The main aim of this article is to show that normalised standard intertwining operator between induced representations of p-adic groups, at a very specific point of evaluation, has an arithmetic origin. This result has applications to Eisenstein cohomology and the special values of automorphic L-functions.

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.

In this paper we study the kernels and the Lp–Lq boundedness properties of some intertwining operators associated to representations of SL(n, R).

In this paper, we determine the residues at poles of standard intertwining operators for parabolically induced representations of an arbitrary connected reductive quansisplit algebraic group over a p-acid field whenever the unipotent radical of the parabolic subgroup is Abelian. We then interpret these residues by means of the theory of endoscopy.