Let X denote a projective variety over an algebraically closed field on which a linear algebraic group acts with finitely many orbits. Then, a conjecture of Soergel and Lunts in the setting of Koszul duality and Langlands' philosophy, postulates that the equivariant derived category of bounded complexes with constructible equivariant cohomology sheaves on X is equivalent to a full subcategory of the derived category of modules over a graded ring defined as a suitable graded Ext. Only special cases of this conjecture have been proven so far. The purpose of this paper is to provide a proof of this conjecture for all projective toroidal imbeddings of complex reductive groups. In fact, we show that the methods used by Lunts for a proof in the case of toric varieties can be extended with suitable modifications to handle the toroidal imbedding case. Since every equivariant imbedding of a complex reductive group is dominated by a toroidal imbedding, the class of varieties for which our proof applies is quite large.

We also show that, in general, there exist a countable number of obstructions for this conjecture to be true and that half of these vanish when the odd dimensional equivariant intersection cohomology sheaves on the orbit closures vanish. This last vanishing condition had been proven to be true in many cases of spherical varieties by Michel Brion and the author in prior work.

令X表示在代数闭合域上的射影变体，线性代数群在有限的许多轨道上作用于该域。然后，在Koszul对偶性和Langlands哲学的背景下的一个Soergel和Lunts猜想，假定在X上具有可构造的等变同调性的有界复合物的等价派生类别等于在某个等级上模的派生类别的完整子类别定义为合适的渐变Ext的环。到目前为止，仅证明了这种猜想的特殊情况。本文的目的是为复杂的还原群的所有射影环形嵌入提供这一猜想的证明。。实际上，我们证明了Lunts在复曲面变体情况下用于证明的方法可以通过适当的修改进行扩展，以处理环形嵌入的情况。由于一个复杂的归约基团的每个等变嵌入都被一个环形嵌入所控制，因此我们证明适用于该类的变体类别很大。

我们还表明，总的来说，这个猜想是真实存在的障碍物数量众多，并且当奇数维等变相交同调在轨道闭合件上消失时，其中一半消失了。在过去的工作中，Michel Brion和作者证明了这种最后消失的条件在许多球形品种的情况下都是正确的。