The generalized Maslov index
The classical Maslov index [cf. \textit{G. Lion} and \textit{M.Vergne}, The Weil representation, Maslov index and theta series (Birkhäuser) (1980; Zbl 0444.22005)] is defined for triples of Lagrangian subspaces of a real symplectic vector space. The invariance by the natural action of the symplectic group, the permutation property and the cocycle condition are the basic properties of the Maslov index. The space of Lagrangian subspaces of a real symplectic vector space is a submanifold, called the Lagrangian manifold, of the Grassmannian of this vector space. It is an example of a space associated to a Euclidean Jordan algebra or symmetric Hermitian space of tube type [cf. \textit{J. Faraut} and \textit{A. Korányi}, Analysis on symmetric cones (Oxford: Clarendon Press) (1994; Zbl 0841.43002)]. Let \(\Omega\) be the cone of a Euclidean Jordan algebra \(J\), and \(\mathbb J\) its complexification. It admits a spectral norm. Denote by \(\mathcal D\) the corresponding open unit ball in \(\mathbb J\). Let \(G\) be the group of holomorphic diffeomorphisms of \(\mathcal D\). Via a Cayley transformation, \(\mathcal D\) is holomorphically equivalent to the domain \(T_{\Omega} = J+i\Omega \subset {\mathbb J}\). Denote by \(S\) the Shilov boundary of \(\mathcal D\). If we take the algebra of real symmetric matrices, its domain \(\mathcal D\) is the Siegel disc, the group \(G\) is isomorphic to the symplectic group and the Shilov boundary, in this case, can be identified with the Lagrangian manifold mentioned above. In [\textit{J.-L. Clerc} and \textit{B. Ørsted}, Transform. Groups 6, No. 4, 303--320 (2001; Zbl 1078.53076) and Asian J. Math. 7, 269--296 (2003)], the author together with B. Ørsted, developed a theory of the Maslov index for such spaces under the additional assumption of ``transversality'', which in the Lagrangian case precisely corresponds to the condition that any two of the three Lagrangian subspaces are transverse. The definition presented in the two papers mentioned above used a passage to the limit which imposed this ``transversality'' condition. In this paper the author uses a more refined passage to the limit - ``\(\Gamma\)-radial convergence'', which permits to define the Maslov index for any triple from \(S\). Technical lemmas are gathered in sections 2 and 3. Section 3 contains also the definition of the index and the proof of the basic three properties. The main result of section 4 is the invariance, up to a constant depending on the dimension, of the Maslov index under transformations induced by homomorphisms of Euclidean Jordan algebras. In section 5 the author shows that his new index coincides with the Maslov index à la Kishiwara defined in [\textit{J.-L. Clerc}, J. Geom. Phys. 49, 21--51 (2004)].
scientific article; zbMATH DE number 2066881