Residues of standard intertwining operators on p-adic classical groups
-
Xiaoxiang Yu
Abstract
We study two basic problems involved in the study of the
standard intertwining operators attached to representations
induced from irreducible unitary supercuspidal representations on
maximal parabolic subgroups of p-adic classical groups. We give a formula to
transfer the integrals over the unipotent radical to orbital integrals under the
adjoint action of the Levi subgroup, and conclude that the residue of such standard
intertwining operators at
Funding statement: This work is partly supported by NSF Grant of China, No. 11271165 and No. 11171343.
We would like to thank the referee for a lot of corrections, suggestions and comments.
References
[1] Arthur J., The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223–293. 10.1215/S0012-7094-88-05612-8Search in Google Scholar
[2] Bröcker T. and Dieck T., Representations of Compact Lie Groups, Grad. Texts in Math. 98, Springer, New York, 1985. 10.1007/978-3-662-12918-0Search in Google Scholar
[3] Cartier P., Representation of p-adic groups: A survey, Automorphic Forms, Representations and L-Functions (Corvallis 1977), Proc. Sympos. Pure Math. 33, Part 1, American Mathematical Society, Providence (1977), 111–155. 10.1090/pspum/033.1/546593Search in Google Scholar
[4] Goldberg D. and Shahidi F., On the tempered spectrum of quasi-split classical groups, Duke Math. J. 92 (1998), 255–294. 10.1215/S0012-7094-98-09206-7Search in Google Scholar
[5] Goldberg D. and Shahidi F., The tempered spectrum of quasi-split classical groups. II, Canad. J. Math. 53 (2001), no. 2, 244–277. 10.4153/CJM-2001-011-7Search in Google Scholar
[6] Goldberg D. and Shahidi F., The tempered spectrum of quasi-split classical groups. III, Forum Math. 26 (2014), no. 4, 1029–1069. 10.1515/form.2011.166Search in Google Scholar
[7] Harish-Chandra , Harmonic Analysis on Reductive p-Adic Groups. Notes by G. Van Dijk, Lecture Notes in Math. 162, Springer, Berlin, 1970. 10.1007/BFb0061269Search in Google Scholar
[8] Koblitz N., p-Adic Numbers, p-Adic Analysis, and Zeta-Functions, 2nd ed., Grad. Texts in Math. 58, Springer, New York, 1984. 10.1007/978-1-4612-1112-9Search in Google Scholar
[9]
Shahidi F.,
On certain
[10] Shahidi F., On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann of Math. (2) 127 (1988), 547–584. 10.2307/2007005Search in Google Scholar
[11] Shahidi F., A proof of Langlands conjecture on Plancherel measures: Complementary series for p-adic groups, Ann. of Math. (2) 132 (1990), 1–41. 10.2307/1971524Search in Google Scholar
[12] Shahidi F., Twisted endoscopy and reducibility of induced representation for p-adic groups, Duke Math. J. 66 (1992), 1–41. 10.1215/S0012-7094-92-06601-4Search in Google Scholar
[13] Shahidi F., The notion of norm and the representation theory of orthogonal groups, Invent Math. 119 (1995), 1–36. 10.1007/BF01245173Search in Google Scholar
[14] Shahidi F., Poles of intertwining operators via endoscopy: The connection with prehomogeneous vector spaces (with an appendix, “Basic endoscopic data”, by Diana Shelstad), Compos. Math. 120 (2000), 291–325. 10.1023/A:1002038928169Search in Google Scholar
[15] Shahidi F., Local coefficients as Mellin transforms of Bessel functions: Towards a general stability, Int. Math. Res. Not. IMRN 2002 (2002), no. 39, 2075–2119. 10.1155/S1073792802204171Search in Google Scholar
[16]
Shahidi F. and Spallone S.,
Residues of intertwining operators for
[17] Silberger A., Introduction to Harmonic Analysis on Reductive p-Adic Groups. Based on Notes of Harish-Chandra, Math. Notes Princeton Univ. Press 23, Princeton University Press, Princeton, 1979. Search in Google Scholar
[18] Spallone S., Residues of intertwining operators for classical groups, Int. Math. Res. Not. IMRN 2008 (2008), Article ID rnn056. 10.1093/imrn/rnn056Search in Google Scholar
[19] Yu X. and Wang D., Norm correspondence on p-adic classical groups, J. Algebra 378 (2013), 22–44. 10.1016/j.jalgebra.2012.12.013Search in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Residues of standard intertwining operators on p-adic classical groups
- Anomalies on codimension growth of algebras
- Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras
- Bounded gaps between primes with a given primitive root, II
- Minimal potential results for the Schrödinger equation in a slab
- The classification of p-nilpotent restricted Lie algebras of dimension at most 4
- Global gradient estimates for nonlinear obstacle problems with nonstandard growth
- Existence of the Bedrosian identity for Fourier multiplier operators
- Projective bundles over small covers and the bundle triviality problem
- Gromov (non-)hyperbolicity of certain domains in ℂN
- The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}
- Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian
Articles in the same Issue
- Frontmatter
- Residues of standard intertwining operators on p-adic classical groups
- Anomalies on codimension growth of algebras
- Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras
- Bounded gaps between primes with a given primitive root, II
- Minimal potential results for the Schrödinger equation in a slab
- The classification of p-nilpotent restricted Lie algebras of dimension at most 4
- Global gradient estimates for nonlinear obstacle problems with nonstandard growth
- Existence of the Bedrosian identity for Fourier multiplier operators
- Projective bundles over small covers and the bundle triviality problem
- Gromov (non-)hyperbolicity of certain domains in ℂN
- The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}
- Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian
