Component groups of centralizers of nilpotents in complex symmetric spaces
Prior work of the second author gave a classification of nilpotent elements in a complex symmetric space generalizing the Bala-Carter classification of nilpotent elements in a complex semi-simple Lie algebra. In this paper, the authors extend this classification in order to give information about the centralizers of nilpotent elements in a complex symmetric space. The statement of the main theorem takes some notation. Let \(\mathfrak g = \mathfrak k \oplus \mathfrak p\) be the Cartan decomposition of a real simple Lie algebra. Let \(G_{\mathbb C}\) be the adjoint group of \({\mathfrak g}_{\mathbb C}\) and \(G\), \(K\), and \(K_{\mathbb C}\) the connected subgroups of \(G_{\mathbb C}\) corresponding to the appropriate Lie algebras. If \(e\) is a nilpotent element of \({\mathfrak p}_{\mathbb C}\), let \(A(e)\) and \(A_k(e)\) be the component groups of \(G_{\mathbb C}^e\) and \(K_{\mathbb C}^e\), respectively. The centralizer in \({\mathfrak g}_{\mathbb C}\) of an elliptic element of \(K_{\mathbb C}\) is called an elliptic pseudo Levi subalgbra. A nilpotent element \(e\) in a reductive (respectively, \(\theta\)-stable reductive) subalgebra \(\mathfrak l\) (respectively, \({\mathfrak l}|\cap{\mathfrak p}_{\mathbb C}\)) of \({\mathfrak g}_{\mathbb C}\) is said to be distinguished (respectively, noticed) in \(\mathfrak l\) if the conditions \(z\in\mathfrak l\) (respectively, \(z\in {\mathfrak l}|\cap{\mathfrak p}_{\mathbb C})\), \(z\) is semisimple, and \([z,e]=0\) imply \(z\) is in the center of \(\mathfrak l\). The first main result says there is a bijection between (1) \(K_{\mathbb C}\) conjugacy classes of pairs \((e,C)\), where \(e\) is nilpotent in \({\mathfrak p}_{\mathbb C}\) and \(C\) is a conjugacy class in \(A_k(e)\); and (2) \(K_{\mathbb C}\) conjugacy classes of triples \((\mathfrak l, \mathfrak q,\overline{M_{K_{\mathbb C}}\cdot e})\), where \(\mathfrak l\) is an elliptic pseudo Levi subalgebra in which \(e\) is noticed and \(\mathfrak q \subset \mathfrak l\), is a \(\theta\)-stable noticed parabolic for \(e\). Here \(\mathfrak q\) has Levi decomposition \(\mathfrak m \oplus \mathfrak u\) and \(M_{K_{\mathbb C}}\) is the connected subgroup of \(L\) with Lie algebra \(\mathfrak m \cap {\mathfrak k}_{\mathbb C}\). Next a three-dimensional subgroup \({\widetilde G}_{\mathbb C}(e)\) of \(G_{\mathbb C}\), called the carrier subgroup of \(e\), is defined. The second main theorem says there is a bijection between \(K_{\mathbb C}\) conjugacy classes of triples \((\mathfrak l, {\mathfrak q}_{\mathfrak l}, W)\) as in the first main result and \({\widetilde G}_{\mathbb C}(e)\) conjugacy classes of pairs \((\widetilde{\mathfrak l}, {\mathfrak q}_{\widetilde {\mathfrak l}})\) where \(\widetilde{\mathfrak l}\) is a pseudo Levi subalgebra of the Lie algebra of \({\widetilde G}_{\mathbb C}(e)\) and \({\mathfrak q}_{\widetilde{\mathfrak l}}\) is a distinguished parabolic of \(\widetilde{\mathfrak l}\) containing \(e\).
scientific article; zbMATH DE number 1539378
scientific article