Armand Borel

Linear algebraic groups

Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.

Continuous cohomology, discrete subgroups, and representations of reductive groups

The Description for this book, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. (AM-94), will be forthcoming.

Stable real cohomology of arithmetic groups

Given a discrete subgroup Γ of a connected real semisimple Lie group G with finite center there is a natural homomorphism

Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups

j_\Gamma ^q:I_G^q \to {H^q}\left( {\Gamma ;c} \right)\quad \left( {q = 0,1, \ldots } \right),

Semisimple groups and Riemannian symmetric spaces

(1) where I G q denotes the space of G-invariant harmonic q-forms on the symmetric space quotient X=G/K of G by a maximal compact subgroup K. If Γ is cocompact, this homomorphism is injective in all dimensions and the main objective of Matsushima in [19] is to give a range m(G), independent of Γ, in which j Γ q is also surjective. The main argument there is to show that if a certain quadratic form depending on q is positive non-degenerate, then any Γ-invariant harmonic q-form is automatically G-invariant. In [3], we proved similarly the existence of a range in which j Γ q is bijective when Γ is arithmetic, but not necessarily cocompact. There are three main steps to the proof: (i) The cohomology of Γ can be computed by using differential forms which satisfy a certain growth condition, “logarithmic growth,” at infinity; (ii) up to some range c(G), these forms are all square integrable; and (iii) use the fact, pointed out in [16] , that for q ≦ m(G), Matsushima’s arguments remain valid in the non-compact case for square integrable forms.

Sheaf Theoretic Intersection Cohomology

We recall that a Riemannian manifold X is symmetric, in the sense of Cartan, if it is connected and if every point x E X is an isolated fixed point of an involutive isometry s, of X. The map s, is then the unique isometry of X leaving x fixed and whose differential at x is -Id. The group I(X) of isometries of X is transitive, and X may be identified with the quotient f(X)“/K of the identity component Z(X)’ of I(X) by a compact subgroup.

Values of indefinite quadratic forms at integral points and flows on spaces of lattices

Supported by the National Science Foundation during 1986–1988 at the Mathematical Sciences Research Institute, Berkeley, and at the Institute for Advanced Study, Princeton.

Commutative subgroups and torsion in compact Lie groups

This article is a collection of letters solicited by the editors of the Bulletin in response to a previous article by Jaffe and Quinn [math.HO/9307227]. The authors discuss the role of rigor in mathematics and the relation between mathematics and theoretical physics.