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By M. Raynaud, T. Shioda

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GEOMETRY. --- DEFINITIONS. 1. If a block of wooden or stone be lower within the sbape represented iu Fjg. 1, it is going to have six fiat faces. each one face of the block is termed a floor; and if those faces are made delicate via sprucing, in order that, whilst a straight-edge is utilized to anyone of them, the directly facet in every little thing will contact the outside, the faces are referred to as aircraft surfaces, or planes.

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21. Suppose n = 1; Lag(1) consists of all straight lines passing through the origin in the symplectic plane (R2n z , − det). When n > 1 the Lagrangian Grassmannian is a proper subset of the set of all n-dimensional planes of (R2n z , σ). Let us study the equation of a Lagrangian plane in the standard symplectic space. In what follows we work in an arbitrary symplectic basis B = {e1 , . . , en } ∪ {f1 , . . , fn } of the standard symplectic space; the corresponding coordinates are denoted by x and p.

26. The mapping ∆ : Sp(n) −→ S 1 defined by ∆(S) = det u where u is the image in U(n, C) of U = S(S T S)−1/2 ∈ U(n) induces an isomorphism ∆∗ : π1 [Sp(n)] ∼ = π1 [U(n, C)] and hence an isomorphism π1 [Sp(n)] ∼ = π1 [S 1 ] ≡ (Z, +). Proof. 24 above and its proof, any loop t −→ S(t) = R(t)eX(t) in Sp(n) is homotopic to the loop t −→ R(t) in U(n). Now S T (t)S(t) = e2X(t) (because X(t) is in sp(n) ∩ Sym(2n, R)) and hence R(t) = S(t)(S T (t)S(t))−1/2 . 42 Chapter 2. 2). Let us next study two useful factorizations of symplectic matrices that be used several times in the rest of this book: the so-called “pre-Iwasawa torization”, reminiscent of the Iwasawa decomposition in Lie group theory, factorization by free symplectic matrices.

3 The Lagrangian Grassmannian Recall that a subset of (E, ω) is isotropic if ω vanishes identically on it. An isotropic subspace of (E, ω) having dimension n = 12 dim E is called a Lagrangian plane. Equivalently, a Lagrangian plane in (E, ω) is a linear subspace of E which is both isotropic and co-isotropic. 15 that there always exists a Lagrangian plane containing a given isotropic subspace: let {e1 , . . , ek } be a basis of such a subspace; complete that basis into a full symplectic basis B = {e1 , .

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