Computing twisted KLV polynomials

1 The Setup {s:setup} The starting point is: a group G, a Cartan involution θ, and another involution σ of finite order, commuting with θ. It is natural to consider the coset σK = {σ ◦ int(k) | k ∈ K} ⊂ Aut(G). Every element of this coset commutes with θ. We’re mainly interested when σ is an involution, especially the case σ = θ. Now fix a pinning P = (H,B, {Xα}), and write δ, ǫ ∈ Aut(G) for the images via the embedding Out(G) →֒ Aut(G) (the image consists of Pdistinguished automorphisms). Then δ, ǫ ∈ Aut(G) commute. We now introduce the usual atlas structure. See [1] for details. Let G = G⋊ 〈δ〉 be the usual extended group; σ acts on it, trivially on δ. Recall X = {ξ ∈ NormGδ(H) | ξ 2 ∈ Z(G)}/H, and X̃ is the numerator. We’ll write x for elements of X , ξ for elements of X̃ , and p : X̃ → X . It is important to distinguish between elements of X and X̃ . For ξ ∈ X̃ , let θξ = int(ξ) ∈ Aut(G), Kξ = G θξ .The restriction of θξ to H only depends on p(ξ) ∈ X , and is denoted θx. It is important to remember θx is only an involution of H, not of G, and Kx is not well defined. It is immediate that σ(X ) = X . After conjugating we can assume θ = int(ξ0) for some ξ0 ∈ X̃ . Let x0 = p(ξ0), and define Xθ = {x ∈ X | x is G-conjugate to x0} (G-conjugacy