The Trace Formula in Invariant Form

Geometric estimates for the trace formula

“…In order to solve this problem, we give a different argument, which works for all T . The resulting geometric side of the trace formula depends polynomially on T (with a slight modification in case of positive characteristic), as already proved in [3] in the sufficiently regular range. The previous lack of knowledge that the explicit expression is convergent for all T was the source of certain complications in [4], which may now be avoided, at least for compactly supported f .…”

“…Of course, it has to be shown that all sums and integrals occurring here are absolutely convergent. We are going to prove this when f is a suitable extension of a function in a certain Schwartz space C p (G(F S ) 1 ), which has been introduced in the case p = 2 in [3], p. 28, and is some kind of tensor product of the spaces introduced by Harish-Chandra. Namely, let Ξ v and σ v be the functions on G(F v ) used in [11] and [13] in the definition of C(G(F v )) (cf.…”

“…Thus, for every P there is a compact set ω 1 ⊂ G(A) such that the support of F P (x, T ) is contained in P (F )N (A)A(A)ω 1 . By Lemma 2.1 of [3], the function Γ P P (H, X) is compactly supported in H ∈ a P P uniformly for X varying in a compact set Ω, say. Hence there is a compact set ω 2 ⊂ A(A) such that the corresponding term on the right-hand side of (iii) is supported on P (F )N (A)A (A)ω 2 ω 1 for X ∈ Ω.…”

Weighted orbital integrals

“…It has been shown ( [3], Lemma 8.1) that the integral converges and defines a tempered distribution J M (m) on G, i.e., a continuous linear functional on C(G). Note that v G is constant equal to 1, so that J G (g) is the ordinary (unweighted) orbital integral.…”

“…These are the main terms in the invariant trace formula [7]. There are two choices which are reciprocal in a certain sense ( [3], p. 7/8, [10], §3). It is analytically easier to modify the weighted orbital integrals.…”