Linear algebraic groups

Linear algebraic groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

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 GL n F for some n > 0 .

The multiplicative group is the linear algebraic group 𝔾 m = F * .

A matrix x M n F 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 > 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 GL n F be an injective homomorphism. An element g G

  1. semisimple if i g is semisimple in GL n F ,
  2. unipotent if i g is unipotent in GL n F .

The resulting notions of semisimple and unipotent elements in g do not depend on the choice of the imbedding i :G GL n .

(Jordan decomposition) Let G be a linear algebraic group and let g G . Then there exist unique g s , g u 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 G is the unique maximal closed connected solvable normal subgroup of G .
  2. The unipotent radical R u G is the unique maximal closed connected unipotent normal subgroup of G .
  3. G is semisimple if R G =1 .
  4. G is reductive if R u G =1 . G is reductive if its Lie algebra is reductive.
  5. G is an (algebraic) torus if G is isomorphic to 𝔾 m × 𝔾 m ( k factors ) for some k > 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 R u G R G G 0 G where R u G is unipotent, R G is solvable, G 0 is connected, G / G 0 is finite, G 0 / R G is semisimple, R G / R u G is a torus and R u G 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 T U where T is a torus and U is unipotent.
  2. If G is a connected reductive group the G = [ G,G] Z , where [ G,G] is semisimple and [ G,G] Z 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 , , G k in G such that G = G 1 G k and G i G 1 G ^ i G k is finite.

Example. If G = GL n then [ G,G] = SL n , Z =Id , and [ G,G] Z = λId| λ n =1 / n .

Structure of a simple algebraic group x α t = e t X α , w α t = x α t x -α t -1 x α t , h α t = w α t w α 1 -1 , U= x α t | α> 0 , T = h α t , N= w α t , B=TU, W=N/T.

The Langlands decomposition of a parabolic is P =M AN where M= A 1 A 2 0 0 A l-1 A l , det A i =1 , A= a 1 Id a 2 Id 0 0 a l-1 Id a l Id , a i > 0 N=Id Id * 0 Id Id , and there is a corresponding decomposition 𝔭 = 𝔪 𝔞 𝔫 at the Lie algebra level.

The Iwasawa decomposition of G =K AN where K= a maximal compact subgroup of   G, A= a 1 a 2 0 0 a l-1 a l , det A =1 N=1 1 * 0 1 1 , and the corresponding Lie algebra decomposition is 𝔤=𝔱 𝔭=𝔱 𝔞 𝔫, where 𝔱= x 𝔤 | θ x=x , 𝔭= x 𝔤 | θ x=-x , 𝔞=a maximal abelian subspace of𝔭, 𝔫=the set of positive roots with respect to𝔞.

The Cartan decomposition of G is G =K AK. The Bruhat decomposition of G is G =B WB.

Let 𝔤 be a semisimple complex Lie algebra.

  1. There is an involutory semiautomorphism σ0 of 𝔤 relative to complex conjugation such that σ0 X α =- X α , σ0 H α =- H α , for all α R . Let G be a Chevalley group over viewed as a (real) Lie group.
  2. There is an analytic automorphism σ of G such that σ x α t = x -α -t , σ h α t = h α t -1 , for all α R ,t .
  3. A maximal compact subgroup of G is K= g G| σ g =g .
  4. K is semisimple and connected.
  5. The Iwasawa decomposition is G=BK.
  6. The Cartan decomposition is G= K AK where A= h H| μ h > 0  for all  μ L .

Let Θ be a PID, k the quotient field, and Θ * the group of units of Θ (examples: Θ = ,Θ =F[t] ,Θ = p . ) If G is a Chevalley group over k and let G Θ be the subgroup of G with coordinates relative to M in Θ. Now let G be a semisimple Chevalley group over k.

  1. The Iwasawa decomposition is G =BK where K= G Θ .
  2. The Cartan decomposition is K A + K where A + = h H| α h Θ  for all  α R + .
  3. If Θ is not a field (in particular if Θ = ) then K is maximal in its commensurability class.
  4. If Θ =p and k = p then K is a maximal compact subgroup in the p -adic topology.
  5. If Θ is a local PID and p is its unique prime then
  6. The Iwahori subgroup I = Up- H Θ U Θ is a subgroup of K.
  7. K= w W I w I .
  8. I w I=Iw U w,Θ with the last component determined uniquely mod U w,p .

Classification theorems

semisimple algebraic groups over   1-1 lattices and root systems complex semisimple Lie groups 1-1 semisimple algebraic groups over  connected reductive algebraic groups over  1-1 compact connected Lie groups G U =maximal compact subgroup of  G semisimple finite center algebraic torus geometric torus connected simply connected Lie groups 1-1 finite dimensional real Lie algebras finite dimensional complex simple Lie algebras 1-1 root systems: 4 infinite families and 5 exceptionals finite dimensional real simple Lie algebras 1-1 12 infinite families and 23 exceptionals

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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