# An Introduction to Symplectic Geometry (Graduate Studies in by Rolf Berndt PDF

M(Lo), because, from M(Lo) = M'(Lo), it follows that M-IM' E GLo. In a similar vein, we can describe another family of interesting subspaces of a given symplectic space.

This question will be considered further below. For now, we give a brief survey. The invertible linear morphisms of R2n, that is, the invertible matrices M, that i) preserve the canonical scalar product, s, form the orthogonal group O(2n); ii) preserve the symplectic standard form, w, form the symplectic group Spn(R); iii) preserve the complex structure J, that is, with MJ = JM, are of the form M- (X -Y YX and form the general linear group GLn(C) via M i-. X + iY. It can be shown that O(2n) n Spn(R) = Spn(R) fl GLn(C) = GLn(C) n O(2n) = U(n).

42. For a real 2n-dimensional symplectic space (V, w), the set 3 (V, w) of positive compatible complex structures on V can be identified with the Siegel upper half plane fin. The identification of the theorem depends on the symplectic basis a of V. )(C D) with ( C D , there is a matrix Z associated to F relative toe with the property that F has the bases e. + eZ as well as e. + eZ. So there is a A E CLn(C) with e. + eZ = (e. + eZ)A. C)Z=e,A+eZA, and thus A=CZ+D and Z=(AZ+B)(CZ+D)-1. From complex function theory, the mapping Z I.