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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 sort, an incredible box of present mathematical learn. within the moment half, Platonov and Yanchevskii survey the constitution of finite-dimensional department algebras, together with an account of diminished K-theory.

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

Example text

This formula specializes when t is replaced by 1 to the original formula T (- l)‘~‘(%v, @,I = 1 f’,,,(1)R,. YQW However the generic version is much stronger, and enables each character of GF coming from a module IH’(X,,,, @,) to be expressed as a linear combination of almost characters R,. e. characters of GF-modules. 5, is an actual character of GF, and this gives further Z-combinations of the almost characters R, which are actual characters of CF. The results obtained by Lusztig about such Z-combinations of almost characters R, can be described as follows.

Each of these equivalence classes gives rise to a representation of the Coxeter group and its associated generic Hecke algebra. We shall now outline these ideas of Kazhdan and Lusztig. Let W be a finite Coxeter group (although similar results can be obtained for infinite Coxeter groups also). Let H = H ZLt1J2,r-1,21(t)be the generic Hecke algebra of W over the ring Z[t1’2, t-‘12] of Laurent polynomials in t’12 with coeflicients in Z. Thus the elements of H are combinations of the basis elements T,, w E W, with coefficients in Z[t112, t-1’2] and with multiplication of basis elements defined as before.

W. Carter 78 I. On the Representation We can now state the generic version of the identity relating l-adic cohomology to I-adic intersection cohomology. This generic identity is given by the formula (We note that although there are two possible extensions 4 of 4, both give the same value on the right hand side). As before, this identity can be used to show that certain combinations of the almost characters Rd give actual characters of GF. One can show that &T,,) is divisible by at least of the Finite c~,,,J by &( T,+) = (- l)f(w)~Fw,~t’i2(r(w)-a~) + higher powers of tl’*.

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