# 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.

**Read Online or Download Representations of finite groups of Lie type PDF**

**Best linear books**

**A first course in linear algebra**

A primary path in Linear Algebra is an creation to the elemental innovations of linear algebra, in addition to an creation to the innovations of formal arithmetic. It starts with structures of equations and matrix algebra sooner than getting into the speculation of summary vector areas, eigenvalues, linear adjustments and matrix representations.

**Measure theory/ 3, Measure algebras**

Fremlin D. H. degree conception, vol. three (2002)(ISBN 0953812936)(672s)-o

**Elliptic Partial Differential Equations**

Elliptic partial differential equations is without doubt one of the major and such a lot energetic parts in arithmetic. In our booklet we examine linear and nonlinear elliptic difficulties in divergence shape, with the purpose of supplying classical effects, in addition to newer advancements approximately distributional recommendations. accordingly the publication is addressed to master's scholars, PhD scholars and someone who desires to commence examine during this mathematical box.

- Direct Integral Theory (Lecture Notes in Pure and Applied Mathematics)
- Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration
- Algebra lineare e primi elementi di geometria (Italian Edition)
- Lineare Algebra II, Edition: version 13 Aug 2014
- Torsion Theories, Additive Semantics, and Rings of Quotients (Lecture Notes in Mathematics)
- Einführung in die lineare Algebra

**Extra resources for Representations of finite groups of Lie type**

**Example text**

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.