MASLOV INDEX AND CLIFFORD ALGEBRAS

The Maslov index of triples of Lagrangean subspaces of a symplectic space over a local field F is considered in the book of G. Lion and M. Vergne [4]. The authors use the Maslov index to construct a central extension of Sp(2n, F) by the Witt group WF of the field F. The cocycle of this extension determines the cocycle of the Weil representation of Sp(2n, F). In the present work we introduce a generalised Maslov index associated to triples of non-zero vectors in a plane. It takes values in a certain non-commutative group N ̃ , which contains K2(F) as a subgroup. The generalised Maslov index can be reduced to this subgroup, giving rise to a central extension of SL(2, F) by K2(F) (the field F is arbitrary). The cocycle of this extension coincides with Matsumoto’s cocycle ([1], [6]). The Witt group (more precisely, its quotient) should be regarded as a “reduction of the group N ̃ (mod 2)”. In this work we give an analogous interpretation of the “reduction of N ̃ (modn)”: it is related to a certain class of Z/n-graded algebras, which generalise Clifford algebras. This work was carried out during my stay at the Steklov Mathematical Institute. I am grateful to Yu.I. Manin for his interest in my work and to A.B. Goncharov for his help with the preparation of the manuscript. 1. Maslov Index (after [4]) The Maslov index arises naturally in the study of invariants of systems of one-dimensional subspaces of a two-dimensional space. Fix the following notation: let F be a field of characteristic char(F) 6 = 2, V = F 2 a two-dimensional vector space over F, B a symplectic form on V, G = SL(2, F) = Aut(V,B). Denote by T