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9) Theorem. 6). Let k be such that 0 < k < and let T = {/lj 10 < j < W2, /lj(l) = W2 /lk(l)}. (a) If /lj E T, then /lj E T and /lj =I- /lj. Also, 0=1- Z[T, L#] = Z[T, A]. (b) The Z-linear mapping from Z[T] to Z[Irr G] which sends the character /lj to Ok LOSi

Let Tl be an isometry from Z[Sl] to Z[Irr G] which extends the restriction ofT to Z[Sl' L#]. Set (Xi -aiXlr = ~() (Xl(1)Xi Xl 1 xi(l)xd T ; this is compatible with previous notation if ai E N. 1) Let (X - aXlt = X - Y, where X E Z[R(X)] and Y is orthogonal to R(X). There is an integer A E Z such that n Y _ TI - aXl - A ' " ai ~ -II '112 XiTI ,=1 X, Z + , where Z E CF( G) is orthogonal to S{'. Proof. Set Y = ax? - 2:;'=1 AiX? + Z with Ai E C and where Z E CF(G) is orthogonal to S{'. For 1 ::; i ::; n, X?

32, it follows that Irr(K) has at most W2 elements left fixed by g. But, since {lOj E Irr(L), Xj is fixed by g. It follows that, if X E Irr(K) is not one of the characters Xj, then X is not fixed by g. b), Ind~ X is irreducible. Moreover, (Ind~ X, {lij) = (X, Xj) = O. Finally, if {l E Irr( L) and if X is an irreducible component of Res~ {l, then {l is a component of Ind~. X, and so is one of the characters {lij or is of the form Ind~ X. 6) Hypothesis. 2). 1). (c) Let H be a normal subgroup of L such that W 2 c H C f{.