Lie algebra of differential oeprators on algebraic variety and its representations

代数簇微分算子的李代数及其表示

• 批准号: 09640030
• 负责人: NISHIYAMA Kyo
• 金额: $ 2.18万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640030/
• 关键词: Lie algebra of Cartan type; Howe duality correspondence; Bernstein degree; associated cycle; representations of Weyl group; Kawanaka invariant; weyl群の表現; Cartan型Lie代数; Schurの相互律; 両側軌道分解; 球部分群; 球関数; 葉層構造; 随伴多様体; 巾零部分環

We have investigated Lie algebras which arise as a ring of (super-) differential operators on a algebraic variety.The most basic Lie algebras of this kind is a Lie algebra of vector fields on a flat affine space. This Lie algebra is called Cartan-type Lie algebra, which is infinite dimensional.In our research, first we study the tensor product of the natural representation of a Cartan type Lie (super-) algebra. The explicit decomposition of the tensor product tells us that there exists a duality between irreducible representations of a Cartan type Lie (super-) algebra and those of the symmetric group, which is similar to the Schur duality. By using symbolic computational system, we verified the duality (or correspondence) explicitly.In the research above, the symmetric group plays an important role, and we had to study its actions on a polynomial ring over ordinary/super variables. In a course of the calculations, we have started studying on invariants of irreducible representations of Weyl groups with A.Gyoja and K.Taniguchi. This invariant is called Kawanaka invariant, and we have gotten complete formulas of the invarinat for Weyl groups of classical type. Though, the formula for Weyl group of type D is far from computable. We have another conjectured formula for this Weyl group, but we cannot prove it yet.On the other hand, as our understanding on the duality went deeper, we became aware of the possibility to express Bernstein degree of certain irreducible representations of a noncompact semisimple Lie group by an integral on a symmetric cone. This discovery lead us to the calculation of associated cycles and a summation formula of stable branching coefficients. However, this part of the research is still in progress.

我们研究了代数簇上的微分算子环的李代数，这类李代数最基本的是平坦仿射空间上的向量场李代数。在我们的研究中，首先研究了Cartan型李（超）代数的自然表示的张量积。张量积的显式分解告诉我们，Cartan型李（超）代数的不可约表示与对称群的不可约表示之间存在一个对偶，类似于Schur对偶。在上述研究中，对称群起着重要的作用，我们必须研究对称群在普通/超变量上的多项式环上的作用。在计算过程中，我们与A.Gyoja和K. Taniguchi一起开始研究Weyl群的不可约表示的不变量。这种不变量称为Kawanaka不变量，并得到了经典型Weyl群的不变量的完整公式。然而，D型Weyl群的公式远不可计算。另一方面，随着我们对对偶性理解的深入，我们意识到可以用对称锥上的积分来表示非紧半单李群的某些不可约表示的伯恩斯坦度。这一发现使我们得以计算相关的圈和稳定分支系数的求和公式。不过，这部分研究仍在进行中。

项目成果

吉野 雄二: "Remarks on depth formula, grade inequality and Auslander Conjecture" Communications in Algebra. 26巻. 3793-3806 (1998)

Yuji Yoshino：“关于深度公式、等级不等式和 Auslander 猜想的评论”通讯代数卷 26. 3793-3806 (1998)

吉野雄二: "Auslander's work on Cohen-Macaulay modules and recent debelopement." Canadian Math.Soc.Conference Proceedings.23巻. 179-198 (1998)

Yuji Yoshino：“Auslander 关于 Cohen-Macaulay 模块的工作和最新发展。”加拿大数学学会会议记录第 23 卷 179-198 (1998)

西山 享: "Dipola rizations in semisimple Lie algebras and homogeneous parak ＼"{a}hler manifolds" Journal of Lie Theory. 9巻. 215-232 (1999)

Toru Nishiyama：“半单李代数和齐次帕拉流形中的偶极化”李理论杂志，第 9 卷，215-232 (1999)

吉野 雄二: "Auslander's Work on Cohen-Macaulay modules and recent developement." Canadian Math.Soc.Conference Proccedings. 23巻. 179-198 (1998)

Yuji Yoshino：“Auslander 关于 Cohen-Macaulay 模块和最新发展的工作。”加拿大数学学会会议记录，第 23 卷，179-198（1998 年）。

西山 享: "Invariants for representations of Weylgroups and two-sided cells" J. Math. Soc. Japan. (未定). 未定 (1998)

Toru Nishiyama：“Weylgroup 和两侧单元的表示的不变量”J. Math。日本（待定）。