Algebras of invariant differential operators on a class of multiplicity free spaces

Details Algebras of invariant differential operators on a class of multiplicity free spaces Algebras of invariant differential operators on a class of multiplicity free spaces

2009-10-29

arxiv.org/abs/0910.5664v1

Comptes Rendus de l Acad\'emie des Sciences - Series I - Mathematics (2009) Rubenthaler

Mathematics

Representation Theory

Scientific paper

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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