Several systems of partial differential operators are associated to each complex simple Lie algebra of rank greater than one. Each system is conformally invariant under the given algebra. The systems so constructed yield explicit reducibility results for a family of scalar generalized Verma modules attached to the Heisenberg parabolic subalgebra of the given Lie algebra. Points of reducibility for such families lie in the union of several arithmetic progressions, possibly overlapping. For classical algebras, enough systems are constructed to account for the first point of reducibility in each progression. The relationship between these results and a conjecture of Akihiko Gyoja is explored. §1. Introduction To describe our results, it is first necessary to set the scene. In the body of the paper we shall work most of the time in an exclusively algebraic framework, but it will be useful here to take a more inclusive viewpoint, mixing the analytic and the algebraic. We shall first attempt to explain the significance of our results and place them in context. To some extent, these remarks may be taken as an introduction to a broader investigation of which this work and [2] are the first fruits. Then we shall draw a map to aid the reader in navigating on the admittedly lengthy journey through the proofs.

The exceptional representations are certain infinite-dimensional projective representations of the general linear group over a local field, somewhat analogous to the Weil representations of the symplectic group. We examine the decomposition of the tensor product of two exceptional representations. Our main results concern the multiplicity with which a given representation may occur in this product and the restrictions imposed upon a representation of the principal series by the assumption that it does occur.  2001 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Les représentations exceptionnelles sont certaines représentations projectives de dimension infinie du groupe linéaire général sur un corps local, apparentées aux représentations de Weil du groupe symplectique. Nous étudions la décomposition du produit tensoriel de deux représentations exceptionnelles. Nos résultats principaux concernent la multiplicité avec laquelle une représentation donnée apparaît dans ce produit, et les restrictions imposées à une représentation de la série principale par le fait qu'elle apparaît dans ce produit.

We give a conceptually simple proof of the square-integrable case of a conjecture of Jacquet concerning distinguished representations of the general linear group over a local field of characteristic zero. The proof is based on consideration of the Rankin-Selberg integral representation of the generalized Asai L-function and utilizes global methods.

Abstract. Let E/F be a quadratic extension of p-adic fields. If π is an admissible representation of GLn(E) that is parabolically induced from discrete series representations, then we prove that the space of Pn(F )-invariant linear functionals on π has dimension one, where Pn(F ) is the mirabolic subgroup. As a corollary, it is deduced that if π is distinguished by GLn(F ), then the twisted tensor L-function associated to π has a pole at s = 0. It follows that if π is a discrete series representation, then at most one of the representations π and π ⊗ χ is distinguished, where χ is an extension of the local class field theory character associated to E/F . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.