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Residues of intertwining operators for SO*6 as character identities

Part of: Lie groups

Published online by Cambridge University Press:  20 April 2010

Freydoon Shahidi  and
Steven Spallone
Freydoon Shahidi
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA (email: shahidi@math.purdue.edu)
Steven Spallone
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73072, USA (email: sspallone@math.ou.edu)
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Abstract

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We show that the residue at s=0 of the standard intertwining operator attached to a supercuspidal representation π⊗χ of the Levi subgroup GL2(F)×E1 of the quasisplit group SO*6(F) defined by a quadratic extension E/F of p-adic fields is proportional to the pairing of the characters of these representations considered on the graph of the norm map of Kottwitz–Shelstad. Here π is self-dual, and the norm is simply that of Hilbert’s theorem 90. The pairing can be carried over to a pairing between the character on E1 and the character on E× defining the representation of GL2(F) when the central character of the representation is quadratic, but non-trivial, through the character identities of Labesse–Langlands. If the quadratic extension defining the representation on GL2(F) is different from E the residue is then zero. On the other hand when the central character is trivial the residue is never zero. The results agree completely with the theory of twisted endoscopy and L-functions and determines fully the reducibility of corresponding induced representations for all s.

Keywords

p-adic representation theoryintertwining operatorscharacter identitiesL-functions

MSC classification

Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 3 , May 2010 , pp. 772 - 794
Copyright
Copyright © Foundation Compositio Mathematica 2010

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