Let A" be a Lie subgroup of the connected, simply connected nilpotent Lie group G, and let f, g be the corresponding Lie algebras. Suppose that a is an irreducible unitary representation of K. We give an explicit direct integral decomposition of IndA^c;o into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between GA and the coadjoint orbits in g* (and sim...

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In the case of the field C, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the field but this will be too coarse to give a differentiable structure. A celebrated example of how partial algebraic and topological data (G a locally euclidean group) determines a differentiable structure (G is a Lie group) is Hilbert’s 5th p...

B. Mazur has considered the question of density in the Euclidean topology of the set of Q-rational points on a variety X defined over Q, in particular for Abelian varieties. In this paper we consider the question of closures of the image of finitely generated subgroups of T (Q) in Γ\T (R) where T is a torus defined over Q, Γ an arithmetic subgroup such that Γ\T (R) is compact. Assuming Schanuel...

Let Γ be a discrete subgroup of a simply connected, solvable Lie group G, such that AdG Γ has the same Zariski closure as AdG. If α : Γ → GLn(R) is any finite-dimensional representation of Γ, we show that α virtually extends to a continuous representation σ of G. Furthermore, the image of σ is contained in the Zariski closure of the image of α. When Γ is not discrete, the same conclusions are t...

There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is natur...

I Koizumi, op. cit., 2, 267. 10 0. Zariski, "The Reduction of the Singularities of an Algebraic Surface," Ann. Math., 40, 639689, 1939; "A Simplified Proof for the Resolution of Singularities of an Algebraic Surface," ibid., 43, 583-593, 1942; S. Abhyankar, "The Theorem of Local Uniformization on Algebraic Surfaces over Modular Fields," Ann. Math. (to appear). "0. Zariski, "Pencils on an Algebr...