Summary: Canad. J. Math. Vol. 54 (4), 2002 pp. 673--693
Local LFunctions for Split Spinor Groups
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 .
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. , ).
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 , , , .
Let M be a (quasi) split connected reductive linear algebraic group defined over