Lie algebra of differential oeprators on algebraic variety and its representations

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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.