Global Solvability of the Casimir Operator

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Let G be a semisimple Lie group, g its Lie algebra. The Killing form on g is a nondegenerate symmetric bilinear form which induces an isomorphism a: g* -. g. The Casimir operator is the unique biinvariant differential operator on G which annihilates constants and whose principal symbol is the pullback of the Killing form under the isomorphism a. Let C denote the Casimir operator and C-(G) the space of infinitely differentiable functions on G. THEOREM. If G is a non-compact semisimple Lie group with finite center, f e C-(G), then there is a u e C-(G) such that Cu = f. Since G is not compact, C is not elliptic but ultrahyperbolic. Analogous questions have been considered by S. Helgason [3] and A. Cerezo and F. Rouviere [1]. Helgason proves local solvability of all biinvariant differential operators and Cerezo and Rouviere prove global solvability of the Casimir operator on complex semisimple Lie groups. The latter authors prove that the imaginary part of the complex Casimir operator on SL(2, C) is not globally solvable. This shows that not every biinvariant operator is globally solvable. Henceforward G will denote a group satisfying the hypotheses of the theorem and g will denote the Lie algebra of left invariant vector fields on G. Proof. The remainder of the paper is devoted to a proof of this theorem. In spite of its length the proof is simple in outline. A result from functional analysis and Hormander's propagation of singularities theorem can be combined to show that it suffices to demonstrate three things.