In a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg’s type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol(G/Γ)<∞. Thus, Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ, and we shall consider how one can define a zeta function ZΓ of Selberg’s type attached to the data (G, K, Γ).

N-ary algebras are modules with a n-fold multiplication which we assume to be associative if nothing else is stated. They are a canonical generalization of binary and ternary associative algebras. Ternary rings were first investigated by Lister [8]. The aim of this note is to show that the Wedderburn structure theory and the usual cohomology for binary associative algebras can be extended to n-ary algebras. For ternary algebras this has been done in [8] and [1]. Moreover analogous results are wellknown for Lie and alternative triple systems, and for ternary Jordan pairs.

In dieser Arbeit werden die Voraussetzungen untersucht, die an einen topologischen Raum (X, ) mit dem System offener Mengen zu stellen sind, damit einige der in der Literatur geläufigen Begriffe der Borelmeβbarkeit reellwertiger Funktionen äquivalent sind.

As for the construction of moduli spaces of stable sheaves, the boundedness of semi-stable sheaves is one of the most important questions which are left unanswered. In the case of dimension one, the boundedness was proved by M. F. Atiyah [1]. When the dimension of the base variety is two and the rank is two, F. Takemoto and D. Mumford showed the boundedness independently ([13]). The author proved in [7] that the boundedness holds for every rank in the case of dimension two, and then D. Gieseker gave another proof of it in [3].

Since the representations of SL(2, C) induced by irreducible unitary representations of appear as the restriction to the Lorentz group of some irreducible unitary representations of the inhomogeneous Lorentz group, the decomposition of the induced representations deserves our investigation. For the representations of SL(2, C) induced by irreducible unitary representations with discrete spin of E(2), the decomposition has been obtained by Mukunda [9]. We hope that our analysis will justify the calculations by Chakrabarti [1], [2] and [3].

Since E. Wigner set up a framework of the relativistically covariant quantum mechanics, several aspects of unitary representations of the Poincaré group have been investigated (see [8], [16]). In this paper it will be shown that some unitary representations of the Poincaré group are irreducible, even if they are restricted to the Poincaré semigroup (Theorem 1, 2 and 3).

Let (, g) be the standard Euclidean space or a Riemannian symmetric space of non-compact type of rank one. Let G be the identity component of the Lie group of all isometries of (, g). Let Γ be a discrete subgroup of G acting fixed point freely on whose quotient manifold MΓ is compact.

A compact simply connected homogeneous complex manifold is called a C-space. A C-space is said to be kählerian if it carries a Kähler metric. It is known (Matsushima [7]) that a kählerian C-space has always an Einstein Kähler metric which is essentially unique.

Let M be an n-dimensional hypersurface immersed in the (n + 1)-dimensional unit sphere Sn+1 with the standard metric by an immersion f. We denote by A the second fundamental form of the immersion / which is considered as a symmetric linear transformation of each tangent space TXM, i.e. for an arbitrary point x of M and the unit normal vector field ξ defined in a neighborhood of x, A is given by where is the covariant differentiation in Sn+i and Thus, A depends on the orientation of the unit normal vector field ξ and, in general, it is locally defined on M.