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Let U be the enveloping algebra of a semisimple complex Lie algebra, ^ a regular maximal ideal of its center. We show that the Hochschild cohomology ring of Uf-^U is just the coinvariant algebra of the Weyl group. This turns out to be almost immediate if one uses localization. Let 0 be a complex semisimple Lie algebra, f) a Cartan subalgebra, U = ?7(fl) D Z the enveloping algebra with its center. Let S = S (f)*) be the regular functions on t) and S^~ C S the maximal ideal of all functions vanishing at 0 € ()• The Weyl group W acts on S and we consider the coinvariant algebra C == S / ( S ~ } ~ ) } / V S , the quotient of S by the ideal generated by all W-invariant functions vanishing at zero. THEOREM 1. Let \ C Z be a regular maximal ideal, i.e. the kernel of a regular central character. Then the Hochschild cohomology ring of U/^U is the coinvariant algebra mth its obvious grading doubled, Hir(u/xU) = c. Remark. I thank Patrick Polo for telling me this problem along with the expected answer, and for his helpful comments on a first version. Proof. By the Hochschild cohomology ring of an associative fc-algebra A we mean just the ring HH^^A) = Ext^^opp(A,A) of self-extensions of A considered as an A ®fc A-module. I now give an outline of the proof. Let X be the variety of all Borel subalgebras of g. Under localization [BB81] the bimodule Vf-^U becomes a P-module M for some 1991 Mathematics Subject Classification. Primary 17B20; Secondary 22E46. ANNALES SCIENTIFIQUES DE L'6coLE NORMALE SUPERIEURE. 0012-9593/96/047$ 4.00/© Gauthier-Villars

The Hochschild cohomology ring of regular maximal primitive quotients of enveloping algebras of semisimple Lie algebras