Dixmier algebras, sheets, and representation theory.

*(English)*Zbl 0854.17010
Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honour of J. Dixmier, Paris/Fr. 1989, Prog. Math. 92, 333-395 (1990).

[For the entire collection see Zbl 0719.00018.]

From the introduction: One of the grand unifying principles of representation theory is the method of coadjoint orbits. After impressive successes in the context of nilpotent and solvable Lie groups, however, the method encountered serious obstacles in the semisimple case. Known examples (like \(SL (2, \mathbb{R})\)) suggested a strong connection between the structure of the unitary dual and the geometry of the orbits, but it proved very difficult to formulate any precise general conjectures that were entirely consistent with these examples.

In the late 1960’s, Dixmier suggested a way to avoid some of these problems. Motivated in part by the theory of \(C^*\)-algebras, he suggested that one should temporarily set aside a direct study of unitary representations and concentrate instead on their annihilators in the universal enveloping algebra. Classification of the annihilators would be a kind of approximation to the classification of the unitary representations themselves. The hope was that this approximation would be crude enough to be tractable, and yet precise enough to provide useful insight into the unitary representations themselves. This hope has been abundantly fulfilled: the two classification problems are now inextricably intertwined, and they continue constantly to shed new light on each other. …

Here is a more detailed outline of the contents of this paper. Section 2 recalls from D. Vogan [CMS Conf. Proc. 5, 281-316 (1986; Zbl 0585.17008)] and W. McGovern [Math. 69, 241-276 (1989; Zbl 0689.17006)] the definition of Dixmier algebras and orbit data, and a corresponding refinement of Conjecture 1.3 (Dixmier): Suppose \(G\) is a complex connected Lie group with Lie algebra \({\mathfrak g}\), and \({\mathcal O}_\mathbb{C}\) is an orbit of \(G\) on \({\mathfrak g}^*\). Then there is attached to \({\mathcal O}_\mathbb{C}\) a completely prime primitive ideal \(I({\mathcal O}_\mathbb{C})\) in \(U({\mathfrak g})\).

(One of McGovern’s results (loc. cit.) is that the main conjecture in the author’s paper cited above is false; and McGovern has since found counterexamples for a revision circulated in an earlier version of this paper. Conjecture 2.3 = Conjecture 1.3 replaced by a conjectural map from orbit data to Dixmier algebras, appears to be consistent with all of his work to date.) Section 3 outlines the extension to orbit data of some of the basic structure theory for coadjoint orbits: Jordan decomposition, parabolic induction, and sheets. In the theory of sheets we find some strong (conjectural!) geometric evidence for the correctness of the general approach to the Dixmier conjecture in the author’s paper cited above: Conjecture 3.24 says that distinct sheets of “orbit data” should be disjoint. The failure of the corresponding fact for sheets of orbits is at the heart of the non-uniqueness problems discovered by Borho and discussed above. Section 4 presents the construction of Dixmier algebras by parabolic induction. Section 5 outlines how these ingredients should fit together to define a Dixmier map for \(G\).

The rest of the paper is devoted to related technical results. In section 6, we relate induction of Dixmier algebras to ordinary induction of Harish-Chandra bimodules. Perhaps the most important consequence is a cohomology vanishing theorem (Corollary 6.16). This generalizes the fact that the higher cohomology of \(G/Q\) with coefficients in the sheaf of differential operators is zero. Section 7 considers the translation principle for induced Dixmier algebras and their modules.

A key tool in all the induction constructions (both for orbit data and for Dixmier algebras) is the notion of equivariant bundles on homogeneous spaces (in the algebraic category). These help to formalize the idea that \(G\)-equivariant constructions on \(G/H\) are equivalent to \(H\)-equivariant constructions at a point. A few of the basic definitions and results are summarized in an appendix for the convenience of the reader.

From the introduction: One of the grand unifying principles of representation theory is the method of coadjoint orbits. After impressive successes in the context of nilpotent and solvable Lie groups, however, the method encountered serious obstacles in the semisimple case. Known examples (like \(SL (2, \mathbb{R})\)) suggested a strong connection between the structure of the unitary dual and the geometry of the orbits, but it proved very difficult to formulate any precise general conjectures that were entirely consistent with these examples.

In the late 1960’s, Dixmier suggested a way to avoid some of these problems. Motivated in part by the theory of \(C^*\)-algebras, he suggested that one should temporarily set aside a direct study of unitary representations and concentrate instead on their annihilators in the universal enveloping algebra. Classification of the annihilators would be a kind of approximation to the classification of the unitary representations themselves. The hope was that this approximation would be crude enough to be tractable, and yet precise enough to provide useful insight into the unitary representations themselves. This hope has been abundantly fulfilled: the two classification problems are now inextricably intertwined, and they continue constantly to shed new light on each other. …

Here is a more detailed outline of the contents of this paper. Section 2 recalls from D. Vogan [CMS Conf. Proc. 5, 281-316 (1986; Zbl 0585.17008)] and W. McGovern [Math. 69, 241-276 (1989; Zbl 0689.17006)] the definition of Dixmier algebras and orbit data, and a corresponding refinement of Conjecture 1.3 (Dixmier): Suppose \(G\) is a complex connected Lie group with Lie algebra \({\mathfrak g}\), and \({\mathcal O}_\mathbb{C}\) is an orbit of \(G\) on \({\mathfrak g}^*\). Then there is attached to \({\mathcal O}_\mathbb{C}\) a completely prime primitive ideal \(I({\mathcal O}_\mathbb{C})\) in \(U({\mathfrak g})\).

(One of McGovern’s results (loc. cit.) is that the main conjecture in the author’s paper cited above is false; and McGovern has since found counterexamples for a revision circulated in an earlier version of this paper. Conjecture 2.3 = Conjecture 1.3 replaced by a conjectural map from orbit data to Dixmier algebras, appears to be consistent with all of his work to date.) Section 3 outlines the extension to orbit data of some of the basic structure theory for coadjoint orbits: Jordan decomposition, parabolic induction, and sheets. In the theory of sheets we find some strong (conjectural!) geometric evidence for the correctness of the general approach to the Dixmier conjecture in the author’s paper cited above: Conjecture 3.24 says that distinct sheets of “orbit data” should be disjoint. The failure of the corresponding fact for sheets of orbits is at the heart of the non-uniqueness problems discovered by Borho and discussed above. Section 4 presents the construction of Dixmier algebras by parabolic induction. Section 5 outlines how these ingredients should fit together to define a Dixmier map for \(G\).

The rest of the paper is devoted to related technical results. In section 6, we relate induction of Dixmier algebras to ordinary induction of Harish-Chandra bimodules. Perhaps the most important consequence is a cohomology vanishing theorem (Corollary 6.16). This generalizes the fact that the higher cohomology of \(G/Q\) with coefficients in the sheaf of differential operators is zero. Section 7 considers the translation principle for induced Dixmier algebras and their modules.

A key tool in all the induction constructions (both for orbit data and for Dixmier algebras) is the notion of equivariant bundles on homogeneous spaces (in the algebraic category). These help to formalize the idea that \(G\)-equivariant constructions on \(G/H\) are equivalent to \(H\)-equivariant constructions at a point. A few of the basic definitions and results are summarized in an appendix for the convenience of the reader.

##### MSC:

17B35 | Universal enveloping (super)algebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |