Isotropy subalgebras of elliptic orbits in semisimple Lie algebras, and the canonical representatives of pseudo-Hermitian symmetric elliptic orbits (Q2468456)

Let \(\mathfrak{g}\) be a real semisimple Lie algebra. An element \(X\in \mathfrak{g}\) is called semisimple if the endomorphism \(\text{ad}_{\mathfrak{g}} X\) on \(\mathfrak{g}\) is semisimple, i.e., diagonalizable over \(\mathbb C\). A semisimple element \(X\in \mathfrak{g}\) is called elliptic (resp. hyperbolic) if the eigenvalues of \(\text{ad} _{\mathfrak{g}} X\) are purely imaginary (resp. real). The complete Jordan decomposition [see \textit{J. Adams} and \textit{D. Vogan} (eds.), Representation theory of Lie groups. Lecture notes from the Graduate summer school program, Park City, UT, USA, July 13--31, 1998. Providence, RI: American Mathematical Society (2000; Zbl 0936.00032)] asserts that each \(X\in \mathfrak{g}\) can be decomposed as \(X = X_e+X_h+X_n\) where \(X_e\) is elliptic, \(X_h\) is hyperbolic, and \(X_n\) is nilpotent and they commute with each other. Let \(G\) be the analytic group of \(\mathfrak{g}\). The adjoint orbit \(\text{Ad} (G) X\) through a semisimple element \(X\) (resp. elliptic element, hyperbolic element, nilpotent element) is called a semisimple orbit (resp. elliptic orbit, hyperbolic orbit, nilpotent orbit). The nilpotent orbits have been studied [see \textit{D. H. Collingwood} and \textit{W. M. McGovern}, Nilpotent orbits in semisimple Lie algebras. New York, NY: Van Nostrand Reinhold Company (1993; Zbl 0972.17008)]. It is well known that an orbit \(\text{Ad} (G)X\) is closed if and only if \(X\in \mathfrak{g}\) is semisimple. [\textit{A. Borel} and \textit{Harish-Chandra}, ``Arithmetic subgroups of algebraic groups'', Ann. Math. (2) 75, 485--535 (1962; Zbl 0107.14804)]. A semisimple orbit is elliptic if and only if it admits a \(G\)-invariant pseudo-Kähler metric [\textit{T. Kobayashi}, Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory, Am. Math. Soc., Transl., 2. Ser. 183, 1--50 (1998); translation from Sūgaku 46, No. 2, 124--143 (1994; Zbl 0895.22006)]. It is also known that the elliptic orbits are in one-to-one correspondence with the \(\text{Ad} (K)\)-orbit on \(\mathfrak{k}\) where \(\mathfrak{g}=\mathfrak{k}+\mathfrak{p}\) is the Cartan decomposition of \(\mathfrak{g}\) (see Adams and Vogan). The first main concern of the paper is to provide a thorough study of elliptic orbits. Since \(\text{Ad} (G)X\) can be identified with \(G/C_G(X)\) as they are diffeomorphic, where \(C_G(X) :=\{g\in G: \text{Ad}(g)X=X\}\) is the isotropy subgroup of \(X\) in \(G\), the authors determines \(C_G(X)\) for any arbitrary elliptic element \(X\) in the algebra level, namely, to determine the centralizer \(\mathfrak{c}_{\mathfrak{g}}(X):=\{Y\in \mathfrak{g}: [X,Y]=0\}\). The problem is then reduced to the determination of the centralizer \(\mathfrak{c}_{\mathfrak{k}}(X)\) in \(\mathfrak{k}\) (\(\mathfrak{k}\) is a maximal compact subalgebra of \(\mathfrak{g}\) containing \(X\)), the semisimple part and the center of the centralizer \(\mathfrak{c}_{\mathfrak{g}_0}(X)\) in \(\mathfrak{g}_u\) (\((\mathfrak{g}_u,\mathfrak{k})\) is the compact dual of the orthogonal symmetric Lie algebra \((\mathfrak{g}, \mathfrak{k})\)). Using the structure theorem, the centralizers of all possible elliptic elements in \(\mathfrak{sl}(4,\mathbb R)\) are worked out, up to inner automorphism. The second main concern is to determine the \(H\)-elements in all twenty-nine simple irreducible pseudo-Hermitian symmetric Lie algebras (an \(H\)-element \(T \in\mathfrak{g}\) is an elliptic element satisfying two conditions (1) \(R\) is the centralizer \(C_G(T)\) of \(T\) in \(G\), and (2) \(\text{ad} _{\mathfrak{g}} T\) induces the complex structure \(J\) for the pseudo-Hermitian symmetric space \(G/R\)). The description is in terms of the dual basis for the simple roots. The method is different from that of Kaneyuki who earlier solved the problem for some specified type of pseudo-Hermitian symmetric spaces. Many of the proofs are intricate and labor intensive.