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Complex Structures Contained in Classical Groups

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Complex Structures Contained in Classical Groups Journal of Lie Theory Volume 8 (1998) 139{152 C 1998 Heldermann Verlag Complex Structures Contained in Classical Groups Christina Birkenhake Communicated by K.-H. Neeb Abstract. For any classical group G let (G)= J G J 2= 1 denote the space of complex C f 2 j − g structures in G . (G) is a symmetric space with finitely many connected components, which C are described explicitly. In particular these components are flag domains in the sense of J. A. Wolf. Moreover it is shown that the components of (G) are parameter spaces for C nondegenerate complex tori with certain endomorphism structure. Let V be a real vector space of finite dimension. A complex structure on V is an endomorphism J of V with J 2 = id . If V admits a complex structure − V J , then necessarily V is of even dimension, say 2n. The pair VJ := (V; J) is a complex vector space of dimension n with respect to the scalar multiplication C V V; (x + iy; v) xv + yJ(v). It is well-known that the set of all complex × ! 7! structures on V R2n is the symmetric space ' (GL(V )) GL2n(R)=GLn(C): C ' Suppose the complex vector space VJ := (V; J) admits a hermitian scalar product H . Then with respect to a suitable basis for V = R2n the hermitian form H is t 0 1n 0 1n given by the matrix J 1n 0 +i 1n 0 . Since Re H is symmetric and positive − − definite, the complex structure J is contained in the symplectic group Sp2n(R) and t 0 1n J 1n 0 is positive definite. Thus the set of hermitian complex vector spaces of − dimension n can be identified with the space 0(Sp2n(R)) of complex structures 2n t 0C 1n on R contained in Sp2n(R) such that J 1n 0 > 0. It is well-known that − 0(Sp2n(R)) is the symmetric space C 0(Sp2n(R)) Sp2n(R)=Un(C): C ' The space 0(Sp2n(R)) admits a further interpretation: it is isomorphic to the Siegel upperC half space and as such parametrizes families of polarized abelian varieties. Quotients of 0(Sp2n(R)) by suitable arithmetic subgroups of Sp2n(R) are moduli spaces of abCelian varieties with certain level structures. It is the aim of this note to generalize these results to arbitrary classical groups, i.e. the groups C ISSN 0949{5932 / $2.50 Heldermann Verlag 140 Birkenhake GLn(R); GLn(C); GLn(H); Op;q(R); On(C); Up;q(C); Up;q(H); Sp2n(R); Sp2n(C) 1 and the antiunitary quaternionic group αUn(H) (here H = C + jC denotes the skew field of Hamiltonian quaternions). Let G be any classical group of the above list. The set of all complex structures contained in G is denoted by (G) := J G J 2 = 1 : C f 2 j − g The group G acts on (G) by conjugation. It will be shown that for G = C GL2n(R); GLn(H); O2p;2q(R); O2n(C); Up;q(H) and Sp2n(C) the set (G) is a G- C orbit. To be more precise Theorem 1. a) The action of G on (G) induces isomorphisms C (GL2n(R)) GL2n(R)=GLn(C) C ' (GLn(H)) GLn(H)=GLn(C) C ' (O2p;2q(R)) O2p;2q(R)=Up;q(C) C ' (O2n(C)) O2n(C)=GLn(C) C ' (Up;q(H)) Up;q(H)=Up;q(C) C ' (Sp2n(C)) Sp2n(C)=GLn(C) C ' b) The spaces (GL2n+1(R)); (Op;q(R)) with p or q odd, and (O2n+1(C)) are C C C empty. For the embeddings of the respective subgroups into G see 1 Remark 1.1. x If G is one of the remaining groups: GLn(C); Up;q(C); Sp2n(R) or αUn(H), the space (G) splits up into finitely many orbits characterized by some index and/or signatureC condition. For this denote 0 1n 2 k(Sp2n(R)) := J (Sp2n(R)) ind 1 −0 J = 2k C f 2 C j R n g 3 r(GLn(C)) := J (GLn(C)) sign(J) = (r; n r) C f 2 C j − g i1p 0 indC( 0 i1q J) = k 4 k;r(Up;q(C)) := J (Up;q(C)) − C 2 C sign(J) = (r; n r) − 5 k(αUn(H)) := J (αUn(H)) ind (Ji) = k . C f 2 C j H g These spaces have the following structure 1 αU ( ) = M GL ( ) M(i1 )tM = i1 , where M denotes quaternionic conjugation of n H f 2 n H j n ng the matrix M . Some authors use the notation SO∗(2n) for this group. 2 indR is the number of negative eigenvalues of a nondegenerate symmetric real matrix. 3 sign(J) = (r; s) if and only if J = A 1 i1r 0 A for some A GL ( ). − 0 i1s n C 4 − 2 indC is the number of negative eigenvalues of a nondegenerate hermitian matrix. 5 t t 1n k 0 For M GLn(H) with M = M define ind M = k if AMA = − for some 2 H 0 1k A GL ( ). − 2 n H Birkenhake 141 n Theorem 2. a) (Sp2n(R)) = k(Sp2n(R)) and C k=0 C k(Sp2n(R)) S Sp2n(R)=Un k;k(C) C ' − n b) (GLn(C)) = r(GLn(C)) and C r=0 C r(GLSn(C)) GLn(C)=(GLr(C) GLn r(C)) C ' × − p+q p+q c) (Up;q(C)) = r=0 k=0;k r q(mod 2) k;r(Up;q(C)) and C ≡ − C r q+2m;r(SUp;q(C))S Up;q(C)=(Ur q+m;q m(C) Up+q r m;m(C)) C − ' − − × − − n d) (αUn(H)) = k(αUn(H)) and C k=0 C S k(αUn(H)) αUn(H)=Un k;k(C) C ' − For the embeddings of the respective subgroups see again 1 Remark 1.1. Within this note the notation (G) refers to any of the spaces ofxTheorems 1 or 2. C∗ In some cases, namely for G = GLn(R); GLn(C); O2n(R) and Sp2n(R), the results of Theorems 1 and 2 are well-known, at least for (G). They are included here C0 for the sake of completeness. The symmetric spaces (G) can be interpreted as open subspaces of proper ∗ subvarieties of certain GrassmannianC varieties. In these terms the theorems seem to be a consequence of Witt's Theorem. However it turns out that this approach does not apply in every case (see [1]). Within this note we restrict ourself to the point of view of complex structures, in order to unify the procedure. As an immediate consequence of the theorems one obtains that every complex structure admits a normal form (see Section 2). In Section 2 also two examples of less immediate applications to Linear Algebra are presented. As a further result it will be shown in Section 3 that the connected components of the symmetric spaces (G) are flag domains in the sense of J. A. Wolf. This C∗ implies for example that they are simply connected submanifolds of smooth pro- jective varieties and that there are certain vanishing theorems for the cohomology of coherent sheaves on (G). C∗ As mentioned above 0(Sp2n(R)) is isomorphic to the Siegel upper half space n . Similarly all spaces C (G) can be interpreted as parameter spaces for nondegen-H C∗ erate complex tori (X; H) with certain level structures. (For the definition of nondegenerate complex tori see Section 4.) This level structure consists of the prescription of an endomorphism structure for (X; H). In this way every nonde- generate complex torus belongs to one of the spaces (G) or a product of these. C∗ Notation: 1n = unit n n matrix; 0 = zero n × n − matrix; n × − 0s;r = zero s r matrix; 1p 0 × − Ip;q := 0 1q − 0 1n O2n(C) is the orthogonal group forthe symmetric bilinear form 1n 0 . O2p;2q(R) is the orthogonal group for the symmetric bilinear form Ip;q 0 . 0 Ip;q 142 Birkenhake 1. The Proofs of Theorems 1 and 2 As already mentioned in the introduction the results of Theorems 1 and 2 are well-known for (GL2n(R)); r(GLn(C)); (O2n(R)) and 0(Sp2n(R)). C C C C Proof. (of Theorem 1 b)): Suppose first G = GL2n+1(R) or O2n+1(C). The determinant of any J (G) is 1. On the other hand the eigenvalues of J r 2 C 2n+1 r n+r+1 are i, so det J = i ( i) − = ( 1) i for some r = 0; : : : ; 2n + 1, · − − a contradiction. Moreover for G = Op;q(R) one uses the fact that every J 2 (Op;q(R)) defines a nondegenerate hermitian form H = Ip;q + iIp;qJ on the C p+q complex vector space VJ = (R ; J). But then 2 indCH = indRRe H = indRIp;q = q, so q is even. Since p + q = 2 dimC VJ is even, p is also even. Proof. (of Theorem 1, case (GLn(H))): This is a consequence of results of C Louise Wolf. Recall the complex representation ρ : GLn(H) , GL2n(C); A+Bj A B ! 7! B A . In [6] it is shown that quaternionic matrices are similar if and only if their− complex representations are similar, and that the eigenvalues of a matrix in im ρ : GLn(H) , GL2n(C) appear as complex conjugate pairs. This implies that f ! g every complex structure J (GLn(H)) is conjugate to i1n GLn(H).
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