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R. Lawther's Centres of centralizers of unipotent elements in simple PDF

By R. Lawther

ISBN-10: 0821847694

ISBN-13: 9780821847695

Quantity 210, quantity 988 (first of four numbers).

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Bob Altemeyer
May, 2010
(http://home. cc. umanitoba. ca/~altemey/)

Extra info for Centres of centralizers of unipotent elements in simple algebraic groups

Sample text

2. Here we constructed Cr of type A1 , again giving subgroups Xγ , and explicit isomorphisms xγ : Ga → Xγ , for γ ∈ {±β}. In each case, we note that X±β is contained in a subsystem 42 8. A COMPOSITION SERIES FOR THE LIE ALGEBRA CENTRALIZER 43 subgroup H, which is a commuting product H = H1 . . Hs where each Hj is of type Aij for some ij ∈ N. 301–302] to find a cocharacter ψj : k∗ → T ∩ πj ( X±β ), such that for all t ∈ k and c ∈ k∗ we have ψj (c)πj (xβ (t))ψj (c)−1 = πj (xβ (c2 t)). If each Hj is of type A1 or A2 , we define hβ = sj=1 ψj .

For those which are irreducible we shall give a high weight vector; for those few which are reducible we shall give generating vectors. 1. Parametrizing a maximal torus of [C ◦ , C ◦ ]. 2 to determine the weights of [C ◦ , C ◦ ] on each Zn,m . 2 that we have [C ◦ , C ◦ ] = C1 . . Cr , a commuting product of simple groups; for each i we have obtained a simple system Π(Ci ) and the corresponding root subgroups, and we shall now give an appropriate maximal torus of Ci . For each β ∈ Π(Ci ), we will define a cocharacter hβ : k∗ → T with the following properties: its image lies in Ci , and for all β ∈ Π(Ci ), t ∈ k and c ∈ k∗ we have hβ (c)xβ (t)hβ (c)−1 = xβ (cnββ t), where the nββ are the integers given by the Cartan matrix of the root system Φ(Ci ).

2 that C ◦ is simple of type A1 , and hence is equal to X±β . Here we define hβ by hβ (c) = hα1 (c6 )hα2 (c9 )hα3 (c12 )hα4 (c18 )hα5 (c15 )hα6 (c10 )hα7 (c5 ); one then checks that im(hβ ) ⊆ Z(L)◦ ⊆ C ◦ , and hβ (c)xβ (t)hβ (c)−1 = xβ (c2 t) for all c ∈ k∗ and t ∈ k as required. If instead G = E8 , we proceed exactly as before. Finally consider the A4 A2 A1 orbit; here G = E8 . 2 that C ◦ is simple of type A1 , and hence is equal to X±β . This time we define hβ by hβ (c) = hα1 (c10 )hα2 (c15 )hα3 (c20 )hα4 (c30 )hα5 (c24 )hα6 (c18 )hα7 (c12 )hα8 (c6 ); one then checks that im(hβ ) ⊆ Z(L)◦ ⊆ C ◦ , and hβ (c)xβ (t)hβ (c)−1 = xβ (c2 t) for all c ∈ k∗ and t ∈ k as required.

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Centres of centralizers of unipotent elements in simple algebraic groups by R. Lawther

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