for each q in 0 ≤ q ≤ ⌊\frac{i+1}{2}⌋. Since P_{−q}^{[a⃗,λ_{i}]}(x) = 0 if a⃗ ∈ A(λ_{i}, q − 1), we have

vector space generated by {P_{−q}^{[a⃗,λ_{i}]}(x) | a⃗ ∈ A(λ_{i}, q)}
= vector space generated by {P_{−q}^{[a⃗,λ_{i}]}(x) | a⃗ ∈ SB_{q}^{λ_{i}} − SB_{q−1}^{λ_{i}}}

for each q in 0 ≤ q ≤ ⌊\frac{i+1}{2}⌋. Then the vector spaces generated by

⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋}\{P_{−q}^{[a⃗,λ_{i}]}(x) | a⃗ ∈ A(λ_{i}, q)\} and ⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋} ⋃_{a⃗∈SB_{q}^{λ_{i}}−SB_{q−1}^{λ_{i}}}\{P_{−q}^{[a⃗,λ_{i}]}(x)\} = ⨁_{j=0}^{n} L_{ij} coincide, and so do the vector spaces generated by

⋃_{i=1}^{n} ⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋}\{P_{−q}^{[a⃗,λ_{i}]}(x) | a⃗ ∈ A(λ_{i}, q)\}  (71)

and the vector space generated by

⨁_{i=1}^{n} ⨁_{j=0}^{n} L_{ij} = ⋃_{i=1}^{n} ⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋} ⋃_{a⃗∈SB_{q}^{λ_{i}}−SB_{q−1}^{λ_{i}}}\{P_{−q}^{[a⃗,λ_{i}]}(x)\}.

Thus we have proved that (68), which is (71), forms a basis of the vector space of G-invariant hyperfunction solution space to the differential equation det(∂^{∗}) det(x)u(x) = 0.

For the differential equation det(x) det(∂^{∗})u(x) = 0, we can prove in the same way that the vector space of G-invariant hyperfunction solution space to the differential equation det(x) det(∂^{∗})u(x) = 0 is generated by (69) since the b_{P} -function of det(x) det(∂^{∗}) is b_{P} (s) = (s)(s + \frac{1}{2}) ⋯ (s + \frac{n−1}{2}).

2. By Proposition 1.5, the elements in

⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋} ⋃_{a⃗∈SB_{q}^{λ_{i}}−SB_{q−1}^{λ_{i}}}\{P_{−q}^{[a⃗,λ_{i}]}(x)\}  (72)

are linearly independent and spans an (n + 1)-dimensional vector space. On the other hand, by Theorem 1.6, each P_{−q}^{[a⃗,λ_{i}]}(x) for a⃗ ∈ SB_{q}^{λ_{i}} − SB_{q−1}^{λ_{i}} is a homogeneous G-invariant hyperfunction of homogeneous degree nλ_{i} = −n(i + 1)/2. This means that it is relatively invariant under the action of GL_{n}(ℝ) corresponding to the character det(g)^{−i−1}. By the main result of [18], the space of relatively invariant hyperfunctions for a fixed character det(g)^{2s} (s ∈ ℂ) is n + 1. Then (72) forms a basis of all relatively invariant hyperfunctions corresponding to the character det(g)^{−i−1}. Then we see that the dimensions of G-invariant hyperfunction solutions to det(∂^{∗}) det(x)u(x) = 0 and det(x) det(∂^{∗})u(x) = 0 are n(n + 1).

5.2. The equations det(x)u(x) = 0.

Let us consider the case of P (x, ∂) = det(x). The total homogeneous degree of det(x) is n and b_{P} (s) = 1. We can prove by our algorithm that the G-invariant solution space of the differential equation det(x)u(x) = 0 is generated by the G-invariant measures on all the singular orbits, i.e., G-orbits contained in det(x) = 0, and hence it is \frac{n(n+1)}{2} -dimensional. The dimension coincides with the number of singular orbits. Here the G-invariant measure on each singular orbit is a relatively invariant hyperfunction. Namely we have the following proposition.

Proposition 5.2. Consider the differential equation det(x)u(x) = 0.

1. The G-invariant hyperfunction solution space to the differential equation det(x)u(x) = 0 is generated by

⋃_{i=1}^{n} \{P_{−⌊ \frac{i+1}{2} ⌋}^{[a⃗,− \frac{i+1}{2} ]}(x) | a⃗ ∈ ℂ^{n+1}\} .  (73)

2. In particular, for i = 1, 2, …, n, the set

\{P_{−⌊ \frac{i+1}{2} ⌋}^{[a⃗,− \frac{i+1}{2} ]}(x) | a⃗ ∈ ℂ^{n+1}\}  (74)

spans an (n + 1 − i)-dimensional vector space generated by the tempered distributions

f(x) ⟼ ∫ f(x)dν_{i}^{j} (f(x) ∈ 𝒮(V ))

(j = 0, 1, …, n − i) where dν_{i}^{j} is the G-invariant measure on

S_{i}^{j} := {x ∈ Sym_{n}(ℝ) | sgn(x) = (j, n − i − j)}.

Proof. 1. We shall prove the first part of the proposition by carrying out Algorithm 3.2-2.

The b_{P} -function is b_{P} (s) ≡ 1 and we have R = R_{1} = \{λ_{i} = −\frac{i+1}{2} | i = 1, …, n} and q_{1} = 1 and q_{2} = 0 in the first step of the procedure. In the second step, we have p_{i} = 0 for all i = 1, …, n. Since all λ_{i} ∈ R are half-integers, we take the standard basis SB^{λ_{i}} = {a⃗_{0}(λ_{i}), …, a⃗_{n}(λ_{i})} of Definition 1.4 as a standard basis at s = λ_{i}. For each a⃗_{j}(λ_{i}) (i = 1, …, n and j = 0, …, n), we have defined o_{ij} := o(a⃗_{j}(λ_{i}), λ_{i}) and o_{ij}^{#q_{1}} := o_{ij}^{#} = o(a⃗_{j}(λ_{i})^{#}, λ_{i} +1). Let SB_{q−1}^{λ_{i}} be the subset of SB^{λ_{i}} consisting of the vectors a⃗_{j}(λ_{i}) such that P^{[a⃗_{j}(λ_{i}),s]}(x) has a pole of order less than q − 1 at s = λ_{i}. If a⃗_{j}(λ_{i}) ∈ SB^{λ_{i}} − SB_{q−1}^{λ_{i}}, then P^{[a⃗_{j}(λ_{i}),s]}(x) has a pole of order q at s = λ_{i} since the possible highest order of P^{[a⃗,s]}(x) at s = λ_{i} is q = ⌊\frac{i+1}{2}⌋. Then we have, by putting q = ⌊\frac{i+1}{2}⌋,

\begin{aligned}o_{ij} & = o_{ij}^{#} + 1 = q if a⃗_{j}(λ_{i}) ∈ SB^{λ_{i}} − SB_{q−1}^{λ_{i}}, \\ o_{ij} & = o_{ij}^{#} ≤ q − 1 if a⃗_{j}(λ_{i}) ∈ SB_{q−1}^{λ_{i}}. \end{aligned}  (75)

Indeed, by Theorem 2.1, Definition 2.3 and the property (44), we see that

(the order of P^{[a⃗_{j}(λ_{i})^{#},s]}(x) at s = λ_{i} + 1)
=(the order of P^{[a⃗_{j}(λ_{i}),s]}(x) at s = λ_{i})

for all a⃗_{j}(λ_{i}) ∈ SB_{q−1}^{λ_{i}} and that the order of P^{[a⃗_{j}(λ_{i})^{#},s]}(x) at s = λ_{i} + 1 is equal to q − 1 for all a⃗_{j}(λ_{i}) ∈ SB^{λ_{i}} − SB_{q−1}^{λ_{i}}. Then we have (75). Thus, for a⃗_{j}(λ_{i}) ∈ SB^{λ_{i}} − SB_{q−1}^{λ_{i}}, we have

−o_{ij} = −(o_{ij}^{#} + 1) = −o_{ij}^{#} + p_{i} − 1 = −q = −⌊\frac{i + 1}{2}⌋,

and L_{ij} in (57) is generated by P_{−⌊ \frac{i+1}{2} ⌋}^{[a⃗_{j}(λ_{i}),λ_{i}]}(x). For a⃗_{j}(λ_{i}) ∈ SB_{q−1}^{λ_{i}}, we have

−o_{ij} = −o_{ij}^{#} > −o_{ij} − 1 = −o_{ij} + p_{i} − 1, and hence L_{ij} in (57) is {0}. Therefore we have

⨁_{\begin{matrix}i=1,...,n \\ j=0,...,n \end{matrix}} L_{ij} = ⨁_{i=1,...,n} \left\{{the vector space generated by \atop \{P_{−⌊ \frac{i+1}{2} ⌋}^{[a⃗_{j}(λ_{i}),λ_{i}]}(x) | a⃗_{j}(λ_{i}) ∈ SB^{λ_{i}} − SB_{q−1}^{λ_{i}}\}}\right\}

= \{ the vector space generated by ⋃_{i=1}^{n}\{P_{−⌊ \frac{i+1}{2} ⌋}^{[a⃗,− \frac{i+1}{2} ]}(x) | a⃗ ∈ ℂ^{n+1}\} \} ,

since P_{−⌊ \frac{i+1}{2} ⌋}^{[a⃗_{j}(λ_{i}),− \frac{i+1}{2} ]}(x) = 0 if a⃗_{j}(λ_{i}) ∈ SB_{q−1}^{λ_{i}}. This is what we wanted to prove.

2. For i = 1, …, n, each element of \{P_{−⌊ \frac{i+1}{2} ⌋}^{[a⃗,− \frac{i+1}{2} ]}(x) | a⃗ ∈ ℂ^{n+1}\} is a homogeneous

G-invariant hyperfunction of homogeneous degree −n(\frac{i+1}{2}) and its support is contained in S_{i}. See Theorem A2. It is proved that such hyperfunctions are given as a linear sum of G-invariant measures on the (n − i + 1) open orbits S_{i}^{j} (j = 1, …, n − i + 1) in S_{i}. See, for example, §4 in [16].

5.3. The equations det(∂^{∗})u(x) = 0.

Similar argument is possible in the case of P (x, ∂) = det(∂). In this case, the total homogeneous degree of P (x, ∂) is −n and we see that b_{P} (s) = ∏_{i=1}^{n}(s + \frac{i−1}{2}). The solution space of det(∂)u(x) = 0 is the Fourier transform of that of det(x)u(x) = 0, and hence it is \frac{n(n+1)}{2} -dimensional and generated by relatively invariant hyperfunctions.

Proposition 5.3. Consider the differential equation det(∂^{∗})u(x) = 0.

1. The G-invariant hyperfunction solution space to the differential equation det(∂^{∗})u(x) = 0 is generated by

⋃_{i=−1}^{n−2} ⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋} \{P_{−q}^{[a⃗,− \frac{i+1}{2} ]}(x) | a⃗ ∈ D_{∗}^{(q)}\} .  (76)

Here, D_{∗}^{(j)} is a vector subspace of ℂ^{n+1} defined by Definition 2.2. The ∗ in D_{∗}^{(j)} is substituted the word half, even or odd according as −\frac{i+1}{2} is a strictly half integer, an even integer or an odd integer , respectively.

2. In particular, for i = −1, 0, 1, …, n − 2,

⋃_{q=0}^{⌊ \frac{i+1}{2} ⌋} \{P_{−q}^{[a⃗,− \frac{i+1}{2} ]}(x) | a⃗ ∈ D_{∗}^{(q)}\}  (77)

spans an (i+2)-dimensional vector space generated by the Fourier transforms of the tempered distributions in (74).

Proof. We follow Algorithm 3.2-1. The first step and the second step of the procedure of Algorithm 3.2-1 are the same as those of Algorithm 3.1. The roots of the b_{P} -function are λ_{i} = −\frac{i+1}{2} (i = −1, 0, …, n−2) and their multiplicity p_{i} is 1. We can determine the generators of the solution space in the same way as the proof of Proposition 5.1. Since they are all half-integers, we can take a standard basis at s = λ_{i}: SB^{λ_{i}} = {a⃗_{0}(λ_{i}), …, a⃗_{n}(λ_{i})} as defined in Definition 2.3. For each a⃗_{j}(λ_{i}) ∈ SB^{λ_{i}}, we define o_{ij} := o(a⃗_{j}(λ_{i}), λ_{i}) and o_{ij}^{#q_{1}} := o_{ij}^{#−1} = o(a⃗_{j}(λ_{i})^{#−1}, λ_{i} − 1). We have only to pick up the vector a⃗_{j}(λ_{i}) satisfying

−o_{ij} ≤ −o_{ij}^{#−1} + p_{i} − 1 = −o_{ij}^{#−1}.  (78)

Since, for i = −1, 0, …, n−2, we see −o_{ij} ≥ −o_{ij}^{#−1} by Theorem 2.1, Definition 2.3 and Theorem A1, (78) means −o_{ij} = −o_{ij}^{#−1}. Namely, we have to choose a⃗_{j}(λ_{i}) satisfying

( order of P^{[a⃗_{j}(λ_{i})^{#−1},s]}(x) at s = λ_{i} − 1)
=( order of P^{[a⃗_{j}(λ_{i}),s]}(x) at s = λ_{i}).

By Theorem 2.1, Definition 2.3 and Theorem A1, we see that this condition is equivalent to that a⃗_{j}(λ_{i}) ∈ D_{∗}^{(q)} with some q = 0, 1, …, ⌊\frac{i+1}{2}⌋. Here, the ∗ in D_{∗}^{(j)} is substituted half, even or odd according as −\frac{i+1}{2} is a strictly half integer, an even integer or an odd integer, respectively. Thus we have the first result.

The second result is easily verified and we omit the proof.

5.4. The equations P (x, ∂)u(x) = δ(x).

We shall find a G-invariant fundamental solution to the homogeneous G-invariant differential operator P (x, ∂). First note that the delta function δ(x) on Sym_{n}(ℝ) is given as δ(x) = (const.) × P_{−⌊ \frac{n+1}{2} ⌋}^{[a⃗,− \frac{n+1}{2} ]}(x) with a vector a⃗ ∈

A(−\frac{n+1}{2}, ⌊\frac{n+1}{2}⌋) which is non-zero in \overline{A(−\frac{n+1}{2}, ⌊\frac{n+1}{2}⌋)}. Henceforth, we fix a⃗ = a⃗_{0} satisfying that

δ(x) = P_{−⌊ \frac{n+1}{2} ⌋}^{[a⃗_{0},− \frac{n+1}{2} ]}(x).  (79)

We are using the procedure of Algorithm 4.1 in the following examples.

Example 5.1. The differential operator P (x, ∂) = det(∂^{∗}) det(x) is an operator of homogeneous degree 0 and hence k = 0 in Algorithm 4.1. The b_{P} -function is b_{P} (s) = (s + 1)(s + \frac{3}{2}) … (s + \frac{n+1}{2}). The function v(x) in Algorithm 4.1 is given by (79) and hence λ = −\frac{n+1}{2} in Algorithm 4.1. Since (s − λ + k) = (s + \frac{n+1}{2}), we have b_{P} (s) = (s + \frac{n+1}{2})b̃(s) where b̃(s) = (s + 1)(s + \frac{3}{2}) … (s + \frac{n}{2}). Then p = 1 in Algorithm 4.1. We have the Taylor expansion b̃(s)^{−1} = ∑_{i=0}^{∞} b_{i}(s + \frac{n−1}{2})^{i} with b_{0} = ((−\frac{n−1}{2})(−\frac{n−2}{2}) ⋯ (−\frac{1}{2}))^{−1} . Then, by the third step of the procedure in Algorithm 4.1, we have

u(x) = ∑_{i+j=−⌊ \frac{n+1}{2} ⌋+1} b_{i}P_{j}^{[a⃗_{0},− \frac{n+1}{2} ]}(x) = ∑_{i+j=−⌊ \frac{n−1}{2} ⌋} b_{i}P_{j}^{[a⃗_{0},− \frac{n+1}{2} ]}(x).

Since i ≥ 0 and P_{j}^{[a⃗_{0},− \frac{n+1}{2} ]}(x) = 0 except for j ≥ −⌊\frac{n+1}{2}⌋, we have

u(x) = b_{0} × P_{−⌊ \frac{n−1}{2} ⌋}^{[a⃗_{0},− \frac{n+1}{2} ]}(x) + b_{1} × P_{−⌊ \frac{n+1}{2} ⌋}^{[a⃗_{0},− \frac{n+1}{2} ]}(x).

This is a G-invariant fundamental solution to P (x, ∂) = det(∂^{∗}) det(x). In this case, b_{1} × P_{−⌊ \frac{n+1}{2} ⌋}^{[a⃗_{0},− \frac{n+1}{2} ]}(x) is not necessary since it is annihilated by P (x, ∂).

Next we consider the differential operator P (x, ∂) = det(x) det(∂^{∗}). It is an operator of homogeneous degree 0 and hence k = 0. The b_{P} -function is b_{P} (s) = s(s + \frac{1}{2}) … (s + \frac{n−1}{2}). The function v(x) is given by (79) and hence λ = −\frac{n+1}{2} . Since (s − λ + k) = (s + \frac{n+1}{2}), we have b_{P} (s) = b̃(s) = s(s + \frac{1}{2}) … (s + \frac{n−1}{2}). Then p = 0. We have the Taylor expansion b̃(s)^{−1} = ∑_{i=0}^{∞} b_{i}(s + \frac{n−1}{2})^{i} with b_{0} = ((−\frac{n+1}{2})(−\frac{n}{2}) ⋯ (−1))^{−1} . Then, by the third step of the procedure, we have

u(x) = ∑_{i+j=−⌊ \frac{n+1}{2} ⌋} b_{i}P_{j}^{[a⃗_{0},− \frac{n+1}{2} ]}(x). Since i ≥ 0 and P_{j}^{[a⃗_{0},− \frac{n+1}{2} ]}(x) = 0 except for j ≥ −⌊\frac{n+1}{2}⌋, we have

u(x) = ((−\frac{n + 1}{2})(−\frac{n}{2}) ⋯ (−1))^{−1} × P_{−⌊ \frac{n+1}{2} ⌋}^{[a⃗_{0},− \frac{n+1}{2} ]}(x).

This is a G-invariant fundamental solution of P (x, ∂) = det(x) det(∂^{∗}).

Example 5.2. We consider the differential operator P (x, ∂) = det(x). It is an operator of homogeneous degree n and hence k = 1. The b_{P} -function is b_{P} (s) = 1. The function v(x) is given by (79) and hence λ = −\frac{n+1}{2} . Since (s − λ + k) = (s + \frac{n+3}{2}), we have p = 0 and b̃(s)^{−1} = 1, and hence b_{0} = 1 and b_{i} = 0 for i > 1. Then, by the third step of the procedure, we have

u(x) = ∑_{i+j=−⌊ \frac{n+1}{2} ⌋+0} b_{i}P_{j}^{[a_{0}^{#−1},− \frac{n+1}{2} −1]}(x) = ∑_{i+j=−⌊ \frac{n+1}{2} ⌋} b_{i}P_{j}^{[a_{0}^{#−1},− \frac{n+3}{2} ]}(x).

Since i ≥ 0 and P_{j}^{[a_{0}^{#1},− \frac{n+3}{2} ]}(x) = 0 except for j ≥ −⌊\frac{n+1}{2}⌋, we have u(x) =

P_{−⌊ \frac{n+1}{2} ⌋}^{[a_{0}^{#1},− \frac{n+3}{2} ]}(x). This is a G-invariant fundamental solution of P (x, ∂) = det(x).

Example 5.3. The differential operator P (x, ∂) = det(∂^{∗}) is an operator of homogeneous degree −n and hence k = −1. The b_{P} -function is b_{P} (s) = s(s + \frac{1}{2}) … (s + \frac{n−1}{2}). The function v(x) is given by (79) and hence λ = −\frac{n+1}{2} . Since (s−λ+k) = (s+ \frac{n−1}{2}), we have b_{P} (s) = (s+ \frac{n−1}{2})b̃(s) where b̃(s) = s(s+ \frac{1}{2}) … (s+ \frac{n−2}{2}). Then p = 1. We have the Taylor expansion b̃(s)^{−1} = ∑_{i=0}^{∞} b_{i}(s + \frac{n−1}{2})^{i} with b_{0} = ((−\frac{n−1}{2})(−\frac{n−2}{2}) ⋯ (−\frac{1}{2}))^{−1} . Then, by the third step of the procedure, we have

u(x) = ∑_{i+j=−⌊ \frac{n+1}{2} ⌋+1} b_{i}P_{j}^{[a_{0}^{#1},− \frac{n+1}{2} +1]}(x) = ∑_{i+j=−⌊ \frac{n−1}{2} ⌋} b_{i}P_{j}^{[a_{0}^{#1},− \frac{n−1}{2} ]}(x).

Since i ≥ 0 and P_{j}^{[a_{0}^{#1},− \frac{n−1}{2} ]}(x) = 0 except for j ≥ −⌊\frac{n−1}{2}⌋, we have

u(x) = ((−\frac{n − 1}{2})(−\frac{n − 2}{2}) ⋯ (−\frac{1}{2}))^{−1} × P_{−⌊ \frac{n−1}{2} ⌋}^{[a_{0}^{#1},− \frac{n−1}{2} ]}(x).

This is a G-invariant fundamental solution to P (x, ∂) = det(∂^{∗}).

A Some preliminary results.

In this section we state the main results of the author’s paper [19], which play crucial roles in this paper. We shall give the statements of the theorems used in this paper for the reader’s convenience without proof.

A1. The exact order of complex power functions.

Using the vectors d^{(k)}[s_{0}] defined in (30), we can determine the exact orders of poles of P^{[a⃗,s]}(x).

Theorem A1. (Exact orders of poles) The exact orders of poles of P^{[a⃗,s]}(x) are computed by the following algorithm.

1. At s = −\frac{2m+1}{2}(m = 1, 2, …), let the coefficient vectors d^{(k)}[−\frac{2m+1}{2}] be as defined in Definition 2.1. The exact pole order P^{[a⃗,s]}(x) at s = −\frac{2m+1}{2}(m = 1, 2, …) is given in terms of the coefficient vector d^{(2k)}[−\frac{2m+1}{2}].

(a) If 1 ≤ m ≤ \frac{n}{2} , then P^{[a⃗,s]}(x) has a possible pole of order not larger than

m.

• If ⟨d^{(2)}[−\frac{2m+1}{2}], a⃗⟩ = 0, then P^{[a⃗,s]}(x) is holomorphic, and the converse is true.

• Suppose integers p is in 1 ≤ p < m. If ⟨d^{(2p+2)}[−\frac{2m+1}{2}], a⃗⟩ = 0 and ⟨d^{(2p)}[−\frac{2m+1}{2}], a⃗⟩ ≠ 0, then P^{[a⃗,s]}(x) has a pole of order p, and the converse is true.

• Lastly, if ⟨d^{(2m)}[−\frac{2m+1}{2}], a⃗⟩ ≠ 0 , then P^{[a⃗,s]}(x) has a pole of order

m, and the converse is true.

(b) If m > \frac{n}{2} , then P^{[a⃗,s]}(x) has a possible pole of order not larger than

n′ := ⌊\frac{n}{2}⌋

• If ⟨d^{(2)}[−\frac{2m+1}{2}], a⃗⟩ = 0, then P^{[a⃗,s]}(x) is holomorphic, and the converse is true.

• Suppose integers p is in 1 ≤ p < n′. If ⟨d^{(2p+2)}[−\frac{2m+1}{2}], a⃗⟩ = 0 and ⟨d^{(2p)}[−\frac{2m+1}{2}], a⃗⟩ ≠ 0, then P^{[a⃗,s]}(x) has a pole of order p, and the converse is true.

• Lastly, P^{[a⃗,s]}(x) has a pole of order n′ if ⟨d^{(n−1)}[−\frac{2m+1}{2}], a⃗⟩ ≠ 0 (when n is odd) or ⟨d^{(n)}[−\frac{2m+1}{2}], a⃗⟩ ≠ 0 (when n is even), and the converse is true.

2. At s = −m(m = 1, 2, …), the coefficient vectors d^{(k)}[−m] are defined in Definition 2.1 with ϵ[−m] = (−1)^{−m+1}. We obtain the exact order at s = −m(m = 1, 2, …) in terms of the coefficient vectors d^{(2k+1)}[−m].

(a) If 1 ≤ m ≤ \frac{n}{2} , then P^{[a⃗,s]}(x) has a possible pole of order not larger than

m.

• If ⟨d^{(1)}[−m], a⃗⟩ = 0, then P^{[a⃗,s]}(x) is holomorphic, and the converse is true.

• Suppose integers p is in 1 ≤ p < m. If ⟨d^{(2p+1)}[−m], a⃗⟩ = 0 and ⟨d^{(2p−1)}[−m], a⃗⟩ ≠ 0 , then P^{[a⃗,s]}(x) has a pole of order p, and the converse is true.

• Lastly, if ⟨d^{(2m−1)}[−m], a⃗⟩ ≠ 0 , then P^{[a⃗,s]}(x) has a pole of order

m, and the converse is true.

(b) If m > \frac{n}{2} , then P^{[a⃗,s]}(x) has a possible pole of order not larger than n′ := ⌊\frac{n+1}{2}⌋

• If ⟨d^{(1)}[−m], a⃗⟩ = 0, then P^{[a⃗,s]}(x) is holomorphic, and the converse is true.

• Suppose integers p is in 1 ≤ p < n′. If ⟨d^{(2p+1)}[−m], a⃗⟩ = 0 and ⟨d^{(2p−1)}[−m], a⃗⟩ ≠ 0 , then P^{[a⃗,s]}(x) has a pole of order p, and the converse is true.

• Lastly, P^{[a⃗,s]}(x) has a pole of order n′ if ⟨d^{(n)}[−m], a⃗⟩ ≠ 0 (when n is odd) or ⟨d^{(n−1)}[−m], a⃗⟩ ≠ 0 (when n is even), and the converse is true.

A2. The exact support of complex power functions.

The exact support of P^{[a⃗,s]}(x) is given by the following theorem.

Theorem A2. (Support of the singular invariant hyperfunctions) Let q be a positive integer. Suppose that P^{[a⃗,s]}(x) has a pole of order p at s = −\frac{q+1}{2} Let

P^{[a⃗,s]}(x) = ∑_{w=−p}^{∞} P_{w}^{[a⃗,− \frac{q+1}{2} ]}(x)(s + \frac{q + 1}{2})^{w}  (80)

be the Laurent expansion of P^{[a⃗,s]}(x) at s = −\frac{q+1}{2} . The support of the Laurent

expansion coefficients P_{w}^{[a⃗,− \frac{q+1}{2} ]}(x) is contained in S if w < 0.

1. Let q be an even positive integer. Then the support of P_{w}^{[a⃗,− \frac{q+1}{2} ]}(x) for w = −1, −2, …, −p is contained in the closure \overline{S_{−2w}}. More precisely, it is given by

Supp(P_{w}^{[a⃗,− \frac{q+1}{2} ]}(x)) = \overline{(  ⋃_{j∈\{0≤j≤n+2w | ⟨d_{j}^{(−2w)}[− \frac{q+1}{2} ],a⃗⟩≠0\}} S_{−2w}^{j}}). (81)

2. Let q be an odd positive integer. Then the support of P_{w}^{[a⃗,− \frac{q+1}{2} ]}(x) for w = −1, −2, …, −p is contained in the closure \overline{S_{−2w−1}}. More precisely, it is given by

Supp(P_{w}^{[a⃗,− \frac{q+1}{2} ]}(x)) = \overline{(  ⋃_{j∈\{0≤j≤n+2w+1 | ⟨d_{j}^{(−2w−1)}[− \frac{q+1}{2} ],a⃗⟩≠0\}} S_{−2w−1}^{j})}. (82)

Here, Supp(−) denotes the support of the hyperfunction in (−).

References

[1] Angeli, Y., “Analyse harmonique sur les cônes satellites, ” Ph.D. thesis, Université Henri Poincaré, 2001.

[2] Ames, G., Fundamental solutions of invariant differential operators on a semisimple Lie group, Pacific J. Math. 196 (2000), 17–44.

[3] Blind, B., Distributions homogènes sur une algèbre de Jordan, Bull. Soc. Math. France 125 (1997), 493–528.

[4] Bogoliubov, N. N., A. A. Logunov, and I. T. Todorov, “Foundation of axiomatic approach in quantum field theory,” (The Japanese translation was

published in 1972 by Tokyo-Tosho Publishers under the title of “Mathematical method of Quantum Field Theory”. The English translation was published in 1975 by W. A. Benjamin, Inc. under the title of “Introduction to Axiomatic

Quantum Field Theory”), Nauka, Moscow, 1969 (Russian).

[5] Bouaziz, A., and N. Kamoun, Équations différentielles invariantes sur les groupes et algèbres de Lie réductifs, Ann. Inst. Fourier (Grenoble) 50 (2000), 1799–1857.

[6] Faraut, J., and A. Korányi, “Analysis on symmetric cones,” Oxford Mathematical Monographs, Oxford University Press, 1994.

[7] Gårding, L., The solution of Cauchy’s problem for two totally hyperbolic differential equations by means of Riesz integrals, Ann. of Math. 48 (1947), 785–826.

[8] Gelfand, I. M., and G. E. Shilov, “Generalized Functions — properties and operations, Generalized Functions,” vol. 1, Academic Press, New York and London, 1964.

[9] Gindikin, S. G., Invariant generalized functions in homogeneous domains, Functional Anal. Appl. 9 (1975), 50–52.

[10] —, “Tube domains and the Cauchy problem,” Translation of Mathematical Monographs, vol. 111, American Mathematical Society, 1992.

[11] Ishi, H., Positive Riesz distributions on homogeneous cones, J. Math. Soc. Japan 52 (2000), 161–186.

[12] Kashiwara, M., T. Kawai, and T. Kimura, “Daisuukaisekigaku no Kiso (Foundations of Algebraic Analysis),” Kinokuniya, Tokyo, 1980 (Japanese), The English translation was published by Princeton Univ. Press in 1985.

[13] Methée, P.-D., Sur les distributions invariantes dans le groupe des rotations de Lorentz, Comment. Math. Helv. 28 (1954), 225–269.

[14] —, Transformée de Fourier de distributions invariantes, C. R. Acad. Sci. Paris Sér. I Math. 240 (1955), 1179–1181.

[15] —, L’equation des ondes avec seconde membre invariante, Comment. Math. Helv. 32 (1957), 153–164.

[16] Muro, M., Microlocal analysis and calculations on some relatively invariant hyperfunctions related to zeta functions associated with the vector spaces of quadratic forms, Publ. Res. Inst. Math. Sci. Kyoto Univ. 22 (1986), 395–463.

[17] —, Singular invariant tempered distributions on regular prehomogeneous vector spaces, J. Funct. Anal. 76 (1988), 317 – 345.

[18] —, Invariant hyperfunctions on regular prehomogeneous vector spaces of commutative parabolic type, Tôhoku Math. J. (2) 42 (1990), 163–193.

[19] —, Singular invariant hyperfunctions on the space of real symmetric matrices, Tôhoku Math. J. (2) 51 (1999), 329–364.

[20] —, An algorithm to compute the b_{P} -functions via Gröbner bases of invariant differential operators on prehomogeneous vector spaces , Sûrikaiseki kenkyûsho Kôkyûroku 1171 (2000), 68–81.

[21] —, Singular Invariant hyperfunctions on the space of complex and quaternion Hermitian matrices, J. Math. Soc. Japan 53 (2001), 589–602.

[22] —, Invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices, J. Funct. Anal., 193 (2002), 346–384.

[23] —, Singular invariant hyperfunctions on the square matrix space and the alternating matrix space, Nagoya Math. J. 169 (2003), 19–75.

[24] Nomura, T., Algebraically independent generators of invariant differential operators on a symmetric cone, J. Reine Angew. Math. 400 (1989), 122– 133.

[25] —, Algebraically independent generators of invariant differential operators on a bounded symmetric domain, J. Math. Kyoto Univ. 31 (1991), 265– 279.

[26] Ricci, R., and E. M. Stein, Homogeneous distribution on spaces of Hermitian matrices, J. Reine. Angew. Math. 368 (1986), 142–164.

[27] Riesz, M., L’intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math. 81 (1949), 1–223.

[28] Sato, M., and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131–170.

[29] Shintani, T., On zeta functions associated with the vector spaces of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 25–65.

Masakazu Muro Gifu University, Yanagito 1-1, Gifu 501-1193,JAPAN muro@cc.gifu-u.ac.jp

Received October 1, 2002 and in final form April 23, 2003