Introduction to chapters IV, V and VI

The study of semi-simple (analytic or algebraic) groups and their Lie algebras leads to the consideration of root systems, Coxeter groups and Tits systems. Chapters IV, V and VI are devoted to these structures.

To orient the reader, we give several examples below.

I. Let $g$ be a complex semi-simple Lie algebra and $h$ a Cartan subalgebra of $g^1$. A root of $g$ with respect to $h$ is a non-zero linear form $\alpha$ on $h$ such that there exists a non-zero element $x$ of $g$ with $[h, x] = \alpha(h)x$ for all $h \in h$. These roots form a reduced root system $R$ in the vector space $h^*$ dual to $h$. Giving $R$ determines $g$ up to isomorphism and every reduced root system is isomorphic to a root system obtained in this way. An automorphism of $g$ leaving $h$ stable defines an automorphism of $h^*$ leaving $R$ invariant, and every automorphism of $R$ is obtained in this way. The Weyl group of $R$ consists of all the automorphisms of $h^*$ defined by the inner automorphisms of $g$ leaving $h$ stable; this is a Coxeter group.

Let $G$ be a connected complex Lie group with Lie algebra $g$, and let $\Gamma$ be the subgroup of $G$ consisting of the elements $h$ such that $\exp_G(2\pi i h) = 1$. Let $R^-$ be the root system in $h$ inverse to $R$, let $Q(R^-)$ be the subgroup of $h$ generated by $R^-$ and let $P(R^-)$ be the subgroup associated to the subgroup $Q(R)$ of $h^*$ generated by $R$ (i.e. the set of $h \in h$ such that $\lambda(h)$ is an integer for all $\lambda \in Q(R)$). Then $P(R^-) \supset \Gamma \supset Q(R^-)$. Moreover, the centre of $G$ is canonically isomorphic to $P(R^-)/\Gamma$ and the fundamental group of $G$ to $\Gamma/Q(R^-)$. In particular, $\Gamma$ is equal to $P(R^-)$ if $G$ is the adjoint group and $\Gamma$ is equal to $Q(R^-)$ if $G$ is simply-connected. Finally, the weights of the finite-dimensional linear representations of $G$ are the elements of the subgroup of $h^*$ associated to $\Gamma$.

II. Let $G$ be a semi-simple connected compact real Lie group, and let $g$ be its Lie algebra. Let $T$ be a maximal torus of $G$, with Lie algebra $t$, and let $X$ be the group of characters of $T$. Let $R$ be the set of non-zero elements $\alpha$ of $X$ such that there exists a non-zero element $x$ of $g$ with $(\mathrm{Ad}\, t).x = \alpha(t)x$ for all $t \in T$. Identify $X$ with a lattice in the real vector space $V = X \otimes_{\mathbf{Z}} \mathbf{R}$; then $R$ is a reduced root system in $V$. Let $N$ be the normaliser of $T$ in $G$; the action

1 In this Introduction, we use freely the traditional terminology as well as the notions defined in Chapters IV, V and VI.

of N on T defines an isomorphism from the group N/T to the Weyl group of R. We have P(R) ⊃ X ⊃ Q(R); moreover, X = P(R) if G is simply-connected and X = Q(R) if the centre of G reduces to the identity element.

The complexified Lie algebra $g_{(C)}$ of $g$ is semi-simple and $t_{(C)}$ is a Cartan subalgebra of it. There exists a canonical isomorphism from $V_{(C)}$ to the dual of $t_{(C)}$ that transforms R into the root system of $g_{(C)}$ with respect to $t_{(C)}$.

III. Let G be a connected semi-simple algebraic group over a commutative field $k$. Let T be a maximal element of the set of tori of G split over $k$ and let X be the group of characters of T (the homomorphisms from T to the multiplicative group). We identify X with a lattice in the real vector space $V = X \otimes_{\mathbf{Z}} \mathbf{R}$. The roots of G with respect to T are the non-zero elements $\alpha$ of X such that there exists a non-zero element $x$ of the Lie algebra $g$ of G with $(\mathrm{Ad}\, t).x = \alpha(t)x$ for all $t \in T$. This gives a root system R in V, which is not necessarily reduced. Let N be the normaliser and Z the centraliser of T in G and let $N(k)$ and $Z(k)$ be their groups of rational points over $k$. The action of $N(k)$ on T defines an isomorphism from $N(k)/Z(k)$ to the Weyl group of R.

Let U be a maximal element of the set of unipotent subgroups of G, defined over $k$ and normalised by Z. Put $P = Z.U$. Then $P(k) = Z(k).U(k)$ and $P(k) \cap N(k) = Z(k)$. Moreover, there exists a basis $(\alpha_1, \ldots, \alpha_n)$ of R such that the weights of T in U are the positive roots of R for this basis; if S denotes the set of elements of $N(k)/Z(k)$ that correspond, via the isomorphism defined above, to the symmetries $s_{\alpha_i} \in W(R)$ associated to the roots $\alpha_i$, the quadruple $(G(k), P(k), N(k), S)$ is a Tits system.

IV. In the theory of semi-simple algebraic groups over a local field, Tits systems are encountered whose Weyl group is the affine Weyl group of a root system. For example, let $G = \mathbf{SL}(n+1, \mathbf{Q}_p)$ (with $n \geq 1$). Let B be the group of matrices $(a_{ij}) \in \mathbf{SL}(n+1, \mathbf{Z}_p)$ such that $a_{ij} \in p\mathbf{Z}_p$ for $i < j$, and let N be the subgroup of G consisting of the matrices having only one non-zero element in each row and column. Then there exists a subset S of $N/(B \cap N)$ such that the quadruple $(G, B, N, S)$ is a Tits system. The group $W = N/(B \cap N)$ is the affine Weyl group of a root system of type $A_n$; this is an infinite Coxeter group.

Numerous conversations with J. Tits have been of invaluable assistance to us in the preparation of these chapters. We thank him very cordially.