Projective representations of real semisimple Lie groups and the gradient map

Abstract

Let G be a real noncompact semisimple connected Lie group and let \(\rho : G \longrightarrow \text {SL}(V)\) be a faithful irreducible representation on a finite-dimensional vector space V over \(\mathbb {R}\). We suppose that there exists a scalar product \(\texttt {g}\) on V such that \(\rho (G)=K\exp ({\mathfrak {p}})\), where \(K=\text {SO}(V,\texttt {g})\cap \rho (G)\) and \({\mathfrak {p}}=\text {Sym}_o (V,\texttt {g})\cap (\text {d} \rho )_e ({\mathfrak {g}})\). Here, \({\mathfrak {g}}\) denotes the Lie algebra of G, \(\text {SO}(V,\texttt {g})\) denotes the connected component of the orthogonal group containing the identity element and \(\text {Sym}_o (V,\texttt {g})\) denotes the set of symmetric endomorphisms of V with trace zero. In this paper, we study the projective representation of G on \({\mathbb {P}}(V)\) arising from \(\rho \). There is a corresponding G-gradient map \(\mu _{\mathfrak {p}}:{\mathbb {P}}(V) \longrightarrow {\mathfrak {p}}\). Using G-gradient map techniques, we prove that the unique compact G orbit \({\mathcal {O}}\) inside the unique compact \(U^\mathbb {C}\) orbit \({\mathcal {O}}'\) in \({\mathbb {P}} (V^\mathbb {C})\), where U is the semisimple connected compact Lie group with Lie algebra \({\mathfrak {k}} \oplus {\textbf {i}} {\mathfrak {p}}\subseteq \mathfrak {sl}(V^\mathbb {C})\), is the set of fixed points of an anti-holomorphic involutive isometry of \({\mathcal {O}}'\) and so a totally geodesic Lagrangian submanifold of \({\mathcal {O}}'\). Moreover, \({\mathcal {O}}\) is contained in \({\mathbb {P}}(V)\). The restriction of the function \(\mu _{\mathfrak {p}}^\beta (x):=\langle \mu _{\mathfrak {p}}(x),\beta \rangle \), where \(\langle \cdot , \cdot \rangle \) is an \(\text {Ad}(K)\)-invariant scalar product on \({\mathfrak {p}}\), to \({\mathcal {O}}\) achieves the maximum on the unique compact orbit of a suitable parabolic subgroup and this orbit is connected. We also describe the irreducible representations of parabolic subgroups of G in terms of the facial structure of the convex body given by the convex envelope of the image \(\mu _{\mathfrak {p}}({\mathbb {P}}(V))\).