格拉斯曼流形的等变量子微分方程与qKZ差分方程的Satake对应研究
On the Satake correspondence for the equivariant quantum differential equations and qKZ difference equations of Grassmannians
AI总结：
本研究建立格拉斯曼流形$G(k,n)$的qDE与qKZ联合系统和射影空间对应系统的Satake对应，推广\textcyr{B}-定理并推导其超几何解的新表达式，还证明该系统斯托克斯基与等变$K$-理论例外类的对应关系。
摘要：
我们研究格拉斯曼流形$G(k,n)$（参数化$\boldsymbol{\text{C}}^n$中$k$维子空间的流形）的等变量子微分方程（qDE）与qKZ差分方程的联合系统。首先，我们建立了$G(k,n)$的该联合系统与射影空间$\boldsymbol{\text{P}}^{n-1}$的对应系统之间的联系。具体而言，我们证明在$G(k,n)$与$\boldsymbol{\text{P}}^{n-1}$等变上同调的合适**Satake等同**下，$G(k,n)$的联合系统与$\boldsymbol{\text{P}}^{n-1}$上同调的第$k$次外幂上的一个微分-差分系统是规范等价的。其次，我们证明如arXiv:1909.06582、arXiv:2203.03039中所述的格拉斯曼流形的\textcyr{B}-定理与Satake等容。这意味着$\boldsymbol{\text{P}}^{n-1}$的\textcyr{B}-定理可通过Satake等同推广到$G(k,n)$。作为推论，我们推导了$G(k,n)$的qDE与qKZ联合系统的多维超几何解的行列式公式与新的积分表示。最后，我们分析了与$G(k,n)$相关的qDE与qKZ方程联合系统的斯托克斯现象。我们证明了解的斯托克斯基对应于$G(k,n)$上等变凝聚层的导出范畴中满例外集合的显式$K$-理论类。此外，我们还证明斯托克斯矩阵等于等变欧拉-庞加莱-格罗滕迪克配对关于这些例外$K$-理论基的格拉姆矩阵。
英文摘要：
We consider the joint system of equivariant quantum differential equations (qDE) and qKZ difference equations for the Grassmannian $G(k,n)$, which parametrizes $k$-dimensional subspaces of $\mathbb{C}^n$. First, we establish a connection between this joint system for $G(k,n)$ and the corresponding system for the projective space $\mathbb{P}^{n-1}$. Specifically, we show that, under suitable \textit{Satake identifications} of the equivariant cohomologies of $G(k,n)$ and $\mathbb{P}^{n-1}$, the joint system for $G(k,n)$ is gauge equivalent to a differential-difference system on the $k$-th exterior power of the cohomology of $\mathbb{P}^{n-1}$. Secondly, we demonstrate that the \textcyr{B}-theorem for Grassmannians, as stated in arXiv:1909.06582, arXiv:2203.03039, is compatible with the Satake identification. This implies that the \textcyr{B}-theorem for $\mathbb{P}^{n-1}$ extends to $G(k,n)$ through the Satake identification. As a consequence, we derive determinantal formulas and new integral representations for multi-dimensional hypergeometric solutions of the joint qDE and qKZ system for $G(k,n)$. Finally, we analyze the Stokes phenomenon for the joint system of qDE and qKZ equations associated with $G(k,n)$. We prove that the Stokes bases of solutions correspond to explicit $K$-theoretical classes of full exceptional collections in the derived category of equivariant coherent sheaves on $G(k,n)$. Furthermore, we show that the Stokes matrices equal the Gram matrices of the equivariant Euler-Poincaré-Grothendieck pairing with respect to these exceptional $K$-theoretical bases.
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