Commutativite´ des ope´rateurs diffe´rentiels sur l'espace des repre´sentations restreintes d'un groupe de Lie nilpotent

Let <f>G</f> be a connected simply connected nilpotent Lie group, <f>K</f> an analytic subgroup of <f>G</f> and <f>π</f> a unitary and irreducible representation of <f>G</f>. We study the algebra <f>Dπ(G)K</f> of differential operators keeping invariant the space of <f>C+∞</f> vectors of <f>π</f> and commuting with the action of <f>K</f> on that space. We show that the commutativity of <f>Dπ(G)K</f> is equivalent to the fact that representation <f>π|K</f> has finite multiplicities. We also study the commutativity of the centralizer of a subalgebra. [Copyright &y& Elsevier]