Rankin–Selberg integrals for principal series representations of GL(n)
-
Dongwen Liu
and
Binyong Sun
Abstract
We prove that the local Rankin–Selberg integrals for principal series representations of the general linear groups agree with certain simple integrals over the Rankin–Selberg subgroups, up to certain constants given by the local gamma factors.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11901466
Award Identifier / Grant number: 11688101
Award Identifier / Grant number: 11621061
Funding source: National Key Research and Development Program of China
Award Identifier / Grant number: 2020YFA0712600
Funding statement: F. Su is supported in part by the National Natural Science Foundation of China (No. 11901466) and the XJTLU Research Development Funding (RDF-19-02-04). B. Sun is supported in part by the National Key R & D Program of China (No. 2020YFA0712600), National Natural Science Foundation of China (No. 11688101, No. 11621061), and Kunpeng Program of Zhejiang Province.
Acknowledgements
We thank the referee for helpful comments.
References
[1] L. V. Ahlfors, Complex Analysis, 3rd ed., Int. Ser. Pure Appl. Math., McGraw-Hill, New York, 1978. Search in Google Scholar
[2] A. Aizenbud and D. Gourevitch, Schwartz functions on Nash manifolds, Int. Math. Res. Not. IMRN 2008 (2008), no. 5, Article ID rnm155. 10.1093/imrn/rnm155Search in Google Scholar
[3] Y. Chen and B. Sun, Schwartz homologies of representations of almost linear Nash groups, J. Funct. Anal. 280 (2021), no. 7, Paper No. 108817. 10.1016/j.jfa.2020.108817Search in Google Scholar
[4] I. M. Gel’fand, M. I. Graev and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, Academic Press, Boston, 1990. Search in Google Scholar
[5] S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Pure Appl. Math. 113, Academic Press, Orlando, 1984. Search in Google Scholar
[6] H. Jacquet, Principal L-functions of the linear group, Automorphic Forms, Representations and L-Functions (Corvallis 1977), Proc. Sympos. Pure Math. 33 Part 2, American Mathematical Society, Providence (1979), 63–86. 10.1090/pspum/033.2/546609Search in Google Scholar
[7] H. Jacquet, Archimedean Rankin–Selberg integrals, Automorphic Forms and L-functions II. Local Aspects, Contemp. Math. 489, American Mathematical Society, Providence (2009), 57–172. 10.1090/conm/489/09547Search in Google Scholar
[8] H. Jacquet, I. I. Piatetskii-Shapiro and J. A. Shalika, Rankin–Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. 10.2307/2374264Search in Google Scholar
[9] S. S. Kudla, Tate’s thesis, An Introduction to the Langlands Program (Jerusalem 2001), Birkhäuser, Boston (2003), 109–131. 10.1007/978-0-8176-8226-2_6Search in Google Scholar
[10] R. P. Langlands, Euler Products, Yale University, New Haven, 1971. Search in Google Scholar
[11] J. Tate, Number theoretic background, Automorphic Forms, Representations and L-Functions (Corvallis 1977), Proc. Sympos. Pure Math. 33 Part 2, American Mathematical Society, Providence (1979), 3–26. 10.1090/pspum/033.2/546607Search in Google Scholar
[12] J. T. Tate, Jr, Fourier analysis in number fields and Hecke’s zeta-functions, Ph.D. Thesis, Princeton University, 1950. Search in Google Scholar
[13] N. R. Wallach, Real Reductive Groups. II, Pure Appl. Math. 132, Academic Press, Boston, 1992. Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On the topological complexity of manifolds with abelian fundamental group
- Reconstructing étale groupoids from semigroups
- Fiberwise linear differential operators
- Foliations with isolated singularities on Hirzebruch surfaces
- On a stronger reconstruction notion for monoids and clones
- Cochain level May–Steenrod operations
- Commutative L-algebras and measure theory
- Rankin–Selberg integrals for principal series representations of GL(n)
- A simple proof of the generalized Leibniz rule on bounded Euclidean domains
- Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
- Seshadri constants on some Quot schemes
- Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
- On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
- C-minimal topological groups
- Cevian properties in ideal lattices of Abelian ℓ-groups
- Study of twisted Bargmann transform via Bargmann transform
Articles in the same Issue
- Frontmatter
- On the topological complexity of manifolds with abelian fundamental group
- Reconstructing étale groupoids from semigroups
- Fiberwise linear differential operators
- Foliations with isolated singularities on Hirzebruch surfaces
- On a stronger reconstruction notion for monoids and clones
- Cochain level May–Steenrod operations
- Commutative L-algebras and measure theory
- Rankin–Selberg integrals for principal series representations of GL(n)
- A simple proof of the generalized Leibniz rule on bounded Euclidean domains
- Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras
- Seshadri constants on some Quot schemes
- Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces
- On spectral and non-spectral problem for the planar self-similar measures with four element digit sets
- C-minimal topological groups
- Cevian properties in ideal lattices of Abelian ℓ-groups
- Study of twisted Bargmann transform via Bargmann transform
