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Towards a Goldberg–Shahidi pairing for classical groups

  • Arnab Mitra and Steven Spallone EMAIL logo
Published/Copyright: July 8, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 2

Abstract

Let G1 be an orthogonal, symplectic or unitary group over a local field and let P=M⁢N be a maximal parabolic subgroup. Then the Levi subgroup M is the product of a group of the same type as G1 and a general linear group, acting on vector spaces X and W, respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M, assuming dim⁡W≤dim⁡X, excluding only the symplectic case with dim⁡W odd. The result is a Weyl-type integration formula for N with applications to the theory of intertwining operators for parabolically induced representations of G1. Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.

Keywords: Integration formula; maximal parabolic; unipotent radical; Langlands–Shahidi method, intertwining operator
MSC 2010: 22E35; 22E50

Communicated by Freydoon Shahidi

Funding statement: During the second half of the project, the first author was supported by a postdoctoral fellowship funded by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev, and wishes to thank the foundation for its financial support.

Acknowledgements

The authors would like to thank Sandeep Varma, Vivek Mallick, Amit Hogadi, Krishna Kaipa, and Freydoon Shahidi for useful conversations. Part of the work was done during the first author’s Ph.D. and appears in his thesis. He would like to thank his advisor Dipendra Prasad for constant encouragement and support. This work was initiated during the second author’s visit to the Tata Institute of Fundamental Research in Mumbai, and it is a pleasure to thank the institute for its support.

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Received: 2017-2-16
Revised: 2017-6-14
Published Online: 2017-7-8
Published in Print: 2018-3-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

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  4. CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
  5. Towards a Goldberg–Shahidi pairing for classical groups
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https://doi.org/10.1515/forum-2017-0033
Keywords for this article
Integration formula; maximal parabolic; unipotent radical; Langlands–Shahidi method, intertwining operator

Articles in the same Issue

  1. Frontmatter
  2. The variance of divisor sums in arithmetic progressions
  3. Characterizing Lie groups by controlling their zero-dimensional subgroups
  4. CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
  5. Towards a Goldberg–Shahidi pairing for classical groups
  6. On the non-existence of cyclic splitting fields for division algebras
  7. Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
  8. Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
  9. Anharmonic solutions to the Riccati equation and elliptic modular functions
  10. A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
  11. Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
  12. On symplectic semifield spreads of PG(5,q2), q odd
  13. From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
  14. Golodness and polyhedral products for two-dimensional simplicial complexes
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