On a Reductive Lie Algebra and Weyl Group Representations
نویسندگان
چکیده
Let V be a finite-dimensional vector space, and let G be a subgroup of GL( V). Set D( V) equal to the algebra of differential operators on V with polynomial coefficients and D( V) G equal to the G invariants in D( V). If 9 is a reductive Lie algebra over C then ~ egis a Cartan subgroup of g, and if G is the adjoint group of 9 then W is the Weyl group of (g, ~) , Harish-Chandra introduced an algebra homomorphism, J, of D(g)G to D(~)w [H3]. J isgiven by the obvious restriction mapping on the subalgebra of invariant polynomials and on the invariant constant coefficient differential operators, and ker J is the ideal, ..Y, of D(g)G consisting of elements that annihilate all G invariant polynomials. In this paper we prove that if 9 has no factor of type E then J is surjective. We also prove that for general g, the homomorphism is surjective after localizing by the discriminant of g. If go is a real form of 9 and if Go is the adjoint group of go then Harish-Chandra has shown that ..Y is precisely the ideal in D(g)G of operators that annihilate all Go invariant distributions on "completely invariant" open subsets of go [H2]. Our first application of our analysis of J is to give a new proof of this important theorem. In light of this theorem the space of Go -invariant distributions on a completely invariant open subset of go is a D(g)G-module that "pushes down" to w w a D(~) -module. To analyze these D(~) -modules we develop a theory analogous to Howe's formalism of dual pairs, proving an equivalence of categories between an appropriate category of D(~)w-modules and the category of all W-modules over C. We show that the D(g)G-module of distributions on go supported in the nilpotent cone of go is (as a D(~)w module) in our category. Thus, to each distribution supported on the nilpotent cone we can associate a (finite dimensional) representation of W. If the distribution is the orbital integral corresponding to a fixed nilpotent element of go then we prove that the representation of W is irreducible and derive a formula for the Fourier transform of the orbital integral in terms of W -harmonic polynomials corresponding
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