On Principal Values on P-Adic Manifolds

In the paper [L] a project for proving the existence of transfer factors for forms of SL(3), especially for the unitary groups studied by Rogawski, was begun, and it was promised that it would be completed by the present authors. Their paper is still in the course of being written, but the present essay can serve as an introduction to it. It deals with SL(2) which has, of course, already been dealt with systematically [L-L], the existence of the transfer factors being easily verified. Thus it offiers no new results, but develops, in a simple context, some useful methods for computing the principal value integrals introduced in [L]. We describe explicitly the Igusa fibering, form and integrand associated to orbital integrals on forms of SL(2), taking the occasion to clarify the relation of this fibering to the Springer-Grothendieck resolution (cf. §3). The Igusa data established, there are two problems: (i) to show that certain principal values are zero, (ii) to compare principal values on two twisted forms of the same variety. To deal with the first we have, in §1, computed directly some very simple principal values on P 1 , and shown that principal values behave like ordinary integrals under standard geometric operations such as fibering and blowing-up. The second problem is dealt with in a similar way, by using Igusa's methods to establish, in a simple case, a kind of comparison principle (Lemma 4.B). The endoscopic groups for a form of SL(2) are either tori or SL(2). For tori the solution of the first problem (Lemma 4.A) leads immediately to the existence of transfer factors, and the hypotheses of [L 1 , pp. 102, 149] are trivially satisfied. If G is anisotropic over F and the endoscopic group is SL(2) the solution of the second problem (Lemma 4.B with κ ≡ 1) and the characterization of stable orbital integrals (cf. [V]) yields the existence of transfer factors as well as the local hypothesis of [L 1 , p. 102]. The principal values which arise for forms of SL(2) are computed without difficulty, but we expressly avoid such calculations. The aim of the project begun in [L], and continued here, is to develop methods for proving the existence of transfer factors which appeal only to geometric techniques of some generality and thus have some prospect of applying to all groups. One encouraging sign is the smoothness with which they …