Residues of intertwining operators for SO^*₆ as character identities

E-ISSN： 1570-5846|146|3|772-794

ISSN： 0010-437x

Source： Compositio Mathematica, Vol.146, Iss.3, 2010-05, pp. : 772-794

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Abstract

We show that the residue at s=0 of the standard intertwining operator attached to a supercuspidal representation πχ 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.