Archimedean zeta integrals on $ {\rm SO}_{2n} \times {\rm GL}_n$ and $ {\rm Sp}_{2n} \times {\rm GL}_n $

@misc{ishii2026archimedean,
  title = {{Archimedean zeta integrals on $ \rm SO_2n \times \rm GL_n$ and $ \rm Sp_2n \times \rm GL_n $}},
  author = {Taku Ishii and Eric Stade},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36489},
  eprint = {2610.36489},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36489},
}

TY  - UNPB
TI  - Archimedean zeta integrals on $ {\rm SO}_{2n} \times {\rm GL}_n$ and $ {\rm Sp}_{2n} \times {\rm GL}_n $
AU  - Taku Ishii
AU  - Eric Stade
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.36489
UR  - https://arxiv.org/abs/2610.36489
AB  - We evaluate the archimedean Rankin-Selberg zeta integrals associated with spherical principal series representations of ${\rm SO}_{2n}(\mathbf R) \times {\rm GL}_n(\mathbf R) $ and $ {\rm Sp}_{2n}(\mathbf R) \times {\rm GL}_n(\mathbf R) $, where $ {\rm SO}_{2n} $ is split and n≥2. We obtain explicit formulas expressing these integrals as quotients of products of gamma functions. Our approach uses Mellin-Barnes representations of spherical Whittaker functions, together with new integral identities for their Mellin transforms and transformations of Barnes integrals. In the even orthogonal case, we establish a recurrence in the rank and evaluate the integral by induction. The symplectic case then follows from an explicit identity relating the two families of zeta integrals.
ER  -