Generic transfer for general spin groups

Abstract

We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group are not self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp 4 to GL 4 follows from our result as a special case.

On the sign of regular algebraic polarizable automorphic representations

“…Our first result shows that this additional hypothesis is in fact superfluous: The result is obvious when N is odd, so from now on we let N = 2n be even. The key ingredient in the proof of theorem is the descent of to a suitable GSpin group, thanks to the work of Asgari-Shahidi [3,4] and Hundley-Sayag [14,15]; there the sign ω v (−1) of interest can be interpreted as a central character, where it is determined by the 'parity' of the corresponding discrete series representation at v.…”

“…Whereas these Galois representations are constructed via a descent to appropriate unitary groups, to understand the parity condition we make use of a descent to quasi-split GSpin groups, using work of Asgari-Shahidi and Hundley-Sayag (see [3,4]), Hundley-2Sayag [14,15].…”

Langlands Functoriality Conjecture

“…Asgari and Shahidi [7], [8] proved that if π is a generic cuspidal representation of GSpin m , then the functoriality is valid for the embedding (5.3). If H = SO(2n + 1), for generic cuspidal representations, Jiang and Soudry [38] proved that the Langlands functorial lift from SO(2n + 1) to GL(2n) is injective up to isomorphism.…”

Infinite Dimensional Groups and Automorphic L-Functions

“…This will be the main theme of the present paper. We refer to [56] for a discussion of the conjecture for more general groups and to [3,4,8,9,30] for the original papers.…”

“…In particular, one needs to use base change, both normal and non-normal cubic [3,37] , as well as a result from theory of K-types [7] which is an appendix to [34]. As a bonus one gets the equality of triple product L-functions L(s, π 1v × π 2v × σ v ) and root numbers ε(s, π 1v × π 2v × σ v , ψ v ) defined by Langlands-Shahidi method [52] with Artin factors for all v. Note that L(s, π 1v × π 2v × σ v ) and ε(s, π 1v × π 2v × σ v , ψ v ) are defined completely by methods of harmonic analysis (cf.…”