Stationary subalgebras in general position for locally strongly effective actions
Let \(G\) be a reductive complex algebraic group, and let \(V\) be a finite-dimensional complex vector space equipped with a linear action of \(G\). For this action, there is a stabilizer in general position \(H\) (briefly, s.g.p.). We denote it by \(G_*(V)\). In invariant theory, a linear representation of an algebraic group \(G\) on a finite-dimensional space \(V\) can have several so-called ``good'' properties. First of all, these are the following properties: freedom of the algebra of invariants, constant dimension of the fibers of the factorization morphism, and finiteness of the family of orbits in each fiber of the factorization morphism. It turns out that the following condition correlates with these properties: the stabilizer in general position of the linear action \(G:V\) is nontrivial. In particular, for connected simple irreducible linear groups, the properties listed above are equivalent. In the general case, this assertion fails. Nevertheless, to clarify the general situation, the study of representations with nontrivial stabilizers in general position is of certain interest. Moreover, if the s.g.p. is nontrivial, then the stabilizers of typical closed orbits are nontrivial (and coincide with the s.g.p., if it is reductive and the group \(G\) is semisimple), and this gives a reduction to evaluate the algebra of invariants by the Luna--Richardson theorem. Denote by \(\mathfrak g\) the Lie algebra of the group \(G\). The Lie algebra \(\mathfrak h\) of the stabilizer in general position \(H = G_*(V)\) is the stationary subalgebra in general position (briefly, s.s.g.p.; we denote it by \({\mathfrak g}_*(V))\). This means that there is an open set \(U\subset V\) such that, for any vector \(v\in U\), the stationary subalgebra \({\mathfrak g}_v = \{\xi\in {\mathfrak g}\,|\,\xi_v= 0\}\) is conjugate to \(\mathfrak h\) with respect to the adjoint action of \(G\). The author somewhat simplifies the problem and study the linear actions \(G:V\) with nontrivial stationary subalgebra in general position. In the paper [\textit{E. M. Andreev, È. B. Vinberg} and \textit{A. G. Èlashvili}, ``Orbits of greatest dimension in semi-sirnple linear Lie groups'', Funkts. Anal. Prilozh. 1, No. 4, 3--7 (1967; Zbl 0176.30301), translation in Funct. Anal. Appl. 1(4), 257--261 (1967)], all connected simple irreducible linear groups with nontrivial stationary subalgebra in general position were classified. Moreover, a necessary condition for the s.s.g.p. to be nontrivial was found in the general case in terms of the index of the representation in question. Recall that by the index \(l_{G,V}\) of a representation \(\rho\) of a simple algebraic group \(G\) on a space \(V\) one means the ratio \(\frac{\text{Tr}\,d\rho(X)^2}{\text{Tr}(\text{ad}X)^2}\), \(X\in\mathfrak g\), which does not depend on \(X\). Here \(d\rho\) stands for the corresponding representation of the Lie algebra \(\mathfrak g\) and ad for the adjoint representation of the Lie algebra \(\mathfrak g\). Let us formulate the result of the above mentioned paper in the following form. Theorem. (Necessary condition for the s.g.p. to be nontrivial). Let \(G\) be a semisimple algebraic group acting on a space \(V\). If the s.s.g.p. is nontrivial, then there is a simple normal subgroup \(H\) of \(G\) such that \(l_{H,V}\leq 1\). In 1972, Èlashvili described all stationary subalgebras in general position for the case in which \(G\) is simple and \(V\) is arbitrary and also if \(G\) is semisimple and \(V\) is irreducible. It is natural to pose the problem to describe the s.s.g.p. in the general case. However, considering arbitrary actions of semisimple algebraic groups, one can readily show that arbitrarily complicated representations with nontrivial s.s.g.p. can be constructed. Namely, consider the representation of the group \[ \text{SL}_{n_1}\times \cdots \times \text{SL}_{n_k}\times G :\mathbb C^{n_1}\otimes V_1\oplus \cdots\oplus\mathbb C^{n_k} \otimes V_k\oplus V, \] where \(G : V_i\) are arbitrary representations, \(G : V\) is a representation with nontrivial s.s.g.p. and \(n_i\geq m_i = \text{dim}\,V_i\) (it is assumed that only the groups SL\(_{n_i}\) and \(G\) nontrivially act on the representation \(\mathbb C^{n_i} \otimes V_i)\). Applying the results of Èlashvili, we see that the s.g.p. of this representation is nontrivial and isomorphic to \[ \text{SL}_{n_1-m_1}\times R_{m_1(n_1-m_1)}\times \cdots\times \text{SL}_{n_k-m_k}\times R_{m_k(n_k-m_k)}\times G_*(V), \] where \(R_N\) stands for the unipotent radical of dimension \(N\). Iterating the above construction, one can produce arbitrarily complicated representations with nontrivial s.s.g.p. For this reason, the author considers actions of a special form only, namely, locally strongly effective actions in the sense of the following definition, rather than all actions: A linear action \(G:V\) of a semisimple group \(G\) is said to be locally strongly effective if every simple normal subgroup of the algebraic group \(G\) acts nontrivially on every irreducible submodule \(W\subseteq V\). Let us explain the origin of this term. In the representation theory for algebraic groups, an action is said to be locally effective if its kernel is finite, and the term ``strongly'' means ``on every submodule.'' The union of these conditions for semisimple groups gives precisely the above definition. The main result of the present paper is a complete description of the locally strongly effective actions of a semisimple algebraic group \(G\) on a space \(V\) with nontrivial stationary subalgebra in general position.
scientific article; zbMATH DE number 5980710
scientific article