Lie algebra of differential oeprators on algebraic variety and its representations

我们研究了作为代数簇上的(超)微分算子环出现的李代数，这类李代数最基本的是平坦仿射空间上的向量场的李代数。这种李代数称为Cartan型李代数，它是无限维的。在我们的研究中，我们首先研究了Cartan型李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着类似于Schur对偶的对偶。在上述研究中，对称群起着重要的作用，我们必须研究它在普通/超变量多项式环上的作用。在计算过程中，我们用A.Gyoja和K.Taniguchi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，我们得到了经典Weyl群的不变式的完整公式。然而，D型Weyl群的公式还远远不能计算。我们有关于这个Weyl群的另一个猜想公式，但我们还不能证明它。另一方面，随着我们对对偶性的理解的深入，我们意识到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Bernstein度是可能的。这一发现使我们得到了相关圈的计算和稳定分支系数的求和公式。然而，这一部分的研究仍在进行中。

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.