By I. I Piatetskii-Shapiro
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Extra resources for Automorphic Functions and the Geometry of Classical Domains (Mathematics and Its Applications)
It is easy to verify that the representation in the form of a sum of a Hermitian form and a symmetric form is unique. Later on we will need semihermitian vector forms, which we define in the following manner: a vector form is said to be semihermitian if each of its components is a semihermitian form. From now on we agree to call a form L(u, v) nondegenerate if L(u, vo) = 0 for all u implies that Vo = O. 0 a bounded region in space Ck, whose points we agree to denote by the letter t. 0. In n-dimensional real space, V is the cone discussed in Section 1.
21) Multiplying (16) by the matrix A, we obtain the relationship A2 = 2iB2A+CA2_ACA from which we obtain v sp A 2 = 2·1 '" ~ v 12 a kk Ak = - k= 1 '" ~ Ak2 k= 1 (22) [here we have used relationship (20)]. It follows from (21) and (22) that spAA* = spA 2 1 = 2ispA. , that aks =f. 0 if k = s = v. We have therefore shown that A= ~(:' ~). -v) x (m-v) matrices, respectively. , that B=t ( Ev o 0) . (27) 0 We conclude from (18) and (27) that the matrix C 1 is skew symmetric. Nz, p(z) = CzHNz, N = (:":).
Let G be a j-algebra of dimension 2, and let r be an element of the algebra G such that [Jr, r] = rand r -+ q, jr -+ p is a normal symplectic representation of the algebra G in the space Z; then the space Z can be represented in the form Z X+jX+Z', and (1) the spaces X +j X llnd Z' are orthogonal, (2) p(x, x') = 0 for any X,X'EX, (3) p(z) = AZ, where A = -t when ZEX, A = t when ZEjX, and A = 0 when ZEZ', and (4) q(x) =jx for XEX, while q(z) = 0 when zejX +Z'. Proof Choose some orthonormal basis in Z.
Automorphic Functions and the Geometry of Classical Domains (Mathematics and Its Applications) by I. I Piatetskii-Shapiro