By Hanspeter Kraft

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Exercise. Show that the centralizer of Tn in GLn is equal to Tn . As is standard we define the center of a group G to be center Z(G) := {g ∈ G | gh = hg for every h ∈ G} = CG (G). The center Z(G) of an algebraic group G is a closed characteristic subgroup of G. 8. Example. Consider the group a A= c Z(G) O2 b ∈ GL2 | At A = E d ⊆ GL2 . Obviously, det(O2 ) = {±1}, and so O2 is not connected. It is easy to determine the center: Z(O2 ) = {±E}. Let SO2 (C) = SO2 := {A ∈ O2 | det A = 1} = O2 ∩ SL2 = { a −b b | a2 + b2 = 1}.

Definition. An element u of an algebraic group G is called unipotent if either u = e or u C+ . If all elements of G are unipotent, then G is called a unipotent group. 4. Example. Clearly, the groups Un are unipotent, as well as every closed subgroup of them. 8). 2). ex If ϕ : G → H is a homomorphism and if u ∈ G is unipotent, then ϕ(u) ∈ H is unipotent. Embedding G into GLn we also see that the set Gu ⊆ G of unipotent elements of G is a closed subset. 6 below). 5. Exercise. Let U be a unipotent group.

Show that X (Tn ) ⊆ O(Tn ) is a C-basis. Exercise. Show that SL2 is generated by U2 and U2− . In particular, the character group X (SL2 ) is trivial. (Hint: U2− U2 ⊆ SL2 is closed and irreducible of dimension 2, U2 U2− U2 is strictly larger than U2− U2 , and therefore dense in SL2 . ) Exercise. lem Z, generated by det : GLn → C∗ . 4 to show that Tn := Tn ∩ SLn is contained in Un− , Un . Since Un− Tn Un ⊆ SLn is dense, we get that X (SLn ) is trivial. ) Exercise. For two algebraic groups H, G we have X (H × G) = X (H) ⊕ X (G) in a canonical way.