Algebras of invariant differential operators on a class of multiplicity free spaces
Résumé
Let G be a connected reductive algebraic group and let G'=[G,G] be its derived subgroup. Let (G,V) be a multiplicity free representation with a one dimensional quotient (see definition below). We prove that the algebra D(V)^{G'} of G'-invariant differential operators with polynomial coefficients on V, is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith as a class of algebras similar to the enveloping algebra U(sl(2)) of sl(2). Our result generalizes the case of the Weil representation, where the associative algebra generated by Q(x) and Q(∂) (Q being a non degenerate quadratic form on V) is a quotient of U(sl(2)) Other structure results are obtained when (G,V) is a regular prehomogeneous vector space of commutative parabolic type.
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