Nikolaus Conference 2023

Speaker: Sören Böhm (Bochum)

Title: Semisimplification of Linear Algebraic Groups and Lie Algebras


j.w. M. Bate, B. Martin, G. Röhrle and L. Voggesberger

Given a subgroup \(H \subseteq \textrm{GL}(V)\), there exists a flag of subspaces \(\mathfrak{F}\) such that each successive quotient in the flag is an irreducible \(H\)-module, by the Jordan-Hölder Theorem. In particular, \(H \subseteq \textrm{stab}_{\textrm{GL}(V)}(\mathfrak{F})\). If we think of \(H\) as a matrix group, then, up to base change, \(H\) is contained in the ''upper-block-diagonal matrices``, where each block gives the action of \(H\) on the corresponding quotient. In general, this \(H\) does not act semisimply on \(V\). If so, then \(H\) is contained in the ''block-diagonal matrices``. However, if this is not the case, we can still define a new group \(H'\) to be the group \(H\), but replace everything above the block-diagonal by zero. Then \(H'\) acts, by definition, semisimply on V, i.e. \(H'\) acts completely reducible on \(V\).

There is no reason, why we cannot replace \(\textrm{GL}(V)\) by any connected reductive linear algebraic group \(G\). This idea goes back to J.P. Serre. He defines: a subgroup \(H \subset G\) is called \(G\)-completely reducible, if for any parabolic subgroup \(P\) containing \(H\), there exists a Levi subgroup \(L\) of \(P\) such that \(H \subseteq L\). In the case of \(G=\textrm{GL}(V)\) this is equivalent to \(H\) being completely reducible. We can also define a '\(G\)-analogue` of \(H'\), as Serre named it in the case \(k=\overline{k}\). For arbitrary \(k\), M. Bate, B. Martin and G. Röhrle called it the \(k\)-semisimplification of \(H\). They proved that for any \(H\) there exists always a \(k\)-semisimplification and it is unique  up to \(G(k)\)-conjugacy, which can be viewed as an analogue of the Jordan-Hölder Theorem.

There are similar notions for the Lie algebra \(\textrm{Lie}(H)\) of \(H\). We proved the analogue of the Bate--Martin--Röhrle result in the Lie algebra case and that there is a natural correspondence between semisimplifications of \(H\) in \(G\) and semisimplifications of \(\textrm{Lie}(H)\) in \(\textrm{Lie}(G)\).

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