Rankin-Selberg duality via gluing

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.