Abstract
Although Kostant's problem was answered negatively by Conze-Berline and Duflo ([5], 6.5) its further investigation is still of considerable interest. This is because the fact that it holds for certain classes of modules and the determination of the precise extent to which it fails for others gives an enormous amount of information on primitive ideals and in particular on the Goldie ranks of primitive quotients. This paper does not review this development in any detail; but concentrates on four examples which form part of the ongoing investigation. Two of these were discussed at the meeting, the third which involves only Goldie rang was developed shortly afterwards following discussions with R.Howe. The fourth example is an application of the Enright functor. The first example shows that for Whittaker modules, Kostant's problem has a negative answer even in type AI, whereas for highest weight modules a negative answer first arises in type B2 (or type C2) as reported by Conze-Berline and Duflo. The second example makes precise this failure in type G2 (and in type B2) and shows that it is an inevitable consequence of a constraint imposed by an additivity principle for Goldie rank. This result perceived in ([15], Sect, 9) has since undergone a considerable extension [18]. The third example shows that in type G2 there is at least one completely prime primitive ideal whose associated variety is the Zariski closure of the unique eight dimensional orbit (in the dual of the Lie algebra). This is part of a general program [18] to determine the scale factors in the Goldie rank polynomials ([17], 1.4). It may also be considered to be part of a program to parametrize the completely prime primitive ideals through the orbit space. In the example the Goldie ranks of all the primitive ideals associated with this eight dimensional orbit can be computed and one finds that there are exactly two which are completely prime. This provides the first unequivocal counterexample to the existence of a natural bijection between these two sets.
Keywords
- Verma Module
- Poisson Algebra
- Zariski Closure
- Primitive Ideal
- High Weight Module
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© 1981 Springer-Verlag
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Joseph, A. (1981). Kostant's problem and goldie rank. In: Carmona, J., Vergne, M. (eds) Non Commutative Harmonic Analysis and Lie Groups. Lecture Notes in Mathematics, vol 880. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090412
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DOI: https://doi.org/10.1007/BFb0090412
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