Rankin-Selberg duality via gluing

2026/07/25 by Bruno da Silveira Dias
#math.RT

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Abstract

We use a gluing procedure introduced by Ginzburg to describe the relative Langlands dual of the hyperspherical Hamiltonian (GL_n × GL_m)-variety T^*(Hom(ℂ^m,ℂⁿ)), 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^*N, for any finite-dimensional representation 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.

Citations

• Pointwise purity, derived Satake, and Symplectic duality
• Functoriality of Coulomb branches
• Strongly tempered hyperspherical Hamiltonian spaces
• Geometric Langlands duality for periods
• Hyperspherical equivariant slices and basic classical Lie superalgebras
• Coulomb branches of 3-dimensional gauge theories and related structures
• The role of Coulomb branches in 2D gauge theory
• Generalizing the MVW involution, and the contragredient
• Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A

Related

• On the relative Langlands duality for Sp_2n \backslash GL_2n+1 (with an appendix by Zeyu Wang)
• Global geometrised Rankin-Selberg method for GL(n)
• Relative Langlands Duality
• Some Singular Examples of Relative Langlands Duality
• Hall algebras of curves, commuting varieties and Langlands duality