270

M . Assem

for 5 € С, Re(5) > О, where \dx\ denotes the Haar measure on F"* normalized such that Op has measure 1. It is known that Z^{s^x^f) is a rational function in ^~* (see [6], [2]). The link between unipotent orbital integrals and Igusa local zeta functions is provided by Ranga Rao's formula ([15]) which holds for any local field of characteristic zero.

Let G denote a connected reductive algebraic group defined over F and К г, a, maximal compact subgroup of G(F) (we assume К is "good" in the sense of Bruhat and Tits when F is non-axchimedean). Let и e G(F) be unipotent and X € g := Lie(G(jP)) such that expX = u (recall that the exponential map is defined, one-to-one, and submersive on an open neighbourhood of 0 in g which is Ad G(F) -invariant and tains all the nilpotent elements). Let M , д(г), г G Z be as in the discussion about P.V.S.'s, and set П2 := ф g(i). It is known that M(jF) acts sub-

•>2

mersively on g(2), hence the M(F) -orbit V(X) of X is open in g(2) with respect to the Hausdorff topology. Moreover, the G(F) -orbit of X is equal to AdK{V{X) + n2) » Finally we introduce Ranga Rao's (^-function as follows. Let B( , ) denote a non-degenerate G(F) -invariant bilinear form on g which coincides with the Killing form on the derived algebra of g. Let X € g(2). Identifying g(—1) with g(l) via S( , ), we may (and do) regard ad(-X") as a linear transformation of g(l). It turns out that det(ad Jf ) =: (p^{X), for some polynomial 9 . Note that (p depends on a choice of basis for g(—1) and g(l). We will choose a basis such that (p has coefficients in Q. It also turns out that tp is г. relative invariant for the RV.S. (M(F), p,g(2)) and is non-trivial iff g(l) ф (0). We normalize measures as follows. When F is archimedean we equip П2 and g(2) with the usual Lebesgue measures, and when F is non-axchimedean we assume that t\2{0f) and q{2){0f) both have measure one. We also normalize the Haar measure on G(F) by requiring that voltmie (K) = 1. Let 0(u) denote the conjugacy class of u in G(F). Then 0{u) is diffeomorphic to Zg(|?)(u)\ G(F) where Zg(F)(^) is the centralizer of u in G(F). Since ZG(/r)(u) is unimodular, 0{u) is endowed with a G(F)-invariant measure which is unique up to a multiplicative positive constant. Let Cc(G(F)) denote the space of complex valued, continuous and compactly supported functions on G(F). For / 6 Cc(G(F)) we shall write /^^^j / for the inte-