Section 1. Unipotent orbits and prehomogeneous spaces 1.1. Ranga Rao data. In this section we introduce the notion of the Ranga Rao data associated to a stable unipotent orbit in a reductive group G defined over F. The main available tool for calculating unipotent orbital integrals is Ranga Rao's formula (cf. [27]). In [3] we explained how this formula can be viewed in the context of prehomogenous vector spaces and how unipotent orbital integrals can be regarded as special values of orbital Igusa zeta functions. Let g denote the Lie algebra of G(F). Stable orbits Throughout the rest of this manuscript, we shall use the following definitions. Let for any y 6 G(F) (not necessarily semi-simple), Ost(y) := {g~lyg : g G(F) and g~lga e G°S(F) for all a e Gal(F/F)} n G(F), where s is the semi-simple part of y in its Jordan decomposition, and G°s is the connected component of the centralizer of s in G (see [14]). Note that if n G G(F) is unipotent then Ost(u) = {g'lug : g G(F)} n G(F). Definition 1.1.1. Let Ost denote a stable unipotent orbit in G(F). The Ranga Rao data associated with Ost consists of an explicit description of the following. (i) The Z-grading g = ® g(z), where g(i) is i-eigenspace of ad(#) , H being the semi-simple member of the t€Z s/2-triplet associated with Ost via the Jacobson-Morozov theorem. (ii) The prehomogeneous vector space (M(F),g(2)), where M is the identity component of the centralizer of H in G. (hi) The Ranga Rao function (p. (iv) The open M(F)-orbits in g(2). We indicate below how some of that data can be obtained. Recall first that the stable unipotent orbits in Sp(2n, F) are parametrized by partitions of 2n, called admissible, such that every odd part appears an even number of times (a partition of m is a sequence of non-negative integer (/ii, ,/Ltfc) such that ^i\ fi2 * Hk a n d /ii + H + Hk = ™\ the integers /i{, 1 i k, are called the parts of the partition. The multiplicity of a part is the number of its appearances in the partition). Assume A = (AJ1, •, A£r) is such a partition, where dj indicates the multiplicity of Aj in A. The conjugacy classes within the stable class corresponding to A are in one to one correspondence with the set of quadratic forms (Qi, ,Qr)i where Qi is an orthogonal (resp. symplectic) form of rank a* if Aj is even (resp. odd) (cf. [31]). We shall realize our symplectic groups as follows. Sp(2n) := {g SL(2n) : Vn 7 = Jn} , where and In = n x n identity matrix. 6
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