On classification of σq-conjugacy classes of a loop group

Let k   be an algebraically closed field and L=k((ϵ))L=k((ϵ)) the field of Laurent series over k. Let G be a connected reductive group over k such that the characteristic of k does not divide the order of the Weyl group of G  . For q∈k×q∈k×, the automorphism of L  , defined by ∑aiϵi↦∑ai(qϵ)i∑aiϵi↦∑ai(qϵ)i, induces an automorphism σqσq of the loop group G(L)G(L). We show that, when q   is not a root of unity, the classification of σqσq-conjugacy classes of G(L)G(L) can be reduced to the classification of unipotent classes of (non-connected) reductive groups. As an application, we recover the classification (of σqσq-conjugacy classes) for G=GLnG=GLn.