Download PDF by Martin W. Liebeck, Gary M. Seitz: Unipotent and nilpotent classes in simple algebraic groups

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By Martin W. Liebeck, Gary M. Seitz

ISBN-10: 0821869205

ISBN-13: 9780821869208

This e-book issues the idea of unipotent components in uncomplicated algebraic teams over algebraically closed or finite fields, and nilpotent components within the corresponding uncomplicated Lie algebras. those themes were a massive zone of research for many years, with functions to illustration idea, personality thought, the subgroup constitution of algebraic teams and finite teams, and the category of the finite easy teams. the main target is on acquiring complete info on classification representatives and centralizers of unipotent and nilpotent components. even supposing there's a large literature in this subject, this publication is the 1st unmarried resource the place such info is gifted thoroughly in all features. moreover, some of the effects are new--for instance, these pertaining to centralizers of nilpotent parts in small features. certainly, the entire procedure, whereas utilizing a few rules from the literature, is novel, and yields many new basic and particular proof about the constitution and embeddings of centralizers

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L. ˜ The Q ˜ − -orbit Q ˜ − P˜ in G/P˜ is open in its closure, Proof (i) Write P˜ − = Q ˜ − P˜ is open dense in G. and by dimension considerations, the closure is G. Hence Q We give the proof for x = e; the unipotent case is essentially the same. Suppose − ˜ − and p ∈ P˜ . Then eq− = ep−1 . But eq− ∈ e + L(Q ˜ − ), ep−1 ∈ eq p = e with q − ∈ Q 34 2. PRELIMINARIES ˜ − )∩L(P˜ ) = 0. It follows that C ˜ − ˜ (e) = C ˜ − (e)C ˜ (e). A similar e+L(P˜ ), and L(Q Q P Q P argument shows that CP˜ (e) = CQ˜ (e)CL˜ (e), so that CQ˜ − P˜ (e) = CQ˜ − (e)CQ˜ (e)CL˜ (e) and the dimension of CQ˜ − P˜ (e) is the sum of the dimensions of the individual centralizers.

Part (ii) is entirely similar, noting that eQ ⊆ e + L(Q)(≥2) . 7. Centralizers of nilpotent elements Continue to assume that G is a simple algebraic group over the algebraically closed field K. In this section we establish a key result (and some refinements of it) on the structure of the centralizers of nilpotent elements in L(G). The result holds in arbitrary characteristic. The section concludes with a result concerning the dimensions of centralizers of nilpotent elements. 25. Let L ¯ . Suppose there exists e be a distinguished nilpotent element of the Lie algebra of L ¯ , such that eT (c) = c2 e for all a 1-dimensional torus T = {T (c) : c ∈ K ∗ } in L c ∈ K ∗ .

Therefore, dim CP (l) = dim P − dim Q = dim L(P ) − dim L(Q). 4 of [26] yields dim CP (l) = dim CL(P ) (l). This implies that [L(P ), l] = L(Q). Now we use an argument of Jantzen to show that all weights of h are even. Write l = i>0 li for li ∈ L(Q)i . Intersect both sides of the equality [L(P ), l] = L(Q) of the last paragraph with L(P )1 + L(P )2 . 3) [L(P )0 , l1 + l2 ] + [L(P )1 , l1 ] = L(P )1 + L(P )2 . Suppose L(P )1 = 0. Then as l1 ∈ L(P )1 we have dim([L(P )1 , l1 ]) < dim(L(P )1 ). 37.

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Unipotent and nilpotent classes in simple algebraic groups and Lie algebras by Martin W. Liebeck, Gary M. Seitz

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