ON STABILITY AND ENDOSCOPIC TRANSFER, SECTION 0 5

one of our aims was to get a feeling for the structure of the stability packets.

There are two main types of calculations used in this article. One is p-adic and the other is a multivariate

residue calculation. Perhaps a feature of the p-adic calculation is a general method to calculate the orbital Igusa zeta

functions. What is needed to apply this method to a general prehomogeneous space (G(F), V ) , G a reductive split

group is to understand the G(OF ^decomposition of V and obtain local density results analogous to those known for

symmetric matrices (cf. [12] and [26]). Although the calculation of orbital Igusa zeta functions are needed for the

arithmetic applications of prehomogeneous spaces, they have not been treated (to the best of our knowledge) in any

systematic fashion. We hope to consider this problem somewhere else.

The residue calculation is used to calculate

fH(l)

with the help of Macdonald's explicit formulae for the Satake

transform and spherical Plancherel measure. The actual calculation is rather long and requires some care in choosing

the order in which the various poles are considered. We have tried to condense the calculation as much as possible

using a set of diagrams.

The organization of this paper is as follows. In Section 1 we introduce the notion of Ranga Rao data, discuss

some elementary aspects of the prehomogeneous space associated to a unipotent orbit, and prove some lemmas about

induction of F-unipotent orbits.

In Section 2, we calculate the Igusa local zeta functions on the space of 2 x 2 and 3 x 3 symmetric matrices,

for all test functions associated with a basis of the spherical Hecke algebra. Part of that calculation is relegated to

Appendix I.

Section 3 deals with residue calculations. Also, the matching results for the orbits associated with the partitions

2

3

1

2(n-3)

a r e g i v e n t h e r e

In Section 4, we state and prove our main results alluded to above. We also discuss some transfer factors.

The discussion in Section 5 is mostly speculative. There, we try to provide some heuristics behind the Conjectures

A, B, and C; as well as pose some new ones. In particular, we suggest the structure of the packets predicted by

Conjecture C and the transfer of certain distributions for classical split groups and we prove the stability of the

distributions associated to the subregular packets in Sp(2n) and SO(2n + 1).

We have provided a second appendix, consisting of some tables for small rank groups that are perhaps helpful

in deciding which orbit is endoscopically induced from which.

Finally, I would like to sincerely thank the Department of Mathematics at the University of Toronto, in particular

Jim Arthur, as well as the Fields Institute and V. Kumar Murty, for the financial help and hospitality. I would also

like to thank Bob Kottwitz for some useful discussions and for informing me about his results ([16]).

Notation. Throughout this article, F will denote a p-adic field of characteristic zero, OF the ring of integers,

and PF the maximal ideal of OF- The order of the residue field OF/PF is denoted by g, and we shall assume

that 9 = 1 mod 4. This assumption is not really necessary since our results will still hold (after the appropiate

reformulations) for any prime power q. The assumption that q is odd is made to reduce the number of F-rational

unipotent orbits lying within a given stable unipotent orbit, and to simplify the density formulae used in Chapter 2.

The assumption that g = 1 mod 4 is made for notational convenience. To formulate our results in the case q = 3 mod 4

requires only a shift in the way we parametrize the unipotent orbits. We also let 7r denote a uniformizing parameter

of Op, and e a Teichmuller representative. Thus 1, e, 7r, en represent the square classes in

Fx.

The absolute value

function | • | is normalized such that |7r| =

q~l.