In this note, we study square integrable representations of a real reductive Lie group with admissible restriction to some reductive subgroup. We give a simple condition which insures admissibility of the restriction, and which allows to compute the branching numbers in a simple explicit manner by means of partition functions, generalizing the multiplicity formulas due to Kostant-Heckman and Hecht-Schmid. We consider also the semi-classical analogue of these results for coadjoint orbits.

We consider the Chebychev semigroup defined on the interval $\left [-1,+1\right ]$ by its Dirichlet form ${\int _{-1}^{+1}}(1-x^2)f^{\prime 2}(x)\, {\frac {dx}{ \pi \sqrt {1-x^2}}}$. We prove, via a method involving probabilistic techniques, a family of inequalities which interpolate between the Sobolev and Poincaré inequalities.

We introduce an operator which enable us to give a simple construction of the isomorphism from the so-called Fock space to the L²-space with respect to a Poisson measure without combinatorial arguments in Schmidt's orthogonalization procedure.

We give a complete description of the integral cohomology ring of the flag manifold E₈/T, where E₈ denotes the compact exceptional Lie group of rank 8 and T its maximal torus, by the method due to Borel and Toda. This completes the computation of the integral cohomology rings of the flag manifolds for all compact connected simple Lie groups.

In this paper, we give some discreteness criteria for a non-elementary Möbius group by using an elliptic element as a test map