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Algebra IX: Finite Groups of Lie Type. Finite-Dimensional by A.I. Kostrikin, I.R. Shafarevich, P.M. Cohn, R.W. Carter,

By A.I. Kostrikin, I.R. Shafarevich, P.M. Cohn, R.W. Carter, V.P. Platonov, V.I. Yanchevskii

The first contribution through Carter covers the idea of finite teams of Lie variety, a big box of present mathematical study. within the moment half, Platonov and Yanchevskii survey the constitution of finite-dimensional department algebras, together with an account of decreased K-theory.

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Extra info for Algebra IX: Finite Groups of Lie Type. Finite-Dimensional Division Algebras (Encyclopaedia of Mathematical Sciences)

Example text

For example, suppose GF has type l&(q). The Weyl group is then dihedral of order 8 and we have f’? ,) x E g9(g). Now we can make CI(‘9) into a commutative ring by the convolution Thus @ C4G(d) We shall also write A(9) = &(9(P)) for convenience. We denote by xccb, the unipotent character in the family 9 corresponding to (x, (r) E A’(9). We next consider the relation between the unipotent characters and the almost characters. We recall that for each q4E I@ we have an almost character R, given by of the Finite + lY3 34(q2 + 11, Ml2 + 1x M4 - 1,‘).

Subsequently a version was described by Deligne in the context of algebraic varieties. We shall need Deligne’s version, since we are concerned with algebraic varieties over an algebraically closed field of characteristic p. For such varieties we consider l-adic intersection cohomology, where 1is a prime different from p. To be precise, it is possible to define, for each algebraic variety X over the field K = ‘FP, sheaves H’(X, @,) of &vector spaces on X called the I-adic intersection cohomology sheaves on X.

In order to explain how this is done we make some general comments about vector bundles for a finite group. Let 99be any finite group acting on a finite set X. A g-vector bundle on X is a set of finite dimensional (C-vector spacesV,, x E X, together with linear maps 0 v, A I/gx satisfying the relations %l = 1 I9gx,g’ o %, = %g’g. There are obvious notions of irreducible g-vector bundles on X, and direct sums of

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