AI 中文总结
该研究借助金兹堡的胶合过程描述特定簇的相对朗兰兹对偶，特别是兰金 - 塞尔伯格情形，证明其对偶同构于特定簇部分，还按建议修改胶合使对偶哈密顿量簇有反辛对合并给出公式。
AI 中文摘要
我们使用金兹堡引入的胶合过程来描述超球面哈密顿量\((\mathrm{GL}_n \times \mathrm{GL}_m)\) - 簇\(T^*(\mathrm{Hom}(\mathbb{C}^m,\mathbb{C}^n))\)的相对朗兰兹对偶，特别是兰金 - 塞尔伯格情形\(m = n\)。我们证明对偶同构于切尔基斯 - 中岛 - 高山弓簇的三角形部分，恢复了中岛的一个结果。按照金兹堡的建议，我们解释如何修改胶合，使得对于复约化群\(G\)的任何有限维表示\(\mathbf{N}\)，\(T^*\mathbf{N}\)的对偶哈密顿量簇自然配备一个反辛对合，并给出兰金 - 塞尔伯格情形下此对合的明确公式。
英文摘要
We use a gluing procedure introduced by Ginzburg to describe the relative Langlands dual of the hyperspherical Hamiltonian $(\mathrm{GL}_n \times \mathrm{GL}_m)$-variety $T^*(\mathrm{Hom}(\mathbb{C}^m,\mathbb{C}^n))$, and in particular the Rankin-Selberg case $m=n$. We show that the dual is isomorphic to the triangle part of Cherkis-Nakajima-Takayama bow varieties, recovering a result of Nakajima. Following a suggestion of Ginzburg, we explain how to modify the gluing so that the dual Hamiltonian variety of $T^*\mathbf{N}$, for any finite-dimensional representation $\mathbf{N}$ of a complex reductive group $G$, is naturally equipped with an anti-symplectic involution, and give an explicit formula for this involution in the Rankin-Selberg case.
Comments18 pages
