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By R. Lawther

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

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In case (i), we find that the T0 -weights occurring in Zn,2 (with multiplicities) are 8, 6, 4, 4, 2, 2, 0, 0, 0, −2, −2, −4, −4, −6, −8. We deduce therefore that Zn,2 has a tilting module summand of high weight 8. We may take as a basis for Zn,2 the 8. A COMPOSITION SERIES FOR THE LIE ALGEBRA CENTRALIZER 47 following vectors, where in each case the subscript denotes the T0 -weight: w8 = e 234321 , 2 w6 = e 123321 − e 123221 , 1 2 w4 = e 122210 + e 122111 , w4 = e 012221 −e 122210 −e 112211 , w2 = e 001111 − e 011111 − e 111110 − e 111100 , w2 = e 001111 +e 011110 +e 111100 , 1 1 w0 = e 000000 , 1 1 0 0 1 1 1 1 1 1 1 w0 = e 100000 + e 010000 , w0 = e 000100 +e 000010 +e 000001 , 0 0 1 0 0 0 0 0 0 0 w−2 = f 111000 − f 011000 − f 001100 − f 001110 , w−2 = f 111000 +f 011100 +f 001110 , 0 1 w−4 = f 012111 + f 012210 , 1 1 w−4 = f 112110 +f 012111 +f 122100 , 1 1 1 w−6 = f 123211 + f 123210 , 1 2 w−8 = f 124321 .

T , one for each simple factor, and the cocharacter which we will associate to e is the product of the τi ; that is, τ (c) = t ∗ i=1 τi (c) for all c ∈ k . 6 we must verify that im(τ ) ⊆ [L, L] ∩ T and τ (c)e = c2 e for all c ∈ k∗ ; the first of these is true by construction, while the second is a simple calculation. At the top of the page for e in §11 we shall represent τ by its labelled diagram, as explained at the beginning of §3; we shall also give the labelled diagram Δ of the nilpotent orbit containing e.

A basis of CL(G) (e)+ and its upper central series. We begin with L(R) = CL(G) (e)+ ; we shall first obtain a basis of this subalgebra. In order to do this we introduce a second grading on L(G), which will lead to a refinement of that defined by the cocharacter τ . We have the torus Z(L)◦ ; write X(Z(L)◦ ) for its character group. For χ ∈ X(Z(L)◦ ) write L(G)χ = {v ∈ L(G) : (Ad t)v = χ(t)v for all t ∈ Z(L)◦ } for the corresponding Z(L)◦ -weight space of L(G); we then have the grading χ L(G) = L(G) .

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