Representations of finite groups of Lie type by François Digne, Jean Michel

By François Digne, Jean Michel

The authors target to regard the fundamental concept of representations of finite teams of Lie style, resembling linear, unitary, orthogonal and symplectic teams. They emphasize the Curtis-Alvis duality map and Mackey's theorem and the implications that may be deduced from it. in addition they speak about Deligne-Lusztig induction. this can be the 1st basic remedy of this fabric in booklet shape and may be welcomed via starting graduate scholars in algebra.

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The analogous statements are clearly true for the special linear, the symplectic, the orthogonal, etc. groups (see chapter 15 for more details). Any embedding of minimal dimension of an algebraic group G into GLn as above defines a standard Frobenius endomorphism on G by restriction of the endomorphism of GLn defined by Tij → Tijq . But there are other examples of rational structures on algebraic groups; for ′ instance the unitary group is GLFn where F ′ is the Frobenius endomorphism defined by F ′ (x) = F ( tx−1 ), with F being the standard Frobenius endomorphism on GLn .

So the natural map GF /HF → (G/H)F is surjective. It is injective since, if x, y ∈ GF are in the same H-coset, then x−1 y is in HF . 14 Example. A trap. , the quotient group of SLn by its centre. 13 that this kind of phenomenon happens only in the case of a quotient by a non-connected group). Indeed, the centre µn of SLn consists of the scalar Rationality, Frobenius endomorphism 39 matrices which are equal to an n-th root of unity times the identity. The image in PSLn of x ∈ SLn is in PSLFn if and only if x.

55 term in the group suggested by its name). We now prove its injectivity. If UlxmV = Ul′ xm′ V, then lxm = l′ uxvm′ for some u ∈ U and v ∈ V since l′ normalizes U and m′ normalizes V. Thus −1 u−1l′ l = x(vm′ m−1 ) ∈ P ∩ xQ. 8 Lemma. Let L and M be respective Levi subgroups of two parabolic subgroups P and Q, and assume they have a common maximal torus. 1 (iii), etc. Proof: We have uV uM lV lM = vU vL mU mL = vU [vL , mU ]mU vL mL and the commutator which appears in this formula is in V ∩ U. 1 (iii) gives the result.

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