2016/05/23 by Lemaire, Bertrand
#22E50 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1605.07076
Let F be a non--Archimedean local field of characteristic ≥ 0, and let G=GL(N,F), N≥ 1. An element γ∈ G is said to be quasi--regular if the centralizer of γ in M(N,F) is a product of field extensions of F. Let G\rm qr be the set of quasi--regular elements of G. For γ∈ G\rm qr, we denote by Oγ the ordinary orbital integral on G associated with γ. In this paper, we replace the Weyl discriminant \vert DG\vert by a normalization factor ηG: G\rm qr→ \Bbb R>0 which allows us to obtain the same results as proven by Harish--Chandra in characteristic zero: for f∈ C^∞\rm c(G), the normalized orbital integral IG(γ,f)=ηG1\over 2(γ)Oγ(f) is bounded on G, and for ε>0 such that N(N-1)ε<1, the function ηG^-1\over 2-ε is locally integrable on G.