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HOMEWORK 8 FOR 18.747, SPRING 2013 DUE FRIDAY, APRIL 12 BY 3PM. In all problems G is a connected linear algebraic group over a field k. (1) Let B ⊂ G a Borel subgroup and set B = G/B. For u ∈ G let Bu = {x ∈ B | u(x) = x}, this is a closed subvariety in Bu . (a) Let G = GLn (n ≥ 3), let u be a unipotent element with dim Ker(u − 1) = 2, dim Ker(u − 1)2 = 3. Show that Bu is a union of n − 1 irreducible components, each isomorphic to P1 . Describe the pattern of intersection of these components. (b) (Optional) Let G = Sp4 . Find a unipotent u ∈ G such that dim(Bu ) = 1. Show that such elements form one conjugacy class. For such an element u show that Gu has three irreducible components, L0 , L1 , L2 , with Li ∼ = P1 and such that L0 intersects L1 , L2 at one point but L1 ∩ L2 = ∅. Find s ∈ G commuting with u such that s permutes L1 with L2 . (2) Let g = su be an element in G with Jordan decomposition. Check that u lies in the connected component of identity of the centralizer of s. (3) Let C be a maximal nilpotent closed subgroup in G such that C is the connected component of identity in its normalizer. Then C is a Cartan subgroup. 1