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Abstract: Consider the enveloping algebra of a complex semisimple Lie algebra . The heart of a prime ideal I of is the center of the total ring of fractions of . This is an extension field of the field of fractions of the center of . Let d be the degree of this field extension. An old problem of J. Dixmier asked whether . A recent paper of the second author [R. Rentschler, A negative answer to the problem of Dixmier on hearts of prime quotients of enveloping algebras, preprint, 2004] gave a negative answer by an example in . The present paper provides many more examples, involving the so-called sheets of primitive ideals introduced and studied by A. Joseph and the first author in [W. Borho, A. Joseph, Sheets and topology of primitive spectra for semisimple Lie algebras, J. Algebra 244 (2001) 76–167]. A sheet corresponds to a prime ideal I which has a heart of degree d. The main result of this paper is that d equals the covering degree of the sheet as introduced in [W. Borho, A. Joseph, Sheets and topology of primitive spectra for semisimple Lie algebras, J. Algebra 244 (2001) 76–167, 8.7]. [Copyright &y& Elsevier]