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

Abstract

We show that the residue at s = 0 of the standard intertwining operator attached to a supercuspidal representation π ⊗ χ of the Levi subgroup GL 2 (F ) × E 1 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 E 1 and the character on E × defining the representation of GL 2 (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 GL 2 (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.

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

“…Most notable is the work of Spallone [33] and the joint work of the second named author and Spallone [30], which give more precise and definitive results in certain cases that have been the subject of our study.…”

On a pairing of Goldberg-Shahidi for even orthogonal groups

“…After these apéritifs, we are able to state the formula for geometric transfer factors in §4.4. We will also prove the Lemma 4.3.1 that supersedes [29,Proposition 8].…”

“…The applicability of Waldspurger's formula is inextricably liked with the first two points. As a consequence, our notations will deviate somehow from those of [8,9,10,29,31]. We hope the reader will be convinced of the flexibility of twisted spaces, especially in the context of classical groups of higher rank.…”

Towards a Goldberg–Shahidi pairing for classical groups

“…These intertwining operators are used to define Langlands-Shahidi L-functions, and the project connects the theory of L-functions to functoriality. We refer the reader to the papers ( [17], [18], [7], [8], [9], [23], [19], [6], [25], [12], [28], [27]) of Goldberg, Shahidi, Wen-Wei Li, Li Cai, Bin Xu, Xiaoxiang Yu, Varma and the second author for details and progress.…”