Jordan canonical form by Chris Kottke

By Chris Kottke

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3, in which we used the pairing x, y = xi yi . e. there is no split maximal torus. The basis 1 1 √ ,√ 2 2 , 1 1 √ , −√ −2 −2 realizes an isomorphism between√SO(2n) and {g ∈ SL(2n) : t g · g = 1}, but only √ after base change to the field Q( 2, −2). It is easy to verify that so2n = a b c − ta : a, b, c ∈ Mn and t b = −b, t c = −c . For a tuple λ = (λ1 , . . , λn ) ∈ Gnm , write diag∗ (λ) = diag(λ1 , . . , λn , λ1−1 , . . , λ−1 n ). Let T = {diag∗ λ : λ ∈ Gnm }; since ZSO(2n) (T ) = T , this is a maximal torus.

Proof. We’ll use two facts. 1. For n n 1, let n T = ker(T − → T ). 7]. 2. For each n 1 invertible in k, the scheme n T is finite ´etale over k. 2] is finite ´etale. Since X is connected, each X → hom(n T, n T ) is constant. By the first fact, we obtain that X is constant. 11. 10 fails for groups that are not tori. Let G = SL(2). Then the map G × G → G given by (g, h) → ghg −1 is a perfectly good family of homomorphisms that is not constant. One has PSL(2) → Aut(SL2 ), with image of index two. The other coset is generated by g → t g −1 .

Let g ∈ G(k). Then f (g)ss = f (gss ) and f (g)u = f (gu ). In other words, the Jordan decomposition is functorial. It follows that if G/k is an affine algebraic group (without a choice of embedding G → GL(n)), then the multiplicative Jordan decomposition within G is well-defined, independent of any choice of embedding. 7 Diagonalizable groups Let k be a field; recall that we are working in the category of fppf sheaves on Schk . The affine line A1 (S) = Γ(S, OS ) is such a sheaf. We have already written Ga for the affine line considered as an algebraic group.

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