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On the classification of primitive ideals for complex classical Lie algebras. III. (English) Zbl 0798.17007
Let \({\mathfrak g}\) be a complex semisimple Lie algebra of classical type and \(U({\mathfrak g})\) its enveloping algebra. The present paper is the third in a series of works by the author [Part I, Compos. Math. 75, 135-169 (1990; Zbl 0737.17003); Part II, ibid. 81, 307-336 (1992; Zbl 0762.17007)] aimed at simplifying the Barbasch-Vogan classification of the primitive spectrum \(\text{Prim} U({\mathfrak g})\) [see D. Barbasch and D. Vogan, Math. Ann. 259, 153-199 (1982; Zbl 0489.22010)] and the conjecture of D. Vogan that primitive ideals can be classified by their so-called generalized \(\tau\)-invariants [Math. Ann. 242, 209-224 (1979; Zbl 0405.17009)]. These last can be calculated in a purely combinatorial way. The first two papers laid the groundwork for the main classification theorem. The author defined standard domino tableaux (an elaboration of Young tableaux), introduced an equivalence relation on them together with a distinguished representative of every equivalence class, and showed how to compute the effect of a Vogan map \(T_{\alpha\beta}\) on tableaux, where \(\alpha\) and \(\beta\) are adjacent simple roots. The present paper completes the proof of the classification theorem in types \(B\) and \(C\); the projected fourth and last paper of the series will take care of type \(D\). In that case Vogan’s conjecture from his 1979 paper cited above is actually false, but admits a correct reformulation using the ideas of that paper.
Recent work of the reviewer is aimed at using domino tableaux to compute a great deal of additional information, viz. the module structure of left cells, their explicit decomposition into irreducible constituents, and multiplication in the asymptotic Hecke algebra.

MSC:
17B35 Universal enveloping (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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References:
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