In this note, by a homogeneous convex domain in Rn we mean a convex domain Ω in Rn containing no complete straight lines on which the group G(Ω) of all affine transformations of Rn leaving Ω invariant acts transitively. Let Ω be a homogeneous convex domain. Then Ω admits a G(©)-invariant Riemannian metric which is called the canonical metric (see [11]). The domain Ω endowed with the canonical metric is a homogeneous Riemannian manifold and we denote by I(Ω) the group of all isometries of it. A homogeneous convex domain Ω is called reducible if there is a direct sum decomposition of thé ambient space Rn = Rn1 × Rn2, ni > 0, such that Ω = Ω1 × 02 with Ω i a homogeneous convex domain in Rni; and if there is no such decomposition, then Ω is called irreducible.

Introduction 1. If denotes the local time of a continuous semi-martingale X at a Bouleau and Yor [1] have obtained a form of Ito’s differentiation formula which states that for absolutely continuous functions F(x)

P. Lévy introduced a notion of Brownian motion with parameter in a metric space (M, d), which is a centered Gaussian system satisfying

Let f(X, T1,…, Tm ) be a polynomial over an algebraic number field k of finite degree. In his paper [2], T. Kojima proved

THEOREM. Let K = Q. if for every m integers t1, …, tm, there exists an r ∈ K such that f(r, t1, …, tm) =), then there exists a rational function g(T1,…,Tm) over Q such that

F(g(T1,…,Tm), T1,…,T)= 0.

Many authors have studied the relationship between nontrivial class numbers h(n) of real quadratic fields and the lack of integer solutions for certain diophantine equations. Most such results have pertained to positive square-free integers of the form n = l2 + r with integer >0, integer r dividing 4l and — l<r<l. For n of this form, is said to be of Richaud-Degert (R-D) type (see [3] and [8]; as well as [2], [6], [7], [12] and [13] for extensions and generalizations of R-D types.)

The asymptotic distribution of eigenvalues has been studied by many authors for the Schrõdinger operators —Δ+V with scalar potential growing unboundedly at infinity. Let N(λ) be the number of eigenvalues less than λ of —Δ + V on L2Rnx ). Under suitable assumptions on V(x), N(λ) obeys the following asymptotic formula:

In connection with a Gaussian system X = {X(x); x ∈ M} called Lévy’s Brownian motion (Definition 1), we shall introduce two integral transformations of special type—one is a generalized Radon transform R on a measure space (M, m), and the other is a dual Radon transform R* on another measure space (H, v) such that H ⊂ 2M , the set of all subsets of M (Definition 2). To each Lévy’s Brownian motion X, there is attached a distance d(x, y):= E[(X(x) — X(y)2] on M having a notable property named L1 -embeddability. The above measure v on H is then chosen to satisfy

where Δ stands for the symmetric difference.

As a continuation of the author’s paper, we shall investigate the null spaces of a dual Radon transform R *, in connection with a Lévy’s Brownian motion X with parameter space (Rn, d). We shall follow the notation used in (I).

One of the most useful theorems in classical representation theory is a result due to N. Ito, which can be stated using the classification of the finite simple groups in the following way.

THEOREM (N. Ito, G. Michler). Let Irr (G) be the set of all irreducible complex characters of the finite group G and q be a prime number. Then if and only if G has a normal, abelian Sylow-q-subgroup.

Let X be the set of all natural numbers and let be the group of all finite permutations of X. The group equipped with the discrete topology, is called the infinite symmetric group. It was discussed in F. J. Murray and J. von Neumann as a concrete example of an ICC-group, which is a discrete group with infinite conjugacy classes. It is proved that the regular representation of an ICC-group is a factor representation of type II1. The infinite symmetric group is, therefore, a group not of type I. This may be the reason why its unitary representations have not been investigated satisfactorily. In fact, only few results are known. For instance, all indecomposable central positive definite functions on , which are related to factor representations of type IIl , were given by E. Thoma. Later on, A. M. Vershik and S. V. Kerov obtained the same result by a different method in and gave a realization of the representations of type II1 in. Concerning irreducible representations, A. Lieberman and G. I. Ol’shanskii obtained a characterization of a certain family of countably many irreducible representations by introducing a particular topology in However, irreducible representations have been studied not so actively as factor representations.

Nous avons étudié en [12] des problèmes d’interpolation dans des espaces de fonctions holomorphes sur un cône ouvert convexe de sommet l’origine dans Cn, la croissance de ces fonctions étant contrôlée à l’infini. Nous nous intéressons maintenant à un espace plus petit en imposant en outre un contrôle de croissance à l’origine dans le cas où le cône est strict.

Let G be a finite group and ℒ(G) the lattice consisting of all subgroups of G. It is well known that ℒ(G) is distributive if and only if G is cyclic (cf. [2, p. 173]). Moreover, the classical result of Iwasawa [8] says that ℒ(G) is pure if and only if G is supersolvable. Here, a finite lattice is called pure if all of maximal chains in it have same length and a finite group G is called supersolvable if ℒ(G) has a maximal chain which consists of normal subgroups of G.

The generalized theta function of a totally imaginary field including n-th roots of unity, which was defined by T. Kubota [2], was introduced in his investigation of the reciprosity law of the n-th power residue. If n = 2, it reduces to the classical theta function. In the case n = 3 for the Eisenstein field, the Fourier coefficients of the cubic theta function, which were explicitly expressed by S.J. Patterson, are essentially cubic Gauss sums [3], Furthermore in the case n = 4 for the Gaussian field those of the biquadratic theta functions, which have been investigated by T. Suzuki [4], haven’t been obtained completely yet.

Let Γ be a fuchsian group of the first kind and assume that Γ contains the element and let x be a unitary representation of Γ of degree 1 such that X(—I) = — 1. Let S1(Γ,X ) be the linear space of cusp forms of weight one on the group Γ with character X. We shall denote by d1 the dimension of the linear space S1(Γ, X). It is not effective to compute the number dl by means of the Riemann-Roch theorem. Because of this reason, it is an interesting problem in its own right to determine the number d1 by some other method (for example,).