Compositio Mathematica

Research Article

Residues of intertwining operators for SO*6 as character identities

Freydoon Shahidia1 and Steven Spallonea2

a1 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA (email:

a2 Department of Mathematics, University of Oklahoma, Norman, OK 73072, USA (email:


We show that the residue at s=0 of the standard intertwining operator attached to a supercuspidal representation πxs2297χ of the Levi subgroup GL2(FE1 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.

(Received March 18 2009)

(Accepted August 12 2009)

(Online publication April 20 2010)

2000 Mathematics Subject Classification

  • 22E50 (primary)


  • p-adic representation theory;
  • intertwining operators;
  • character identities;
  • L-functions


The first author was supported by NSF Grant DMS-0700280.