On Isotropic Bessel Models for Principal Series Representations of So

Introduction. Whittaker models have been crucial in the development of the theory of automorphic forms. Within the Langlands-Shahidi method, Whittaker models give rise to the definition of local coefficients [Sh1,Sh2], which proved critical in establishing the proof of Langlands’s conjecture relating local L–functions to Plancherel measures for generic representations [Sh3]. This, in turn, led to the recent breakthroughs in functoriality [K,KS], the transfer of generic cusp forms from classical groups to general linear groups [CKPSS1,CKPSS2], and progress on the local Langlands conjecture for SO2n+1 [JS]. This raises the question of whether other models can be used to obtain similar results for the non-generic spectrum. In [FG] we used the generalized Bessel models of [GPSR] to give some preliminary results on local coefficients for non-generic representations of certain classical groups. There, the analog of Rodier’s Theorem is established, subject to a minimality condition, and this leads to results on local coefficients of the type in [Sh1]. In order to carry the philosophy of the LanglandsShahidi method forward for non-generic representations of these groups, several questions need to be answered. In particular, the minimality condition has to be relaxed or removed, the analog of the results of [Sh2] need to be established for these models, and global methods will need to be exploited to obtain results relating Plancherel measures and L–functions. In addressing the questions on local coefficients for Bessel models of groups over archimedean fields, several issues arise, including the existence and uniqueness of such models. Here we address the existence of Bessel models for principal series of G = SOn,n+1(R). By the Casselman Embedding Theorem, every irreducible representation is a subquotient of such a principal series. Thus, we wish to investigate to what extent principal series have generalized Bessel models. We show that every such representation has a Bessel model with respect to a very particular character of a subgroup of SO1,2(R). In the language [FG] this model is rank one, and as remarked there [loc cit, Remark 2.10], the minimality condition is automatically satisfied. Thus, one can hope to carry out the steps outlined above. The Rodier result can be establsihed for these models, i.e., one can establish that the dimension of the space of Bessel models with compatible parameters is preserved under