By Patrick Kornprobst

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**Example text**

The centralizer in G of 1) is the group Na(VI , . ch G1J for everyone of the remaining classical groups contains one of these centralizing a subspace of codimension 2, 3, 2, 2, respectively. In the last two cases, the matrices = ~ G(1J) G1J = Na{VI , ... , Vi} = Na(VI , ... 10 54 so that G'D acts faithfully as permutation group on the t spaces in 'D. We say that G stabilizes 'D if G = G1J and that G centralizes 'D if G = G(1J). 11. Suppose tbat G :::; GL(V) and tbat G centralizes tbe subspace de- composition 'D = {Vi, ...

8. We now prove the two 'if' assertions simultaneously. Assume that pt/l is equivalent to p*, where 1/J E Aut(F) and IV)I ::; 2. 1). Since G7) is transitive, we have Ai = Aj for all i,j, which is to say c acts as a scalar on all of V. 1. • We conclude this chapter with some relatively easy results concerning quasiequivalent representations, and the relationship between representations and non-degenerate forms. GL(V, F) is an absolutely irreducible represen- only if p is equivalent to the dual p*.

But clearly there are singular vectors in V\ (el' ... 5). Also = 1,2), n contains the along with the element k of order 3 acting as el I-t el + II + e2 + 12, e2 I-t el + 12, II I-t el, to verify that COL(V)( (g, h, k)) = F*. When 12 I-t el + e2. Then we leave it to the reader = -, let el, II, x, y be a standard basis as COL(V)( (rxry, r e1 +It 1'x, r e1 +xTx)) = F*, and € by induction on n. 9, so we prove only (i). 13). The case n ~ Sp4(q) is left as an exercise. 3). We complete the proof *' n~(q).