Four multiplicative cohomology theorems

I try to find natural statement and proof of the de Rham Theorem and of other cohomology theorems.Comment: 18 pages, Te

A naive question about quantum groups

The category O of BGG can be thought of as a category of sheaves over the flag variety F in the sense that the algebra E of self-extensions of the trivial object of O is isomorphic to the cohomology algebra of the flag variety. A deformation of O' - giving rise to a "new" algebra E' - can be thought of as a (possibly noncommutative) deformation F' of F. The mythic variety F', being a deformation of F, should have the same homotopy type as F, and E' should therefore be isomorphic to E.Comment: 4 pages, Te

A simple question about a complicated object

Let n and k be positive integers with and k < n. Then of course SU(k,1) is contained into SU(n,1). Moreover, which is less clear - but proved by Khoroshkin -, the representation theory of SU(k,1) at the generalized infinitesimal character of the trivial module can be fully (and even Ext-fully) embedded into that of SU(n,1). Here is the obvious bet: This embedding is implemented by the cohomological induction functor. I conjecture that a similar phenomenon occurs whenever SU(k,1) is a Levi factor of a theta stable parabolic subalgebra of a reductive group.Comment: 4 pages, Te

Integral Congruences

To each i, j belonging to some set of integers, attach the integer a(i,j). Are there integers x(i) such that x(j)-x(i) is congruent to a(i,j) mod (i,j)? A necessary condition is that a(i,j)+a(j,k) be congruent to a(i,k) mod (i,j,k). This condition is sufficient.Comment: 9 pages, LaTeX. Results have been improve

About a Theorem of Cline, Parshall and Scott

We give a simple proof of a Theorem of Cline, Parshall and Scott about the category O of BGG and suggest an analog for Harish-Chandra modules.Comment: 6 pages, LaTeX. Related material is available at http://www.iecn.u-nancy.fr/~gaillar

A Hodge Theorem for Noncompact Manifolds

If M is a riemannian manifold, then the inclusion of the complex of coclosed harmonic forms into the de Rham complex induces a linear isomorphism in cohomology. If M has at most countably many connected components, this linear isomorphism is a Frechet isomorphism.Comment: For the last version of this text and additional comments, go to http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Hodgegaillard/, or type hodgegaillard on Googl

The Gauss-Dirichlet Orbit Number

Dirichlet computed in some particular cases the number of equivalence classes of representations of a nonzero integer by a representative system for the integral binary quadratic forms of a given discriminant. We complete this computation.Comment: I changed one word in the abstrac

The functional equation of the zeta function of a global field

We write down the functional equation of the zeta function of a global field. This equation is implicit in Weil's ``Basic Number Theory''.Comment: 2 pages, LaTe

Statement of the Alexandru Conjecture

The Vogan Conjectures (sometimes called Kazhdan-Lusztig Conjectures) say that a certain algorithm works both on the category of BGG modules and on the category of Harish-Chandra modules. The Alexandru Conjecture tries to uncover the general property common to these two categories which makes Vogan's algorithm work.Comment: 19 pages, Te

Matrix exponentials

We give a formula for matrix exponentials and partial fraction decompositions.Comment: 1 page, LaTe