Chevalley group

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

A Chevalley group is a linear algebraic group over some field, related to a semi-simple complex Lie algebra, in the following way: Let $\def\fg{ {\mathfrak g}}\fg$ be a semi-simple Lie algebra over $\C$, let $\def\fh{ {\mathfrak h}}\fh$ be its Cartan subalgebra, let $\def\S{\Sigma}\S$ be a root system of $\fg$ with respect to $\fh$, let $\def\a{\alpha}\{\a_1,\dots,\a_k\}\subset \S$ be a system of simple roots, let $\{H_{\a_i} (1\le i\le k\}); X_\a (\a\in\S)\}$ be a Chevalley basis of the algebra $\fg$, and let $\fg_\Z$ be its linear envelope over $\Z$. Let $\def\phi{\varphi}\phi$ be a faithful representation of the Lie algebra $\fg$ in a finite-dimensional vector space $V$. It turns out that there is a lattice in $V$ (i.e. a free Abelian subgroup a basis of which is the basis of the space $V$) which is invariant with respect to all operators $\phi(X_\a)^m/m!$ ($\a\in\S$, $m$ is a natural number). If $k$ is an arbitrary field and if $V^k=M\otimes k$, then, for $\a\in\S$, homomorphisms $x_\a : k^+ \to {\rm GL}(V^k)$ of the additive group $k^+$ of $k$ into ${\rm GL}(V^k)$ are defined and given by the formulas

$$x_\a(t) = \sum_{m=0}^\infty t^m\frac{\phi(X_\a)^m}{m!}.$$ The subgroups $\def\fX{ {\rm X}}\fX_\a = {\rm Im}\; x_\a$, $\a\in\S$, generate in ${\rm GL}(V^k)$ some subgroup $G_k$, which is called the Chevalley group related to the Lie algebra $\fg$, the representation $\phi$ and the field $k$. If $\phi = {\rm ad}$ (the adjoint representation), the Chevalley groups were defined by C. Chevalley in 1955 (see [Ch]).

If $K$ is an algebraically closed field containing $k$, then a Chevalley group $G_K$ is a connected semi-simple linear algebraic group over $K$, defined and split over the prime subfield $k_0\subseteq k$. Its Lie algebra is isomorphic to $\fg_\Z\otimes K$. The group $G_k$ is the commutator subgroup of the group $G_K(k)$ of points of $G_K$ that are rational over $k$. Any connected semi-simple linear algebraic group over $K$ is isomorphic to one of the Chevalley groups. The algebraic groups $G_K$ (and $G_k$ as abstract groups) depend only on the lattice $\def\G{\Gamma}\G_\phi\subset \fh^*$ generated by the weights of the representation $\phi$. If $\G_\phi$ coincides with the lattice of roots $\G_0$, then $G_K$ is called the adjoint group, and if $\G_\phi=\G_1$ (the lattice of weights, see Lie group, semi-simple), then $G_K$ is called a universal or simply-connected group. If $G_K$ is universal, then $G_k=G_K(k)$.

The Chevalley group $G_K$ always coincides with its commutator subgroup. The centre of $G_K$ is finite. For example, the centre $Z$ of the universal group $G_K$ is isomorphic to ${\rm Hom}(\G_1/\G_0, k^*)$, and the corresponding adjoint group is isomorphic to $G_k/Z$ and has trivial centre.

If the algebra $\fg$ is simple, then the adjoint Chevalley group $G_k$ is simple, except in the following cases: $|k| = 2$ and $\fg$ is a Lie algebra of type $A_1, B_2$ or $G_2$; or $|k| = 3$ and $\fg$ is a Lie algebra of type $A_1$. Other series of simple groups can be obtained when considering the subgroups of fixed points of certain automorphisms of finite order of Chevalley groups (so-called torsion groups).

If the field $k$ is finite, then the order of the universal group $G_k$ can be computed by the formula

$$|G_k| = q^N \prod_{l=1}^r(q^{d_i} -1),$$ where $q = |k|$, $d_i$ ($i=1,\dots,r$) are exponents of the Lie algebra $\fg$, i.e. the degrees of the free polynomials on $\fh$, generating the algebras, that are invariant with respect to the Weyl group, and $N=\sum_{i=1}^r(d_i - 1)$ is the number of positive roots.

There is a well-developed theory of rational linear representations of the Chevalley groups $G_k$ over an infinite field $k$. It is reduced to the case of an algebraically closed field, and in the latter case coincides with the theory of rational representations (cf. Rational representation) of semi-simple algebraic groups. If $\fg$ is simple, $G_k$ is the universal Chevalley group over the infinite field $k$, and $\def\s{\sigma}\s$ is a non-trivial irreducible finite-dimensional representation of $G_k$ (as an abstract group) over an algebraically closed field $K$, then there exists a finite set of imbeddings $\phi_i:k\to K$ and a set of rational representations $\rho_i$ of the groups $G_{\phi_i}(k)$ such that $\s = \otimes_i\rho_i\circ \phi_i$. Concerning representations of Chevalley groups, see also [St], [BoCaCuIwSpSt], [Hu2].

References

[BoCaCuIwSpSt]	A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.), Seminar on algebraic groups and related finite groups, Lect. notes in math., 131, Springer (1970) Zbl 0192.36201
[Ca]	R.W. Carter, "Finite groups of Lie type. Conjugacy classes and complex characters", Wiley (Interscience) (1985) MR0794307 Zbl 0567.20023
[Ch]	C. Chevalley, "Sur certains groupes simples" Tôhoku Math. J., 7 : 1–2 (1955) pp. 14–66 MR0073602 Zbl 0066.01503
[Hu]	J.E. Humphreys, "Introduction to Lie algebras and representation theory", Springer (1972) MR0323842 Zbl 0254.17004
[Hu2]	J.E. Humphreys, "Ordinary and modular representations of Chevalley groups", Springer (1976) MR0453884 Zbl 0341.20037
[St]	R.G. Steinberg, "Lectures on Chevalley groups", Yale Univ. Press (1968) MR0466335 Zbl 1196.22001
[St2]	R. Steinberg, "Variations on a theme of Chevalley" Pacific J. Math., 9 (1959) pp. 875–891 MR0109191 Zbl 0092.02505