Involutions of reductive Lie algebras in positive characteristic

Abstract

Let G be a reductive group over a field k of characteristic = 2, let g = Lie(G), let θ be an involutive automorphism of G and let g = k⊕p be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group G θ on p is well understood, since the well-known paper of Kostant and Rallis [17]. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic.Among other results, we prove that the variety N of nilpotent elements of p has a dense open orbit, and that the same is true for every fibre of the quotient map p → p/ /G θ . However, we show that the corresponding statement for G, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of N , extending a result of Sekiguchi for k = C. Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants k[p] k . morphism π : P → P/ /K is false; finally, we apply a theorem of Skryabin to describe the ring k[p] K i , where K i is the i-th Frobenius kernel of K.A torus A in G is θ-split if θ(a) = a −1 for all a ∈ A. It was proved by Vust that the set of maximal θ-split tori are K-conjugate. Let a be a toral algebra contained in p. If a is maximal such, then by abuse of terminology we say that a is a maximal torus of p.Lemma 0.1. Let a be a maximal torus of p. Then z g (a) ∩ p = a, and there exists a unique maximal θ-split torus A of G such that Lie(A) = a.

Exposing relative endoscopy in unitary symmetric spaces

“…Since θJ = Jθ, the involution θ on Res E/F GL 4 gives a well-defined involution on U 4 . For computational purposes, it is necessary to write down explicitly the F points of the −1-eigenspace g 1 in terms of matrices.…”

“…We prove a fundamental lemma for (U (4), θ) where θ is an involution such that U(4) θ ∼ = U(2) × U(2) and when γ is of the form γ = diag(x, y, −y, −x) with x = ±y ∈ F × . Motivated by the usual fundamental lemma for unitary groups, we define for a nontrivial κ : D(I γ ) → C × the endoscopic symmetric space to be (H, θ H ) = (U 2 , σ )×(U 2 , σ ) where σ : U 2 → U 2 is such that U σ 2 ∼ = U 1 ×U 1 .…”

“…Indeed, it maps to σ → g −1 σ (g) = (πI 4 , π −1 I 4 ). We remark to the unwary reader that σ here acts on the right of E ⊗ F E, which on E ⊕ E is the same (a, b) → (b, a).…”

“…Hence, σ → (πI 4 , π −1 I 4 ) is a nontrivial cocycle because π is not a square in E. We set γ stc = Ad(B)γ . Now let κ : D → {±1} be a character.…”

“…This is equivalent to the orbit G 0 ·x having minimal dimension. For this and further facts, the reader is referred to the paper [4]. and its Lie algebra is the functor given by…”