Symmetry Characterization Theorems for Homogeneous Siegel Domains - - PowerPoint PPT Presentation

Symmetry Characterization Theorems for Homogeneous Siegel Domains Takaaki Nomura

Kyushu University (Professor Emeritus) Osaka City University Advanced Mathematical Institute 7 September, 2019 Young Mathematicians Workshop on Several Complex Variables 2019 (at OCU)

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Siegel domains (history) — Introduced by Piatetski-Shapiro (1957) for his study of automorphic forms — Needed an unbounded realization of the unit ball B2N ⊂ CN — Just a moment’s consideration on Shilov boundary convinces us that it cannot be of the half-space type RN + iΩ ⊂ CN if N > 1: The Shilov boundary of B2N is the sphere S2N−1, whereas the Shilov boundary of RN + iΩ is RN. — Piatetski-Shapiro gave an example of non-symmetric homogeneous Siegel domains in C4 and C5 (1959). — This solved a problem posed by E. Cartan (1935), since Siegel domains are holomorphically equivalent to bounded domains. — By E. Cartan: in C2 and C3, every homogeneous bounded domain is symmetric. — Recall that a domain D ⊂ CN is symmetric

def

⇐⇒ ∀z0 ∈ D, ∃φ0 ∈ Hol(D) s.t.     

(1) z0 is an isolated fixed point of φ0, (2) φ0 ◦ φ0 = IdD.

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• E. Cartan’s conjecture

The discovery of non-symmetric bounded domains would have to be based on a fresh idea. — turned out to be correct. — Note that Cartan never wrote that every homogeneous bounded domains was symmetric, the false conjecture spread by someone who did not read or did not understand what Cartan wrote in French subjonctif. Now we know a lot of non-symmetric homogeneous Siegel domains. Basic Question How do we characterize symmetric domains among homoge- neous Siegel (or bounded) domains?

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There are already many works. Just list some. . . — A. Borel, J.-L. Koszul, J. Hano (50’s) : if the domain is homogeneous under a (semisimple or unimodular) Lie group — I. Satake, J. Dorfmeister (late 70’s) in terms of the defining data of Siegel domains — J. E. D’Atri and Dotti Miatello (1983) by the non-positivity of the sectional curvature of the Bergman metric — K. Azukawa (1985) by the number of distinct eigenvalues of the curvature operator — J. Vey (1970) by the existence of a discrete subgroup Γ ⊂ Hol(D) acting on D properly s.t. ΓK = D for a compact subset K ⊂ D. — N (2001) by the commutativity of the Berezin transform with the Laplace–Beltrami op. — Xu Yichao (1979), N (2003) by the harmonicity of the Poisson kernel defined a là Hua. Today’s talk Symmetry characterization related to Cayley transforms

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Siegel domains (definition) — V : a finite-dimensional real vector space (with a norm), — Ω : an open convex cone in V, We assume that Ω is regular (contains no entire line). — W := VC: the complexification of V, — w → w∗ : the conjugation in W w.r.t. the real form V, — U : another finite-dimensional complex vector space, — Q : a Hermitian sesqui-linear map U × U → W (complex linear in the first variable, conjugate linear in the second), We assume that Q is Ω-positive. Thus we have

Q(u′,u) = Q(u,u′)∗, Q(u,u) ∈ Cl(Ω) \ {0} (0 ∀u ∈ U).

Definition 1

D(Ω,Q) := {(u,w) ∈ U × W ; w + w∗ − Q(u,u) ∈ Ω}.

Remarks (1) We take a generalized right half-space instead of an upper half-space. (2) We do not exclude the possibility U = {0}, in which case D(Ω,Q) = Ω + iV.

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• D = D(Ω,Q) is said to be homogeneous if Hol(D) acts on D transitively.

Remark Since D is holomorphically equivalent to a bounded domain, Hol(D) is a finite-dimensional Lie group (H. Cartan).

• We always assume that our Siegel domain is homogeneous.

Examples (1) V = R, Ω = R>0, U = {0}, W = VC = C. In this case D = R>0 + iR is the right half-space in C. The Cayley transform

w → z := w − 1 w + 1 maps D onto the unit disk {z ∈ C ; z < 1}.

(2) V = Sym(n,R) : the real vector space of n × n real symmetric matrices,

Ω = P(n,R) : the positive-definite matrices in Sym(n,R), U = {0}, W = VC = Sym(n,C).

In this case D = P(n,R) + Sym(n,R) is the Siegel right half-space. The Cay- ley transform w → z := (w − e)(w + e)−1 (e is the identity matrix) maps the Siegel right half-space onto the Siegel disk D := {z ∈ Sym(n,C) ; e − z∗z ≫ 0}.

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(3) V = R, Ω = R>0, U = Cm, W = VC = C, Q(u1,u2) := 2 u1 · u2. In this case

D(Ω,Q) = {(u,w) ∈ Cm × C ; Re w − ∥u∥2 > 0}.

The Cayley transform

(u,w) → ( 2 u w + 1 , w − 1 w + 1 )

maps D(Ω,Q) onto the unit ball in Cm+1 = Cm × C.

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Piatetski-Shapiro algebra (normal j-algebra) We know {homogeneous Siegel domains} ⇄ {Piatetski-Shapiro algebras}

g : a split solvable Lie algebra (ad is triangularizable over R) J : a linear operator on g with J2 = −Idg. ω : a linear form on g.

Then (g, J,ω) (or simply g) is a Piatetski-Shapiro algebra if (1) [x, y] + J[Jx, y] + J[x, Jy] = [Jx, Jy], (2) ⟨ x | y ⟩ω := ⟨ω,[Jx, y]⟩ defines a J-invariant inner product on g.

• Linear forms ω that satisfy (2) are said to be admissible.
• The linear form β on g defined by

⟨β, x⟩ := tr(ad(Jx) − J ad(x))

is admissible called the Koszul form. This corresponds to the real part of the Hermitian inner product (up to a positive number multiple) coming from the Bergman metric on D.

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Structure of a Piatetski-Shapiro algebra

(g, J,ω): a Piatetski-Shapiro algebra, ⟨ x | y ⟩ω : J-invariant inner product.

Let n := [g,g] ; the derived ideal of g. and put a := n⊥ w.r.t. ⟨ · | · ⟩ω. Then, g = a + n with [a,n] ⊂ n. For α ∈ a∗, we put

nα := {x ∈ n ; [a, x] = ⟨α,a⟩x (∀a ∈ a)}.

Then, ∃∆ ⊂ a∗

\ {0} (♯∆ < +∞) s.t. nα {0} (∀α ∈ ∆) and g = a + ∑

n∈∆

nα.

We can choose a basis H1,. . . , Hr of a so that with Ej := −JHj (∈ n) we have

[Hj,Ek] = δjkEk. Let α1,. . . ,αr be the basis of a∗ dual to H1,. . . , Hr.

Then, the elements of ∆ are (not all possibilities occur)

1 2 (αk ± αj) (j < k), αk (k = 1,. . . ,r), 1 2 αk (k = 1,. . . ,r).

Moreover, nαk = REk (∀k = 1,. . . ,r). Define E∗

k ∈ g∗ by requiring

⟨E∗

k,Ek⟩ = 1,

E∗

k = 0 on a and on nα (α αk).

We set for s = (s1,. . . ,sr) ∈ Rr

E∗

s := s1E∗ 1 + · · · + srE∗ r .

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We write s > 0 if s1 > 0,. . . ,sr > 0. Proposition 2 The set of the admissible linear forms on g coincides with

a∗ + {E∗

s ; s > 0}.

Thus we only have to consider E∗

s (s > 0) for the inner product on g, and we put

⟨ x | y ⟩s := ⟨E∗

s,[Jx, y]⟩.

Let

g(0) := a ⊕ ∑

m>k

n(αm−αk)/2, g(1/2) :=

r

∑

k=1

nαk/2, g(1) :=

r

∑

j=1

nα j ⊕

r

∑

m>k

n(αm+αk)/2.

Then, [g(i),g(j)] ⊂ g(i + j). In particular, g(0) is a Lie subalgebra. Moreover, Jg(1) = g(0), Jg(1/2) = g(1/2). In fact,

JEj = Hj, Jn(αk+α j)/2 = n(αk−α j)/2 (k > j), Jnαk/2 = nαk/2.

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Siegel domains defined by a Piatetski-Shapiro algebra

(g, J,ω) : our Piatetski-Shapiro algebra, G(0) =: expg(0), E := E1 + · · · + Er ∈ nα1 ⊕ · · · ⊕ nαr ⊂ g(1). G(0) acts on V := g(1), and the orbit Ω := G(0)E through E is a regular open

convex cone, and G(0) acts on Ω simply transitively.

W := VC, w → w∗ : the conjugation in W relative to V. U := (g(1/2),−J) : the vector space g(1/2) made complex by −J. Q(u,u′) := 1 2 ([Ju,u′] − i[u,u′]) (u,u′ ∈ U)

turns out to be complex sesqui-linear (C-linear in the first variable, conjugate linear in the second) Hermitian map U × U → W, and Ω-positive.

• We assume that our Siegel domain D = D(Ω,Q) is defined by these data Ω,Q.
• nD := g(1) + g(1/2) is a 2-step nilpotent Lie algebra.

ND := exp nD acts on D as follows:

writing elements of ND by n(a,b) (a ∈ g(1), b ∈ g(1/2)), e have

n(a,b) · (u,w) = (u + b, w + ia + 1

2Q(b,b) + Q(u,b))

G(0) acts on V, and hence on W = VC. Also G(0) acts on U complex linearly.

• Thus we have the action of G := expg = G(0) ⋉ ND on D by C-affine auto.

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Compound power functions

G(0) = expg(0) is a semidirect product A ⋉ N(0), where A = expa, N(0) = exp n(0), n(0) := ∑

m>k

n(αm−αk)/2.

For each s ∈ Rr, let χs be the one-dimensional representation of A defined by

χs ( exp

r

∑

j=1

t jHj ) = exp ( r ∑

j=1

sjt j ) ,

and extend it to a one-dimensional representation of G(0) by setting identically equal to 1 on N(0). Fix a base point e := (0,E) ∈ D ⊂ U × W, we have diffeomorphisms

G ∋ g → g · e ∈ D, G(0) ∋ h → hE ∈ Ω.

For every s ∈ Rr, define a function ∆s on Ω by ∆s(hE) := χs(h) (h ∈ G(0)).

• ∆s is relatively invariant : ∆s(hx) = χs(h)∆s(x) (h ∈ G(0), x ∈ Ω).

Theorem 3 (Gindikin 1975, Ishi 2000) The functions ∆s are analytically continued to holomorphic functions on Ω + iV.

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(1) Bergman kernel κ(z1, z2) of D is written as (up to a positive number multiple)

κ(z1, z2) = ∆−2d−b (w1 + w∗

2 − Q(u1,u2))

(zj = (uj,wj) ∈ D),

where

dk := tr adg(1)(Hk) = tr [Hk, g(1)], bk := tr adg(1/2)(Hk).

and d = (d1,. . . ,dr), b = (b1,. . . ,br). (2) The characteristic function ϕ of Ω:

ϕ(x) := ∫

Ω∗ e−⟨λ,x⟩ dλ

(x ∈ Ω). ϕ(x) = ∆−d(x) up to a positive number multiple.

(3) Szegö kernel S(z1, z2) (the reproducing kernel of the Hardy space H2(D) over

D) is ∆−d−b (w1 + w∗

2 − Q(u1,u2)) up to a positive number multiple.

H2(D) is the Hilbert space of holomorphic functions F on D s.t. ∥F∥2 = sup

t∈Ω

∫

U

dm(u) ∫

V

F ( u, t + 1 2 Q(u,u) + ix ) 2 dx < +∞.

Remark (1) D : symmetric =⇒ d1 = · · · = dr and b1 = · · · = br. (2) (D’Atri–Dotti 1983) Irreducible D is quasi-symmetric

⇐⇒ (a) dim n(αk+α j)/2 is indep. of k, j and (b) dim nαm/2 is indep. of m.

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Cayley transforms Recall that for the case of right half-plane → the unit disk, we have z = w − 1

w + 1 .

Observe that w − 1

w + 1 = 1 − 2 w + 1 . What we need is the denominator (w + 1)−1.

• Recall the following formula for x,v ∈ Sym(n,R), and x ≫ 0:

− d dt log det(x + tv)−1

• t=0 = tr(x−1v).

Definition 4 For x ∈ Ω and s > 0, we define Is(x) ∈ V∗ by

⟨Is(x),v⟩ = − d dt log ∆−s(x + tv) (v ∈ V).

• Is(hx) = h · Is(x) (h ∈ G(0), x ∈ Ω).

In particular, Is(λx) = λ−1Is(x) (λ > 0).

• Let Ω∗ := {ξ ∈ V∗ ; ⟨ξ,v⟩ > 0 for ∀x ∈ Cl(Ω)

\ {0}}. Ω∗ is called the dual cone of Ω.

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Proposition 5 (N 2003DGA) Suppose s > 0. (1) For any x ∈ Ω, we have Is(x) ∈ Ω∗, and Is gives a bijection of Ω onto Ω∗. (2) Is(E) = E∗

s .

(3) Is is analytically continued to a rational map W → W∗. (4) Is : Ω + iV → Is(Ω + iV) is a biholomorphic bijection. Theorem 6 (Kai–N 2005)

Is(Ω + iV) = Ω∗ + iV∗ ⇐⇒ s is a positive multiple of d and Ω is selfdual.

In this case we have s1 = · · · = sr.

• Ω is said to be selfdual if Ω∗ is transferred to V by means of an appropriate inner

product, then Ω∗ = Ω.

• Cs(w) := Is(E) − 2Is(w + E) (w ∈ W). Recall that Is(E) = E∗

s .

• Cs is holomorphic on a domain containing Cl(Ω + iV).

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Definition 7 For z = (u,w) ∈ D, we define

Cs(z) = (2⟨Is(w + E), Q(u,·)⟩, Cs(w))

• Note U ∋ u′ → ⟨Is(w + E), Q(u,u′)⟩ ∈ C is a conjugate linear form on U.

Remark (1) For symmetric domains, our Cayley transform with the parameter

s1 = · · · = sr is essentially the inverse transform defined by Korány–Wolf (1965).

(2) The parameter s = d corresponds to Penney’s Cayley transform (1996). Theorem 8 (N 2003DGA) (1) Cs is a birational and biholomorphic bijection of D onto Cs(D). (2) Cs(D) is bounded in U† ⊕ W∗, where U† denotes the complex space of all conjugate linear forms on U. Theorem 9 (Kai 2007)

Cs(D) is convex if and only if D is symmetric and s satisfies s1 = · · · = sr.

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Convex realization of a homogeneous bounded domain Harish-Chandra (1956):

• Every non-cpt Hermitian symm. space is realized as a bdd domain in CN.
• This is given without using classification.
• In particular, Harish-Chandra gave two exceptional symmetric bounded

domains which E. Cartan did not treat.

• Harish-Chandra’s realization turned out to be a convex set.
• This is proved first by Hermann (1963), ant it is in fact an open unit ball w.r.t

some (Banach) norm.

• Nowadays we have a more elementary description in terms of Hermitian Jordan

triple system (JTS), and the Harish-Chandra realization is the open unit ball of the spectral norm of the JTS. For domains equivalent to tube domains (Jordan algebra case), see the book of Faraut–Korányi .

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Mok-Tsai’s Theorem (1992) Let D be an irreducible symmetric bounded domain of rank ≧ 2. If D is a convex set, then D is affinely equivalent to the Harish-Chandra realization. Gindikin’s conjecture If a bounded homogeneous domain is a convex set, it is symmetric. Ishi–Kai’s representative domain of a homogeneous bounded domain (2010) In the spirit of Bergman (1929), Ishi and Kai introduced the representative domain

D0 of a given homogeneous bounded domain D by using the Bergman kernel of

• D. The domain can be considered as a generalization to non-symmetric case of

Harish-Chandra realization. They showed that D0 coincides, up to a positive num- ber multiple, with my Cayley transform image C2d+b(D) of the corresponding Siegel domain realization D of D. Restatement of Kai’s 2007 result Let D be a homogeneous bounded domain. Then, its representative domain D0 is convex if and only if D is symmetric.

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Other symmetry characterization theorems related to Cayley transforms Recall the J-invariant inner product ⟨ · | · ⟩s on g. By the differential of the diffeomor- phic orbit map G ∋ g → g · e ∈ D, this norm is transferred to a Hermite inner product on the tangent space Te(D) = U + W, and further transferred naturally to a Hermitian inner product on U† + W∗ which we denote by (· | ·)s. The corresponding norm is denoted by ∥ · ∥s. Theorem 10 (N 2001TG) Suppose D is irreducible. Then,

• C2d+b(g · e)

s = C2d+b(g−1 · e) s (∀g ∈ G)

if and only if D is symmetric and s > 0 is positive number multiple of 2s + b.

• This theorem is used to prove that the Berezin transform on D commutes with the

Laplace–Beltrami operator Ls on D defined by using the metric ⟨ · | · ⟩s if and only if

D is symmetric and s is a positive number multiple of 2s + b (N 2001DGA)

Theorem 10 can be rephrased as follows.

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Theorem Suppose D is irreducible. Then,

∥h · 0∥s = ∥h−1 · 0∥s

for ∀h ∈ C2d+b ◦ G ◦ C −1

2d+b

if and only if D2d+b := C2d+b(D) is symmetric and s is a positive number multi- ple of 2d + b.

r

• r

1 1

• 1
• 1

P Q

SU(1,1) acts on D := { z < 1} by (α β β α ) · z := αz + β βz + α

.

a(t) := (cosh t sinh t sinh t cosh t )

,

A := {a(t) ; t ∈ R}, n(ξ) := (1 − iξ iξ −iξ 1 + iξ )

,

N := {n(ξ) ; ξ ∈ R}. G := N A is the Iwasawa subgroup of SU(1,1). P : n(ξ)a(t) · 0 = n(ξ) · tanh t ∈ N · r (r := tanh t) : green circle. Q : (n(ξ)a(t))−1 · 0 = n(−e−2tξ)a(−t) · 0 ∈ N · (−r): green circle. g · 0 = g−1 · 0 (∀g ∈ S) ⇐⇒ D is symmetric.

But this is trivial for g =

(α β β α ) ∈ SU(1,1): g · 0 = β α , g−1 · 0 = − β α .

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In the following theorem Σ denotes the Silov boundary of D. It is known that

Σ = {(u,w) ∈ U × W ; 2 Re w = Q(u,u)}.

If D = Ω + iV (U = {0}), then Σ = iV. Take Ψs ∈ g s.t. tr ad(x) = ⟨ x | Ψs ⟩s (∀x ∈ g). We know that Ψs ∈ a. Next, we put αs :=

r

∑

j=1

sjαj ∈ a∗.

Theorem 11 (N 2003JFA) Suppose D is irreducible. Then,

• Cd+b(ζ)
• 2

s = ⟨αd+b,Ψs⟩

(∀ζ ∈ Σ)

if and only if D is symmetric and s is a positive number multiple of d + b. The equality in Theorem 11 represents that the image of Σ under Cd+b lies on a sphere centered at the origin. Theorem 11 is used to prove that the Poisson-Hua kernel P(z,ζ) :=

S(z,ζ) 2 S(z, z) (z ∈ D, ζ ∈ Σ) satisfies LsP(·,ζ) = 0 (∀ζ ∈ Σ) if and only if D is symmetric and s

is a positive number multiple of d + b.

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References cited (not all)

• S. Bergman (Bergman1929), Über die Existenz von Repräsentantensbereichen

in der Theorie der Abbildung durch Paare von Funktionen zweier komplexen Verän- derlichen, Math. Ann., 102 (1929), 430–446.

• J. E. D’Atri and I. Dotti Miatello (D’Atri–Dotti1983), A characterization of bounded

symmetric domains by curvature, Trans. Amer. Math. Soc., 276 (1983), 531–540.

• S. Gindikin (Gindikin1975), Invariant generalized functions in homogeneous

domains, Funct. Anal. Appl., 9 (1975), 50–52.

• J. Faraut and A. Korányi (Faraut–Koranyi1994) Analysis on symmetric cones,

Clarendon Press, Oxford, 1994.

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VI, Amer. J. Math., 78 (1956), 564–628.

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symmetric domains II, Math. Ann., 151 (1963), 143–149.

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• H. Ishi and C. Kai (Ishi–Kai2010), The representative domain of a homogeneous

bounded domain, Kyushu J. Math., 64 (2010), 35–47.

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• C. Kai (Kai2007), A characterization of symmetric Siegel domains by convexity of

Cayley transform images, Tohoku Math. J., 59 (2007), 101–118.

• C. Kai and T. Nomura (KaiN2005), A characterization of symmetric cones through

pseudoinverse maps, J. Math. Soc. Japan, 57 (2005), 195–215.

• A. Korányi and J. Wolf (Koranyi–Wolf1965), Realization of Hermitian symmetric

spaces as generalized half-planes, Ann. of Math., 81 (1965), 265–288.

• N. Mok and I.-H. Tsai (Mok–Tsai1992) Rigidity of complex realizations of irre-

ducible bounded symmetric domains of rank ≧ 2, J. Reine Angew. Math., 431 (1992), 91–122.

• T. Nomura (N2001TG), A characterization of symmetric Siegel domains through a

Cayley transform, Transform. Groups, 6 (2001), 227–260.

• T. Nomura (N2001DGA), Berezin transforms and Laplace–Beltrami operators on

homogeneous Siegel domains, Diff. Geom. Appl., 15 (2001), 91-106.

• T. Nomura (N2003JFA), Geometric norm equality related to the harmonicity of the

Poisson kernel, J. Funct. Analysis, 198 (2003), 229–267.

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domain parametrized by admissible linear forms, Diff. Geom. Appl., 18 (2003), 55– 78.

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• R. Penney (Penney1996), The Harish-Chandra realization for non-symmetric do-

mains in Cn, In: Topics in geometry in memory of Joseph D’Atri (ed. S. Gindikin), Birkhäuser, Boston, 1996, 295–313. — Papers not in the above list are included in References of my article

• T. Nomura, Focusing on symmetry characterization theorems for homogeneous

Siegel domains, Sugaku Expositions, 23 (2010), 47–67. Unfortunately this is not yet of open access, and the preprint version is availabe at my webpage in Kyushu University (may not be permanent),

http://www2.math.kyushu-u.ac.jp/˜tnomura/nomurappt.pdf