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ON THE COMMUTATIVITY OF THE ALGEBRA OF INVARIANT DIFFERENTIAL OPERATORS ON

By Certain Nilpotent, Homogeneous Spaces, Hidénori Fujiwara, Gérard Lion and Salah Mehdi

Abstract

Abstract. Let G be a simply connected connected real nilpotent Lie group with Lie algebra g, H a connected closed subgroup of G with Lie algebra h and β ∈ h ∗ satisfying β([h, h]) = {0}. Let χβ be the unitary character of H with differential 2 √ −1πβ at the origin. Let τ ≡ IndG Hχβ be the unitary representation of G induced from the character χβ of H. We consider the algebra D(G, H, β) of differential operators invariant under the action of G on the bundle with basis H\G associated to these data. We consider the question of the equivalence between the commutativity of D(G, H, β) andthe finite multiplicities of τ. Corwin and Greenleaf proved that if τ is of finite multiplicities, this algebra is commutative. We show that the converse is true in many cases. 1. Notations and formulation of the question Let G be a simply connected connected real nilpotent Lie group with Lie algebra g and H a connected closed subgroup of G with Lie algebra h. Forl ∈ g ∗,wedenote by g(l) the Lie algebra of the stabilizer G(l) ofl under the co-adjoint action Ad ∗ of G on g ∗. For β ∈ h ∗ satisfying β([h, h]) = {0}, the homomorphism β induces a character χβ of H with 2 √ −1πβ as differential at the origin. We then form the unitary induced representation τ ≡ Ind G H χβ of G in Hτ realized, in the usual way, as the completion of a vector subspace of C ∞ (G, H, β), namely the vector space of the C ∞ complex functions f on G satisfying the following covariance relation: (1.1) f(hg) =χβ(h)f(g) ∀h ∈ H ∀g ∈ G. The action of G is given by right translations

Year: 2001
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.4539
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