Towards the Jantzen conjecture
Towards the Jantzen conjecture
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走向 Jantzen 猜想
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发表时间：
1980
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通讯作者：
A. Joseph
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作者：
A. Joseph
设g是复半单李代数，U（g）是它的包络代数，Prim U（g）是U（g）的本原理想集，B是g的Cartan子代数.对于An-1型g单（Cartan记法），Jantzen [3]，5.9证明了每个素数U（g）纤维投射到g* 中固定的正则积分中心特征标和固定的幂零轨道上的基数正好是对称群Sn的适当不可约表示的维数。在这里，它建议，适当的制定这一猜想一般g涉及的维数fj* 上的多项式的某些子空间，确定的维数的不可约有限维表示的抛物子代数g。它对An-i型的Jantzen猜想的还原实质上是Garnir [14]的一个组合结果。然后通过对诱导模的有限同态的研究（给出了一些独立的结果），将Jantzen猜想归结为两个公开问题。第一个涉及的主要系列，并会给一个下界（涉及上述子空间的尺寸）的基数，每个定期整纤维。在An-i的情况下，这只是Sn中对合的数量，并且与Duflo的上界[13]，II.2一致。第二个是Borho [1]，3.3的问题，只要[21]，4.3的最后一部分成立（例如在An-1 [25]，4.1中），就会固定相关的幂零轨道。
Let g be a complex semisimple Lie algebra, U(g) its enveloping algebra, Prim U(g) the set of primitive ideals of U(g) and b a Cartan subalgebra for g. For g simple of type An-l (Cartan notation), Jantzen [3], 5.9 conjectured that the cardinality of each Prim U(g) fibre projecting onto a fixed regular integral central character and onto a fixed nilpotent orbit in g* is just the dimension of the appropriate irreducible representation of the symmetric group Sn. Here it is suggested that the appropriate formulation of this conjecture for general g involves the dimensions of certain subspaces of polynomials on fj* which determine the dimensions of the irreducible finite dimensional representations of parabolic subalgebras of g. Its reduction to the Jantzen conjecture for type An-i is essentially a combinatorial result of Garnir [14]. Then through a careful study of ad A finite homomorphisms of induced modules (which gives some results of independent interest) the Jantzen conjecture is reduced to two open questions. The first involves the principal series and would give a lower bound (involving the dimensions of the above-mentioned subspaces) on the cardinality of each regular integral fibre. In case An-i this is just the number of involutions in Sn and coincides with Duflo’s upper bound [13], II.2. The second is a problem of Borho [1], 3.3 which whenever the last part of [21], 4.3 holds (for example in type An-1 [25], 4.1) fixes the associated nilpotent orbit.