论文标题
shalika细菌用于$ gl_n $中的驯服元素
Shalika germs for tamely ramified elements in $GL_n$
论文作者
论文摘要
脱离了椭圆形霍尔代数在fock空间上的作用，我们为tamibified的常规半圣经元素的shalika细菌提供了组合公式，$ gl_n $ $ gl_n $ ynonarchimedean本地领域。作为副产品，我们计算A型弹簧纤维的重量多项式和A型和轨道上的轨道积分的重量，并被腐烂的常规半神经元素。我们猜想$γ$的Shalika细菌对应于某些准代码的纸条$ \ Mathcal {f}_γ$的圆环定位权重，以$ \ MATHBB {a}^2 $上的Hilbert of Hilbert方案，从而找到了他们的几何解释。作为推论，我们以平面曲线压实的雅各布人的点数$ q $获得了多项式，以及Cherednik-Danilenko猜想的虚拟版本在其贝蒂数字上。我们的结果还为ORS猜想提供了进一步的证据，这些ORS与代数结的压缩的Jacobians和Homfly-PT不变性。
Degenerating the action of the elliptic Hall algebra on the Fock space, we give a combinatorial formula for the Shalika germs of tamely ramified regular semisimple elements $γ$ of $GL_n$ over a nonarchimedean local field. As a byproduct, we compute the weight polynomials of affine Springer fibers in type A and orbital integrals of tamely ramified regular semisimple elements. We conjecture that the Shalika germs of $γ$ correspond to residues of torus localization weights of a certain quasi-coherent sheaf $\mathcal{F}_γ$ on the Hilbert scheme of points on $\mathbb{A}^2$, thereby finding a geometric interpretation for them. As corollaries, we obtain the polynomiality in $q$ of point-counts of compactified Jacobians of planar curves, as well as a virtual version of the Cherednik-Danilenko conjecture on their Betti numbers. Our results also provide further evidence for the ORS conjecture relating compactified Jacobians and HOMFLY-PT invariants of algebraic knots.
