A note on affine quotients and equivariant double fibrations

Equivariant Hirzebruch classes and Molien series of quotient singularities

We study properties of the Hirzebruch class of quotient singularities $\mathbb{C}^n/G$, where $G$ is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of $G$ under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of $G$. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4…

GL(2)-表示中轨道的等变类

Equivariant classes of orbits in GL(2)-representations

本研究计算GL(2)-表示中轨道的等变基本类，并将其应用于求解对应椭圆纤维化和射影直线自映射的轨道闭包次数，为相关代数几何问题提供了计算结果。

仿射韦尔群与伪反射群的粗商

The coarse quotient for affine Weyl groups and pseudo-reflection groups

本文研究了仿射韦尔群与伪反射群的粗商，并通过逐点准则证明了sheaves降维到GIT商的条件，推广了Lonergan的最新结果。

The derived category of a GIT quotient

Given a quasiprojective algebraic variety with a reductive group action, we describe a relationship between its equivariant derived category and the derived category of its geometric invariant theory quotient. This generalizes classical descriptions of the category of coherent sheaves on projective space and categorifies several results in the theory of Hamiltonian group actions on projective manifolds. This perspective generalizes and provides new insight into examples of derived equivalences between birational varieties. We provide a criterion under which two different GIT quotients are deri…

Springer theory via the Hitchin fibration

In this paper, we translate the Springer theory of Weyl group representations into the language of symplectic topology. Given a semisimple complex group G, we describe a Lagrangian brane in the cotangent bundle of the adjoint quotient g/G that produces the perverse sheaves of Springer theory. The main technical tool is an analysis of the Fourier transform for constructible sheaves from the perspective of the Fukaya category. Our results can be viewed as a toy model of the quantization of Hitchin fibers in the Geometric Langlands program.

Homotopy principles for equivariant isomorphisms

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. When is there an equivariant biholomorphism of $X$ and $Y$? A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $ϕ$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $ϕ$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $Φ\colon X\to Y$, inducing $…