Some infinite dimensional representations of reductive groups with Frobenius maps

In this paper,we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps.In particular,a few classical results of Steinberg and Deligne&Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.     

Regular representations of discrete groups

Kaniuth [3] has given necessary and sufficient conditions for the regular representation of a discrete group to be type I or type II. We give a non-measure-theoretic proof of his results, plus additional information on the structure of the W^∗ algebra generated by the regular representation.     

Some remarks on faster convergent infinite series

A structure on terms of faster convergent series is studied in the paper. Necessary and sufficient conditions for the existence of faster convergent series with different types of terms are found. A faster convergence criteria for certain Kummer??s series is proved in this paper.     

Some remarks on just-inpinitb groups

Let B(?) be the group of homeomcrphisms of the real line having bounded support. Higman has shown that (?) is simple. Schreier conjectured in 1928 that for any simple group, the outer automorphism groiftp is solvable. But it turns out that not only is Outer(?) non-sol table, but its derived series stabilizes after one step. Thus, although the counterexample is infinite, solvability fails as it must in finite groups (rather than because the derived series descends ad infinitum). For another (new) simple group G, Outer(G) is its own derived group. A countable counterexample is also given. Most important among the new simple groups is the group of diffeomorphisms (differentiable, but not necessarily C¹) of the real line having bounded support.     

Some remarks on linear groups

Three problems concerning linear groups are solved.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 635–638, December, 1968.     

Constructing irreducible representations of discrete groups

The decomposition of unitary representations of a discrete group obtained by induction from a subgroup involves commensurators. In particular Mackey has shown that quasi-regular representations are irreducible if and only if the corresponding subgroups are self-commensurizing. The purpose of this work is to describe general constructions of pairs of groups Γ₀ with Γ its own commensurator in Γ. These constructions are then applied to groups of isometries of hyperbolic spaces and to lattices in algebraic groups.     

Some remarks on Rademacher's theorem in infinite dimensions

We provide an infinite dimensional version of Rademacher's theorem in a linear space provided with a bounded Radon measure . The underlying concepts of the Lipschitz property and differentiability hold -almost everywhere and only in the linear subspace of directions along which is quasiinvariant. The particular case where (X, ) is the Wiener space (and for which the subspace of quasiinvariance coincides with the Cameron-Martin space) was proved in Enchev and Stroock (1993).Partially supported by the fund for promotion of research at the Technion.     

Some remarks on conjugate closed groups

Let G be a group. In this note we define conjugate closed groups, which are briefly called CC — Groups. These groups form a proper subclass of T — Groups. We prove that if G = Z(G) × H, then G is conjugate closed if and only if H is conjugate closed. We also show that a finite group G is semisimple, conjugate closed and perfect if and only if it is a direct product of non-abelian and simple groups.     

Some remarks on isologic extensions of groups

In this paper we study the concept of -isologisms among the -marginal extensions of groups, with respect to a given variety of groups . We also give some equivalent conditions under which two extensions are -isologic. Received: 9 January 2002     

Some remarks on the degrees of faithful irreducible representations

We study a class of finite p-groups admitting faithful irreducible complex representations of distinct degrees. In particular, for every prime p, we produce an example of such a p-group having minimal order p ⁸.Dedicated to A. Wagner on the occasion of his 60th birthdayLavoro eseguito nell'ambito dei finanziamenti M.U.R.S.T.     

On irreducible representations of discrete Heisenberg groups

We prove that the irreducible representations with finite weight of discrete nilpotent groups of class 2, including the discrete Heisenberg groups, are monomial representations and obtain their classification for the group Heis(3, ?).     

On holomorphic representations of discrete Heisenberg groups

We define a class of monomial not necessary unitary representations of discrete Heisenberg groups. Their moduli space is a complex-analytic manifold. There exist characters of the representations which are automorphic forms on the moduli space.