Let G be a real reductive Lie
group, 𝜃 a Cartan involution of G, and B = N0A0M01 a fixed minimal
parabolic subgroup of G. Fix H a 𝜃-stable Cartan subgroup of G, and assume
that if H = H+H− is the decomposition of H relative to 𝜃, then H−⊂ A0.
Let χ ∈H+, then following Harish-Chandra, we introduce the functions of
type II(χ). It is known that the positive chamber A0(B) can be covered by
sectors A0(B|Q), where Q varies over the maximal parabolic subgroups of G
which are standard with respect to B. Let Q = NM be such a maximal
parabolic for which the split component of M can be conjugated by an element
of G into H−. We show that given a function φ(x,λ) of type II(χ) there
exists an asymptotic expansion (along Q) for this function with the following
properties: first the partial sums uniformly approximate the function as λ varies
over H−, and x varies over A0(B|Q), and second, the terms in this sum
are essentially functions of type II(χ) on an approximate Levi subgroup of
M.
