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- Tags: - #todo/untagged - Refs: - Overview of reductive groups: https://gauss.math.yale.edu/~il282/group_stuff.pdf#page=5 - - Links: - root system

Reductive groups

Let \(G\) be an affine algebraic group and \(H\leq G\) a subgroup.

Summary

Reducibility

• A subgroup \(H\leq G\) is irreducible iff \(H\) is not contained in any proper parabolic.
  • Idea: for \(\rho: G\to \operatorname{GL}(V)\) a classical representation, \(V\) is irreducible iff the only \(G{\hbox{-}}\)invariant subspaces of \(V\) are \(0\) and \(V\) itself.
• \(H\) is completely reducible iff for every \(P\) with \(H \subseteq P\leq G\), there is a Levi subgroup \(L \leq P\) with \(L\supseteq H\).
  • Idea: semisimple. A representation \(\rho:G\to \operatorname{GL}(V)\) is completely reducible iff \(V\) decomposes as a direct sum of irreducibles (\(G{\hbox{-}}\)invariant subspaces).
  • Note that irreducible implies completely reducible.
• A representation \(\rho: X\to G\) is irreducible (resp. completely reducible) if its image \(\rho(X) \leq G\) is reducible (resp. completely reducible).
• Let \({ \operatorname{Ad} }: G\to \operatorname{GL}({\mathfrak{g}})\) be the adjoint representation, then a representation \(\rho\) is \({ \operatorname{Ad} }{\hbox{-}}\)irreducible iff the composed representation \({ \operatorname{Ad} }\circ \rho: G\to \operatorname{GL}({\mathfrak{g}})\) is irreducible.

Borels and Parabolics

• A subgroup \(B \leq G\) is called a Borel subgroup if it is maximal (w.r.t. inclusion) of all connected solvable subgroups of \(G\).
  • All Borels are conjugate.
  • One could define \(P\leq G\) to be parabolic iff \(G/P\) is a projective variety.
  • For \(G=\operatorname{GL}_n\), parabolics are stabilizers of partial flags
• A subgroup \(P\leq G\) is parabolic iff \(P\) contains a Borel subgroup, iff the homogeneous space \(G/P\) is a complete variety.
  • So a Borel is a minimal parabolic, and \(G/B\) is the maximal quotient that is a complete variety.
  • Every parabolic is conjugate to a standard parabolic.

Levis

• A Levi subgroup is the centralizer of a subtorus.
  • For \(G = \operatorname{GL}_n\), Levis are stabilizers of ordered direct sum decompositions \(k^n = \bigoplus_i V_i\).
  • Levis are connected and reductive
  • For every Levi \(L\) there is a parabolic \(P\) such that \(P = L \rtimes R_u(P)\).
• Standard parabolics and Levis:
  • For \(P = P_J\) a standard parabolic, \(G/P\) is a smooth projective variety.

Reductive groups

• A matrix \(M\in {\mathfrak{gl}}_n(k)\) is unipotent iff \(\operatorname{Spec}_\lambda(M) = \left\{{1}\right\}\), ie all eigenvalues are 1.
• \(M\) is quasi-unipotent iff \(M^n\) is unipotent for some \(n\), iff \(\operatorname{Spec}_\lambda(M) \in \mu_\infty(k)\), i.e. all eigenvalues are roots of unity.
• The identity component of the intersection of all Borel subgroups is the radical of \(G\), denoted \begin{align*}RG \coloneqq\qty{\bigcap_{B\leq G \text{ Borel}} B}^0\end{align*}
• The subgroup \(RG^u \leq RG\) of unitpotent elements is the unipotent radical of \(G\).
• If \(RG = 1\) then \(G\) is reductive.

Split

For Chevalley groups

Standard Borels

Levi decomposition

Opposite parabolics

Characters