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Canad. J. Math. Vol. 54 (4), 2002 pp. 673--693 Local LFunctions for Split Spinor Groups

Summary: Canad. J. Math. Vol. 54 (4), 2002 pp. 673--693
Local L­Functions for Split Spinor Groups
Mahdi Asgari
Abstract. We study the local L­functions for Levi subgroups in split spinor groups defined via the
Langlands­Shahidi method and prove a conjecture on their holomorphy in a half plane. These results
have been used in the work of Kim and Shahidi on the functorial product for GL 2 ×GL 3 .
1 Introduction
The purpose of this work is to prove a conjecture on the holomorphy of local Lang­
lands L­functions defined via the Langlands­Shahidi method in split spinor groups.
These local factors appear in the Euler products of global automorphic L­functions
and information about their holomorphy is frequently exploited in order to prove
results about the analytic properties of global objects. In particular, in a recent im­
portant work, H. Kim and F. Shahidi have used some cases of our result here in order
to handle some local problems in their long­awaited result on the existence of sym­
metric cube cusp forms on GL 2 (cf. [11], [12]).
Apart from trace formula methods, two methods have been suggested to study
these factors: the Rankin­Selberg method which uses ``zeta integrals'' and the
Langlands­Shahidi method which uses ``Eisenstein series''. Our focus in this work
is on the latter [13], [16], [18], [20].
Let M be a (quasi) split connected reductive linear algebraic group defined over


Source: Asgari, Mahdi - Department of Mathematics, Oklahoma State University


Collections: Mathematics