Linear algebraic groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 April 2010

Linear algebraic groups

A linear algebraic group is an affine algebraic variety $G$ which is also a group such that multiplication and inversion are the morphisms of algebraic varieties.

The following fundamental theorem is reason for the terminology linear algebraic group.

If $G$ is a linear algebraic group then there is an injective morphism of algebraic groups $i:G\to {\mathrm{GL}}_n\left(F\right)$ for some $n\in {\mathbb{Z}}_{>0}.$

The multiplicative group is the linear algebraic group ${𝔾}_m=F^{*}.$

A matrix $x\in M_n\left(F\right)$ is

1. semisimple if it is conjugate to a diagonal matrix,
2. nilpotent if all its eigenvalues are 0, or, equivalently, if $x^n=0$ for some $n\in {\mathbb{Z}}_{>0},$
3. unipotent if all its eigenvalues are 1, or, equivalently, if $x-1$ is nilpotent.

Let $G$ be a linear algebraic group and let $i:G\to {\mathrm{GL}}_n\left(F\right)$ be an injective homomorphism. An element $g\in G$

1. semisimple if $i\left(g\right)$ is semisimple in ${\mathrm{GL}}_n\left(F\right),$
2. unipotent if $i\left(g\right)$ is unipotent in ${\mathrm{GL}}_n\left(F\right).$

The resulting notions of semisimple and unipotent elements in $g$ do not depend on the choice of the imbedding $i:G\to {\mathrm{GL}}_n\left(ℂ\right).$

(Jordan decomposition) Let $G$ be a linear algebraic group and let $g\in G.$ Then there exist unique $g_s,g_u\in G$ such that

1. $g_s$ is semisimple,
2. $g_u$ is unipotent,
3. $g=g_s{g}_u=g_u{g}_s.$

Let $G$ be a linear algebraic group.

1. The radical $R\left(G\right)$ is the unique maximal closed connected solvable normal subgroup of $G.$
2. The unipotent radical $R_u\left(G\right)$ is the unique maximal closed connected unipotent normal subgroup of $G.$
3. $G$ is semisimple if $R\left(G\right)=1.$
4. $G$ is reductive if $R_u\left(G\right)=1.$ $G$ is reductive if its Lie algebra is reductive.
5. $G$ is an (algebraic) torus if $G$ is isomorphic to ${𝔾}_m\times \dots {𝔾}_m$ ($k$ factors ) for some $k\in {\mathbb{Z}}_{>0}.$
6. A Borel subgroup of $G$ is a maximal connected closed solvable subgroup of $G^0.$

Let $G$ be a linear algebraic group and let $G^0$ be the connected component of the identity in $G.$ Then $$1\subseteq R_u\left(G\right)\subseteq R\left(G\right)\subseteq G^0\subseteq G$$ where $R_u\left(G\right)$ is unipotent, $R\left(G\right)$ is solvable, $G^0$ is connected, $G/G^0$ is finite, $G^0/R\left(G\right)$ is semisimple, $R\left(G\right)/R_u\left(G\right)$ is a torus and $R_u\left(G\right)$ is unipotent.

A linear algebraic group is simple if it has no proper closed connected normal subgroups. This implies that proper normal subgroups are finite subgroups of the center.

Let $G$ be an algebraic group.

1. If $G$ is nilpotent the $G\cong TU$ where $T$ is a torus and $U$ is unipotent.
2. If $G$ is a connected reductive group the $G=[G,G]Z^{\circ },$ where $[G,G]$ is semisimple and $[G,G]\cap Z^{\circ }$ is finite.
3. If $[G,G]$ is semisimple the $G$ is an almost direct product of simple groups, ie there are closed normal subgroups $G_1,\dots ,G_k$ in $G$ such that $G=G_1•\dots •G_k$ and $G_i\cap \left(G_1\dots {\hat{G}}_i\dots G_k\right)$ is finite.

Example. If $G={\mathrm{GL}}_n\left(ℂ\right)$ then $$[G,G]={\mathrm{SL}}_n\left(ℂ\right),Z^{\circ }=ℂ•\mathrm{Id},\text{and}[G,G]\cap Z^{\circ }=\left\{\lambda •\mathrm{Id}|{\lambda }^n=1\right\}\cong \mathbb{Z}/n\mathbb{Z}.$$

Structure of a simple algebraic group $$x_{\alpha }\left(t\right)=e^{t{X}_{\alpha }},w_{\alpha }\left(t\right)=x_{\alpha }\left(t\right)x_{-\alpha }\left(t^{-1}\right)x_{\alpha }\left(t\right),h_{\alpha }\left(t\right)=w_{\alpha }\left(t\right)w_{\alpha }{\left(1\right)}^{-1},$$ $$U=⟨x_{\alpha }\left(t\right)|\alpha >0⟩,T=⟨h_{\alpha }\left(t\right)⟩,N=⟨w_{\alpha }\left(t\right)⟩,B=TU,W=N/T.$$

The Langlands decomposition of a parabolic is $P=MAN$ where $$M=\left(\begin{array}{ccccc}{A}_1& & & & \\ & A_2& & 0& \\ & & \dots & & \\ & 0& & A_{l-1}& \\ & & & & A_l\end{array}\right),\det \left(A_i\right)=1,$$ $$A=\left(\begin{array}{ccccc}{a}_1\mathrm{Id}& & & & \\ & a_2\mathrm{Id}& & 0& \\ & & \dots & & \\ & 0& & a_{l-1}\mathrm{Id}& \\ & & & & a_l\mathrm{Id}\end{array}\right),a_i>0$$ $$N=\left(\begin{array}{ccccc}\mathrm{Id}& & & & \\ & \mathrm{Id}& & *& \\ & & \dots & & \\ & 0& & \mathrm{Id}& \\ & & & & \mathrm{Id}\end{array}\right),$$ and there is a corresponding decomposition $𝔭=𝔪\oplus 𝔞\otimes 𝔫$ at the Lie algebra level.

The Iwasawa decomposition of $G=KAN$ where $$K=\text{a maximal compact subgroup of} G,$$ $$A=\left(\begin{array}{ccccc}{a}_1& & & & \\ & a_2& & 0& \\ & & \dots & & \\ & 0& & a_{l-1}& \\ & & & & a_l\end{array}\right),\det \left(A\right)=1$$ $$N=\left(\begin{array}{ccccc}1& & & & \\ & 1& & *& \\ & & \dots & & \\ & 0& & 1& \\ & & & & 1\end{array}\right),$$ and the corresponding Lie algebra decomposition is $$𝔤=𝔱\oplus 𝔭=𝔱\oplus 𝔞\oplus 𝔫,\text{where}\begin{array}{c}𝔱=\left\{x\in 𝔤|\theta x=x\right\},𝔭=\left\{x\in 𝔤|\theta x=-x\right\},\\ 𝔞=\text{a maximal abelian subspace of}𝔭,\\ 𝔫=\text{the set of positive roots with respect to}𝔞.\end{array}$$

The Cartan decomposition of $G$ is $G=KAK.$ The Bruhat decomposition of $G$ is $G=BWB.$

Let $𝔤$ be a semisimple complex Lie algebra.

1. There is an involutory semiautomorphism ${\sigma }_0$ of $𝔤$ relative to complex conjugation such that $${\sigma }_0\left(X_{\alpha }\right)=-X_{\alpha },{\sigma }_0\left(H_{\alpha }\right)=-H_{\alpha },\text{for all}\alpha \in R.$$ Let $G$ be a Chevalley group over $ℂ$ viewed as a (real) Lie group.
2. There is an analytic automorphism $\sigma$ of $G$ such that $$\sigma x_{\alpha }\left(t\right)=x_{-\alpha }\left(-\right)$$t , σ h α t = h α t -1 , for all α∈ R ,t∈ ℂ.
3. A maximal compact subgroup of $G$ is $$K=\left\{g\in G|\sigma \left(g\right)=g\right\}.$$
4. $K$ is semisimple and connected.
5. The Iwasawa decomposition is $G=BK.$
6. The Cartan decomposition is $G=KAK$ where $$A=\left\{h\in H|\mu \left(h\right)>0\text{ for all }\mu \in L\right\}.$$

Let $\Theta$ be a PID, $k$ the quotient field, and ${\Theta }^{*}$ the group of units of $\Theta$ (examples: $\Theta =\mathbb{Z},\Theta =F\left[t\right],\Theta ={\mathbb{Z}}_p.$ ) If $G$ is a Chevalley group over $k$ and let $G_{\Theta }$ be the subgroup of $G$ with coordinates relative to $M$ in $\Theta .$ Now let $G$ be a semisimple Chevalley group over $k.$

1. The Iwasawa decomposition is $G=BK$ where $$K=G_{\Theta }.$$
2. The Cartan decomposition is $K{A}^{+}K$ where $$A^{+}=\left\{h\in H|\alpha \left(h\right)\in \Theta \text{ for all }\alpha \in R^{+}\right\}.$$
3. If $\Theta$ is not a field (in particular if $\Theta =\mathbb{Z}$ ) then $K$ is maximal in its commensurability class.
4. If $\Theta ={\mathbb{Z}}_p$ and $k={\mathbb{Q}}_p$ then $K$ is a maximal compact subgroup in the $p$ -adic topology.
5. If $\Theta$ is a local PID and $p$ is its unique prime then
1. The Iwahori subgroup $I=U_p^{-}{H}_{\Theta }{U}_{\Theta }$ is a subgroup of $K.$
2. $K=\underset{w\in W}{\cup }IwI.$
3. $IwI=Iw{U}_{w,\Theta }$ with the last component determined uniquely mod $U_{w,p}.$

Classification theorems

$\begin{array}{ccc}\left\{\text{semisimple algebraic groups over }ℂ\right\}& \stackrel{1-1}{\leftrightarrow }& \left\{\text{lattices and root systems}\right\}\\ \left\{\text{complex semisimple Lie groups}\right\}& \stackrel{1-1}{\leftrightarrow }& \left\{\text{semisimple algebraic groups over} ℂ\right\}\\ \left\{\text{connected reductive algebraic groups over} ℂ\right\}& \stackrel{1-1}{\leftrightarrow }& \left\{\text{compact connected Lie groups}\right\}\\ G& \mapsto & U=\text{maximal compact subgroup of} G\\ \text{semisimple}& \mapsto & \text{finite center}\\ \text{algebraic torus}& \mapsto & \text{geometric torus}\\ \left\{\text{connected simply connected Lie groups}\right\}& \stackrel{1-1}{\leftrightarrow }& \left\{\text{finite dimensional real Lie algebras}\right\}\\ \left\{\text{finite dimensional complex simple Lie algebras}\right\}& \stackrel{1-1}{\leftrightarrow }& \left\{\text{root systems: 4 infinite families and 5 exceptionals}\right\}\\ \left\{\text{finite dimensional real simple Lie algebras}\right\}& \stackrel{1-1}{\leftrightarrow }& \left\{\text{12 infinite families and 23 exceptionals}\right\}\end{array}$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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