Rankin-Selberg duality via gluing
Abstract
We use a gluing procedure introduced by Ginzburg to describe the relative Langlands dual of the hyperspherical Hamiltonian (GLn × GLm)-variety T*(Hom(ℂm,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. Using this result, we give a new proof of a theorem of Braverman-Finkelberg-Nakajima describing the regular perverse sheaf in type A. Following a suggestion of Ginzburg, we explain how to modify the gluing so that the dual Hamiltonian variety of T*N, for any gluable 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.
Related
- A Rankin-Selberg Integral for SU2,2
- Archimedean Modular Symbols for Rankin-Selberg Convolutions of GLn(ℂ)×GLn(ℂ) and GLn(ℂ)×GLn+1(ℂ)
- Rankin--Selberg integrals of opposite conductor--one newforms
- Regularized spectral expansion of a Rankin--Selberg period and moments
- Weyl Bounds for GL2× GL2 Rankin--Selberg L-functions via Short Spectral Moments
BibTeX
@misc{dias2026rankinselberg,
title = {{Rankin-Selberg duality via gluing}},
author = {Bruno da Silveira Dias},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.22965},
eprint = {2607.22965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.22965},
}
RIS
TY - UNPB
TI - Rankin-Selberg duality via gluing
AU - Bruno da Silveira Dias
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2607.22965
UR - https://arxiv.org/abs/2607.22965
AB - 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,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. Using this result, we give a new proof of a theorem of Braverman-Finkelberg-Nakajima describing the regular perverse sheaf in type A. Following a suggestion of Ginzburg, we explain how to modify the gluing so that the dual Hamiltonian variety of T^*N, for any gluable 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.
ER -