论文标题
不良的滑轮，尼尔植物赫森伯格品种和模块化定律
Perverse sheaves, nilpotent Hessenberg varieties, and the modular law
论文作者
论文摘要
我们考虑了通过用某些其他矢量束在标志品种上替换cotangent束，考虑了简单谎言代数的弹簧锥的弹簧分辨率的概括。我们表明，Springer Sheaf的类似物具有直接汇总的相交共同体学套管，而Springer对应关系中会出现。这些一般地图的纤维是Nilpotent Hessenberg品种，我们基于De Concini，Lusztig和Procesi建立的技术来研究其几何形状。例如，我们表明这些纤维在奇数程度上具有消失的共同体学。这导致了对双重图片的几种影响，我们考虑地图概括了整个谎言代数的刺激性分辨率。特别是我们能够证明Brosnan的猜想。 随着我们的变化，相应的尼尔植物黑森伯格品种的共同体通常满足我们称之为几何模块化法的关系，这也起源于de concini，lusztig和procesi的作品。我们将这种关系与类型$ a $联系起来，由guay-paquet定义的组合模块化定律，该法律通过某些对称函数满足并推断出该连接的某些后果。
We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotangent bundle with certain other vector bundles over the flag variety. We show that the analogue of the Springer sheaf has as direct summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques established by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This leads to several implications for the dual picture, where we consider maps that generalize the Grothendieck-Springer resolution of the whole Lie algebra. In particular we are able to prove a conjecture of Brosnan. As we vary the maps, the cohomology of the corresponding nilpotent Hessenberg varieties often satisfy a relation we call the geometric modular law, which also has origins in the work on De Concini, Lusztig, and Procesi. We connect this relation in type $A$ with a combinatorial modular law defined by Guay-Paquet that is satisfied by certain symmetric functions and deduce some consequences of that connection.
