arXiv subjects
Publications and source records attributed to Pierre-Yves Gaillard.
Is a Verma module transformed into another Verma module by a selfequivalence? The answer is affirmative and the proof suggests a notion of standard object in the category of Harish-Chandra modules that coincides often, but not always, with the usual one.
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.
I try to find natural statement and proof of the de Rham Theorem and of other cohomology theorems.
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.
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.
We prove an explicit Chinese Remainder Theorem for one variable polynomials with complex coefficients, and derive some consequences.
The groupoid attached to the action of PSL(2,Z) on the irrational reals by linear fractional transformations is free.
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.
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.
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.
We write down the functional equation of the zeta function of a global field. This equation is implicit in Weil's ``Basic Number Theory''.
We give examples of pairs (G1,G2) where G1 is a Grothendieck category and G2 a full Grothendieck subcategory of G1, the inclusion G2 --> G1 being denoted i, for which R^+i : D^+G2 --> D^+G1 (or even Ri : DG2 --> DG1) is a full embedding. This yields generalizations of some results of Bernstein and Lunts, and of Cline, Parshall and Scott.
We give a formula for matrix exponentials and partial fraction decompositions.
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.