李代数模表示中若干问题的研究
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摘要
本文研究李代数模表示理论中的相关问题.主要考虑了素特征的代数闭合域上阶化Cartan型李代数不可约模的确定、Verma模的支柱簇的确定,以及秩一的基本Cartan型李代数幂零轨道的具体构造与几何信息,并由此给出更一般W系列Cartan型李代数幂零轨道的基本性质与特征.同时,就一般限制李代数的表示,本文从包络代数的本原理想角度,给出了一些新的结论.具体如下：
1.设R=21(m；n)是一个除幂代数,L=X(m;n),X∈{W,S,H)是特征p>0的代数封闭域F上的阶化Cartan型李代数系列中的广义Jacobson Witt代数或特殊代数或哈密尔顿代数.在广义限制李代数意义下,L的任一单模都唯一对应于一个(广义)特征函数χ.当χ的高度ht(χ) 2.决定了特征p>3的代数封闭域上秩一的基本Cartan型李代数Witt代数在自同构群作用下的幂零轨道.对比于典型李代数情形下幂零轨道个数的有限性,在Witt代数情形下,有无限个幂零轨道.给出了所有幂零轨道的代表元以及每个轨道的维数.我们同时也得到Jacobson-Witt代数有无限个幂零轨道.对于其他Cartan型李代数,我们猜想有类似的结论.
3.研究了Cartan型李代数的支柱簇.对于小Verma模以及具有半单特征的一类模的支柱簇给出了一些描述.
4.给出了有限维限制李代数的任一不可约模所对应的“中心特征”理想在包络代数中所生成的理想的余维数的一个估计.刻画了最大维数的单模所对应的本原理想.在简约代数群G的李代数情形下,对一类所谓的G-不变的理想给了‘些刻画.
In this dissertation, we study related problems in modular representation theory of Lie algebras. We mainly consider the determination of irreducible modules of graded Lie algebras of Cartan type over algebraically closed fields of prime characteristic, the determination of the support varieties of Verma mod-ules and the precise construction and geometry of nilpotent orbits in the basic Cartan type Lie algebra of rank one, from which we also give the basic property and characterization of nilpotent orbits for algebras of type W in the Cartan type series. Furthermore, we give some new results on representations of general restricted Lie algebras from the view of primitive ideals of enveloping algebras. More precisely:
1. Let R= (?)(m;n) be the divided power algebra and L= X(m;n), X∈{W, S, H} be a generalized Jacobson-Witt algebra, special algebra or Hamil-tonian algebra in the graded Cartan type series over an algebraically closed field F of characteristic p> 0. In the generalized restricted Lie algebra setting, any simple module of L corresponds to a unique (generalized) p-characterχ.When the height ht(χ) ofχis no more than min{pni-pni-1| i=1,…, m}-2+δxw, simple modules of L with p-characterχare determined. This is done by intro-ducing a "modified" induced module structure and thereby endowing induced module with the so-called (?)-module structure. The so-called category (?) for the generalized Jacobson-Witt algebras by Skryabin will be constructed on induced modules of a more natural class of generalized restricted Lie algebras. Moreover, this construction is applicable to all four series of Cartan type Lie algebras.(1) We prove that all irreducible representations of L with characterχsatisfying the above condition are induced from irreducible submodules of the maximal subal-gebra L0, modulo some exceptional cases. The exceptional cases happen to theχof height lower than 1. Simple modules in the exceptional cases were deter-mined mainly by Guang-Yu Shen, Nai-Hong Hu and so on. For the case that ht(χ)= -1, simple exceptional modules were determined by Shen [66] for types W,S,H and by Hu [25] for type K (see also [21,19,20]). When ht(χ)= 0 and X= W, S, we precisely construct simple exceptional modules in this dissertation via a complex of "modified" induced modules, and their dimensions are also ob-tained. For the case that ht(χ)= 0 and X= H, simple exceptional modules were determined by Pu and Jiang [59]. For type K, we can also introduce the category (?) and "modified" induced representations. But unlike the other three series of Cartan type Lie algebras, we can not strictly prove that those "modified" induced modules belong to the category (?) due to the fact that the graded structure of the Contact algebra does not inherit from the gradation of the generalized Jacobson-Witt algebra. However, by some concrete computation, we could conjecture that this holds. Then parallel to the other series of Cartan type Lie algebras, we can also conjecture that all simple modules of the Contact algebra with p-characterχsuch that ht(χ)< min{pni-pni-1|i= 1,…, m} - 2 are "modified" induced modules except the exceptional cases. This conjecture is true for the restricted Contact algebra by Zhang's work [100] (I would like to give some explanation as follows:One needs to handle each case of the four classes of Cartan type Lie algebras respectively. Until now, there is no unified method to deal with them in an axiomatic way).
2. The nilpotent orbits of the Witt algebra W1, which is the basic Cartan type Lie algebra of rank 1, are determined under the automorphism group over an algebraically closed field F of characteristic p> 3. In contrast with a finite number of nilpotent orbits in a classical simple Lie algebra (cf. [31]), there is an infinite number of nilpotent orbits in W1. A set of representatives of nilpotent orbits, as well as their dimensions, are clearly presented. We also obtain that there are infinitely many nilpotent orbits in the Jacobson-Witt algebras. For the other Cartan type Lie algebras, we conjecture the same results.
3. Support varieties for Lie algebras of Cartan type are studied. We give some description of the support varieties for the so-called baby Verma modules and a class of modules with semisimple characters.
4. For an arbitrary restricted Lie algebra g, we give an estimate of the codi-mension of the ideal of U(g) generated by the so-called central character ideal associated with an irreducible g-module. Moreover, we describe the primitive ideals corresponding to simple modules of maximal dimension. For the case of Lie algebras of reductive algebraic groups, we further give some description on the so-called G-invariant ideals.
1.设R=21(m；n)是一个除幂代数,L=X(m;n),X∈{W,S,H)是特征p>0的代数封闭域F上的阶化Cartan型李代数系列中的广义Jacobson Witt代数或特殊代数或哈密尔顿代数.在广义限制李代数意义下,L的任一单模都唯一对应于一个(广义)特征函数χ.当χ的高度ht(χ)
3.研究了Cartan型李代数的支柱簇.对于小Verma模以及具有半单特征的一类模的支柱簇给出了一些描述.
4.给出了有限维限制李代数的任一不可约模所对应的“中心特征”理想在包络代数中所生成的理想的余维数的一个估计.刻画了最大维数的单模所对应的本原理想.在简约代数群G的李代数情形下,对一类所谓的G-不变的理想给了‘些刻画.
In this dissertation, we study related problems in modular representation theory of Lie algebras. We mainly consider the determination of irreducible modules of graded Lie algebras of Cartan type over algebraically closed fields of prime characteristic, the determination of the support varieties of Verma mod-ules and the precise construction and geometry of nilpotent orbits in the basic Cartan type Lie algebra of rank one, from which we also give the basic property and characterization of nilpotent orbits for algebras of type W in the Cartan type series. Furthermore, we give some new results on representations of general restricted Lie algebras from the view of primitive ideals of enveloping algebras. More precisely:
1. Let R= (?)(m;n) be the divided power algebra and L= X(m;n), X∈{W, S, H} be a generalized Jacobson-Witt algebra, special algebra or Hamil-tonian algebra in the graded Cartan type series over an algebraically closed field F of characteristic p> 0. In the generalized restricted Lie algebra setting, any simple module of L corresponds to a unique (generalized) p-characterχ.When the height ht(χ) ofχis no more than min{pni-pni-1| i=1,…, m}-2+δxw, simple modules of L with p-characterχare determined. This is done by intro-ducing a "modified" induced module structure and thereby endowing induced module with the so-called (?)-module structure. The so-called category (?) for the generalized Jacobson-Witt algebras by Skryabin will be constructed on induced modules of a more natural class of generalized restricted Lie algebras. Moreover, this construction is applicable to all four series of Cartan type Lie algebras.(1) We prove that all irreducible representations of L with characterχsatisfying the above condition are induced from irreducible submodules of the maximal subal-gebra L0, modulo some exceptional cases. The exceptional cases happen to theχof height lower than 1. Simple modules in the exceptional cases were deter-mined mainly by Guang-Yu Shen, Nai-Hong Hu and so on. For the case that ht(χ)= -1, simple exceptional modules were determined by Shen [66] for types W,S,H and by Hu [25] for type K (see also [21,19,20]). When ht(χ)= 0 and X= W, S, we precisely construct simple exceptional modules in this dissertation via a complex of "modified" induced modules, and their dimensions are also ob-tained. For the case that ht(χ)= 0 and X= H, simple exceptional modules were determined by Pu and Jiang [59]. For type K, we can also introduce the category (?) and "modified" induced representations. But unlike the other three series of Cartan type Lie algebras, we can not strictly prove that those "modified" induced modules belong to the category (?) due to the fact that the graded structure of the Contact algebra does not inherit from the gradation of the generalized Jacobson-Witt algebra. However, by some concrete computation, we could conjecture that this holds. Then parallel to the other series of Cartan type Lie algebras, we can also conjecture that all simple modules of the Contact algebra with p-characterχsuch that ht(χ)< min{pni-pni-1|i= 1,…, m} - 2 are "modified" induced modules except the exceptional cases. This conjecture is true for the restricted Contact algebra by Zhang's work [100] (I would like to give some explanation as follows:One needs to handle each case of the four classes of Cartan type Lie algebras respectively. Until now, there is no unified method to deal with them in an axiomatic way).
2. The nilpotent orbits of the Witt algebra W1, which is the basic Cartan type Lie algebra of rank 1, are determined under the automorphism group over an algebraically closed field F of characteristic p> 3. In contrast with a finite number of nilpotent orbits in a classical simple Lie algebra (cf. [31]), there is an infinite number of nilpotent orbits in W1. A set of representatives of nilpotent orbits, as well as their dimensions, are clearly presented. We also obtain that there are infinitely many nilpotent orbits in the Jacobson-Witt algebras. For the other Cartan type Lie algebras, we conjecture the same results.
3. Support varieties for Lie algebras of Cartan type are studied. We give some description of the support varieties for the so-called baby Verma modules and a class of modules with semisimple characters.
4. For an arbitrary restricted Lie algebra g, we give an estimate of the codi-mension of the ideal of U(g) generated by the so-called central character ideal associated with an irreducible g-module. Moreover, we describe the primitive ideals corresponding to simple modules of maximal dimension. For the case of Lie algebras of reductive algebraic groups, we further give some description on the so-called G-invariant ideals.
引文
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