Page - 349 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 Asymmetric space isaRiemannianmanifold,where thereversalof thegeodesics iswelldefined andisan isometry. Formally, expp(u) → expp(−u) isan isometry foreach ponthemanifold,whereu isavector in the tangentspaceat p, and expp theRiemannianexponentialapplicationat p. Inother words, thesymmetryaroundeachpointisanisometry.Hn isasymmetricspace(see[20]). Thestructure ofasymmetric spacecanbestudied through its isometrygroupandtheLiealgebraof its isometry group. Thepresentworkwillmakeuseof theCartanandIwasawadecompositionsof theLiealgebra ofSp(n,R) (see [22]). Let sp(n,R)be theLiealgebraofSp(n,R). GivenA,BandC three realn×n matrices, letdenote ( A B C −At ) =(A,B,C).Wehave sp(n,R)={(A,B,C)|BandC symmetric} . TheCartandecompositionofsp(n,R) isgivenby sp(n,R)= t⊕p, where t={(A,B,−B)|B symmetricandA skew-symmetric} , p={(A,B,B)|A,B, symmetric} . (1) TheIwasawadecomposition isgivenby sp(n,R)= t⊕a⊕n, where a={(H,0,0)|Hdiagonal} , n={(A,B,0)|Aupper triangularwith0onthediagonal ,B symmetric} . It canbeshownthat p=∪k∈KAdk(a), (2) whereAd is theadjoint representationofSp(n,R). 2.2. TheSiegelDisk The Siegel disk Dn is the set of complex matrices {Z|I−Z∗Z≥0}, where ≥ stands for the Loewner order (see [24] for details on the Loewner order). Recall that for A and B two Hermitianmatrices,A≥Bwithrespect to theLoewnerordermeans thatA−B ispositivedefinite. The transformation Z∈Hn → (Z− iI)(Z+ iI)−1∈Dn is an isometry between the Siegel upper half space and the Siegel disk. Let C = ( I −iI I iI ) . The application g ∈ Sp(n,R) → CgC−1 identifies the set of isometries ofHn and ofDn. Thus, it canbeshownthatamatrixg= ( A B A B ) ∈Sp(n,C)acts isometricallyonDnby g.Z=(AZ+B)(AZ+B)−1, 349