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N}, j > i], n(n−1) implying that Un ∼ is irreducible. Since G is irreducible, Gu is irre=K 2 ducible as well. We proceed to describe the orbit closure relation on Gu : Elements of Gu are conjugate if and only if their Jordan normal forms coincide. The latter up to reordering are uniquely described by the sizes λ1 , . . , λl ∈ N, for some l ∈ N, of the Jordan blocks Jλi (1) ∈ Kλi ×λi with respect to the l eigenvalue 1 ∈ K occurring. We have i=1 λi = n and we may assume that λ1 ≥ · · · ≥ λl ≥ 1; then the conjugacy class associated to λ := [λ1 , .

N ] n. Then µ is called to k k dominate λ, if for all k ∈ {1, . . , n} we have i=1 λi ≤ i=1 µi ; we write λ µ. 25). d) We have λ max µ if and only if µ = [λ1 , . . , λr−1 , λr + 1, λr+1 , . . , λs−1 , λs − 1, λs+1 , . . 26): II Algebraic groups 32 If λ max µ, let r := min{i ∈ {1, . . , n}; λi = µi } and r < s := min{k ∈ k k {r + 1, . . , n}; i=1 λi = i=1 µi } ≤ n. Hence we have λr < µr , and µr ≤ µr−1 = λr−1 if r > 1, as well as λs > µs ≥ µs+1 ≥ λs+1 . This yields λ ν := [λ1 , . . , λr−1 , λr + 1, λr+1 , .

The morphism Bn → Tn : [aij ] → diag[a11 , . . , ann ] restricts to the morphism G → Gs : g → gs , hence µ−1 : G → Gs × Gu : g → [gs , gs−1 g] is a morphism. 1) Lemma. Let G be an algebraic group acting morphically on V , and let x ∈ V . Then we have dim(xG) = dim(xG◦ ) and dim(CG (x)) = dim(CG◦ (x)), as well as dim(G) = dim(CG (x)) + dim(xG). Proof. 22). Since CG◦ (x) ≤ CG (x) is a closed subgroup of finite index, we have dim(CG (x)) = dim(CG◦ (x)). Hence to show the last assertion, we may assume that G is connected.

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Algebraic Groups by Jürgen Müller


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