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Description of observable subgroups of linear algebraic groups. (Russian) Zbl 0663.20043

Let \(G\) be a linear algebraic group over an algebraically closed field of characteristic zero. If we are given a rational finite dimensional linear representation of \(G\) \(\theta: G\to \text{GL}(V)\), then \(V\) is a rational finite dimensional \(G\)-module (\(gv=\theta (g)v\), \(v\in V\), \(g\in G\)). An algebraic subgroup \(H\) of \(G\) is called an observable subgroup in \(G\) if any rational \(H\)-module \(W\) is a submodule of some rational \(G\)-module \(V\). The author studies observable subgroups of linear algebraic groups. Let \(G\) be a connected group. A subgroup \(H\) of \(G\) is called a quasiparabolic subgroup in \(G\) if it is the stabilizer of the highest weight vector of some irreducible rational \(G\)-module \(V\). An algebraic connected subgroup \(H\) of \(G\) is called a subparabolic subgroup in \(G\) if \(H_ u\subset Q_ u\) where \(H_ u\) and \(Q_ u\) are the unipotent radicals of \(H\) and of some quasiparabolic subgroup \(Q\), respectively.
The main results of the paper under review are the following.
Theorem 1. Let \(G\) be a connected linear algebraic group and \(L\) a connected linear algebraic subgroup of \(G\). Then the following conditions are equivalent:
1. \(L\) is a subparabolic subgroup in \(G\);
2. \(L\) is an observable subgroup in \(G\).
Theorem 2. Let \(G\) be as above, \(K\) a connected algebraic subgroup of \(G\), \(T, K_ u\) the maximal torus and the unipotent radical of \(K\), respectively, and \(L\) the radical of \(K\), then the following conditions are equivalent:
1. \(K\) is an observable subgroup in \(G\);
2. \(T\cdot K_ u\) is an observable subgroup in \(G\);
3. \(L\) is an observable subgroup in \(G\).
Let \(K\) be a solvable subgroup of \(G\). One says that \(K\) is radical if \(K=T\cdot U\) where \(T\) is the maximal torus and \(U\) is the unipotent radical of some quasiparabolic subgroup \(Q\) of \(G\).
Theorem 3. The class of connected algebraic radical subgroups of a connected algebraic group \(G\) coincides with the class of connected observable subgroups in \(G\).
Reviewer: V. Yanchevskij

MSC:

20G05 Representation theory for linear algebraic groups
20G15 Linear algebraic groups over arbitrary fields
20E07 Subgroup theorems; subgroup growth
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