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 C(G)={J∈G | J^{2}=−1} denote the space of complex structures in G. C(G) is a symmetric space with finitely many connected components, which 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 C(G) are parameter spaces for 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_{V} . If V admits a complex structure J, then necessarily V is of even dimension, say 2n. The pair V_{J} := (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 structures on V ≃ ℝ^{2n} is the symmetric space

C(GL(V)) ≃ GL_{2n}(ℝ)/GL_{n}(ℂ).

Suppose the complex vector space V_{J} := (V, J) admits a hermitian scalar product H. Then with respect to a suitable basis for V = ℝ^{2n} the hermitian form H is given by the matrix ^{t}J( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} )+i( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ). Since Re H is symmetric and positive definite, the complex structure J is contained in the symplectic group Sp_{2n}(ℝ) and tJ( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ) is positive definite. Thus the set of hermitian complex vector spaces of dimension n can be identified with the space C_{0}(Sp_{2n}(ℝ)) of complex structures on ℝ^{2n} contained in Sp_{2n}(ℝ) such that ^{t}J( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ) > 0. It is well-known that C_{0}(Sp_{2n}(ℝ)) is the symmetric space

C_{0}(Sp_{2n}(ℝ)) ≃ Sp_{2n}(ℝ)/U_{n}(ℂ).

The space C_{0}(Sp_{2n}(ℝ)) admits a further interpretation: it is isomorphic to the Siegel upper half space and as such parametrizes families of polarized abelian varieties. Quotients of C_{0}(Sp_{2n}(ℝ)) by suitable arithmetic subgroups of Sp_{2n}(ℝ) are moduli spaces of abelian varieties with certain level structures. It is the aim of this note to generalize these results to arbitrary classical groups, i.e. the groups

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag

GL_{n}(ℝ), GL_{n}(ℂ), GL_{n}(ℍ), O_{p,q}(ℝ), O_{n}(ℂ), U_{p,q}(ℂ), U_{p,q}(ℍ), Sp_{2n}(ℝ), Sp_{2n}(ℂ)

and the antiunitary quaternionic group αU_{n}(ℍ) ^{1} (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

C(G) := {J ∈ G| J^{2} = −1}.

The group G acts on C(G) by conjugation. It will be shown that for G = GL_{2n}(ℝ), GL_{n}(ℍ), O_{2p,2q}(ℝ), O_{2n}(ℂ), U_{p,q}(ℍ) and Sp_{2n}(ℂ) the set C(G) is a Gorbit. To be more precise

Theorem 1. a) The action of G on C(G) induces isomorphisms

\begin{aligned}C(GL_{2n}(ℝ)) & ≃ & GL_{2n}(ℝ)/GL_{n}(ℂ) \\ C(GL_{n}(ℍ)) & ≃ & GL_{n}(ℍ)/GL_{n}(ℂ) \\ C(O_{2p,2q}(ℝ)) & ≃ & O_{2p,2q}(ℝ)/U_{p,q}(ℂ) \\ C(O_{2n}(ℂ)) & ≃ & O_{2n}(ℂ)/GL_{n}(ℂ) \\ C(U_{p,q}(ℍ)) & ≃ & U_{p,q}(ℍ)/U_{p,q}(ℂ) \\ C(Sp_{2n}(ℂ)) & ≃ & Sp_{2n}(ℂ)/GL_{n}(ℂ) \end{aligned}

b) The spaces C(GL_{2n+1}(ℝ)), C(O_{p,q}(ℝ)) with p or q odd, and C(O_{2n+1}(ℂ)) are empty. For the embeddings of the respective subgroups into G see §1 Remark 1.1. If G is one of the remaining groups: GL_{n}(ℂ), U_{p,q}(ℂ), Sp_{2n}(ℝ) or αU_{n}(ℍ), the space C(G) splits up into finitely many orbits characterized by some index and/or signature condition. For this denote

C_{k}(Sp_{2n}(ℝ)) := \{J ∈ C(Sp_{2n}(ℝ))| ind_{ℝ} (( \begin{matrix}0 & −1_{n} \\ 1_{n} & 0 \end{matrix} )J) = 2k\}^{2}

C_{r}(GL_{n}(ℂ)) := {J ∈ C(GL_{n}(ℂ))| sign(J) = (r, n − r)} ^{3}

C_{k,r}(U_{p,q}(ℂ)) := \{J ∈ C(U_{p,q}(ℂ)) | \begin{matrix}ind_{ℂ}(( \begin{matrix}i1_{p} & 0 \\ 0 & −i1_{q} \end{matrix} )J) = k \\ sign(J) = (r, n − r) \end{matrix} \}^{4}

C_{k}(αU_{n}(ℍ)) := {J ∈ C(αU_{n}(ℍ))| ind_{ℍ}(Ji) = k}^{5}.

These spaces have the following structure Theorem 2. a) C(Sp_{2n}(ℝ)) = ⋃_{k=0}^{n} C_{k}(Sp_{2n}(ℝ)) and

1 αU_{n}(ℍ) = \{M ∈ GL_{n}(ℍ)|M(i1_{n})^{t}\overline{M} = i1_{n}\}, where \overline{M} denotes quaternionic conjugation of the matrix M. Some authors use the notation SO^{∗}(2n) for this group.

2 ind_{ℝ} is the number of negative eigenvalues of a nondegenerate symmetric real matrix.

3 sign(J) = (r, s) if and only if J = A^{−1}( \begin{matrix}i1_{r} & 0 \\ 0 & −i1_{s} \end{matrix} )A for some A ∈ GL_{n}(ℂ).

4 ind_{ℂ} is the number of negative eigenvalues of a nondegenerate hermitian matrix.

5For M ∈ GL_{n}(ℍ) with ^{t}\overline{M} = M define ind_{ℍ}M = k if ^{t}AM\overline{A} = ( \begin{matrix}1_{n−k} & 0 \\ 0 & −1_{k} \end{matrix} ) for some A ∈ GL_{n}(ℍ).

C_{k}(Sp_{2n}(ℝ)) ≃ Sp_{2n}(ℝ)/U_{n−k,k}(ℂ)

b) C(GL_{n}(ℂ)) = ⋃_{r=0}^{n} C_{r}(GL_{n}(ℂ)) and

C_{r}(GL_{n}(ℂ)) ≃ GL_{n}(ℂ)/(GL_{r}(ℂ) × GL_{n−r}(ℂ))

c) C(U_{p,q}(ℂ)) = ⋃_{r=0}^{p+q} ⋃_{k=0,k≡r−q(mod 2)}^{p+q} C_{k,r}(U_{p,q}(ℂ)) and

C_{r−q+2m,r}(U_{p,q}(ℂ)) ≃ U_{p,q}(ℂ)/(U_{r−q+m,q−m}(ℂ) × U_{p+q−r−m,m}(ℂ))

d) C(αU_{n}(ℍ)) = ⋃_{k=0}^{n} C_{k}(αU_{n}(ℍ)) and

C_{k}(αU_{n}(ℍ)) ≃ αU_{n}(ℍ)/U_{n−k,k}(ℂ)

For the embeddings of the respective subgroups see again §1 Remark 1.1. Within this note the notation C_{∗}(G) refers to any of the spaces of Theorems 1 or 2.

In some cases, namely for G = GL_{n}(ℝ), GL_{n}(ℂ), O_{2n}(ℝ) and Sp_{2n}(ℝ), the results of Theorems 1 and 2 are well-known, at least for C_{0}(G). They are included here for the sake of completeness.

The symmetric spaces C_{∗}(G) can be interpreted as open subspaces of proper subvarieties of certain Grassmannian 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 C_{∗}(G) are flag domains in the sense of J. A. Wolf. This implies for example that they are simply connected submanifolds of smooth projective varieties and that there are certain vanishing theorems for the cohomology of coherent sheaves on C_{∗}(G).

As mentioned above C_{0}(Sp_{2n}(ℝ)) is isomorphic to the Siegel upper half space H_{n}. Similarly all spaces C_{∗}(G) can be interpreted as parameter spaces for nondegenerate 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 nondegenerate complex torus belongs to one of the spaces C_{∗}(G) or a product of these.

Notation:

1_{n} = unit n × n − matrix,
0_{n} = zero n × n − matrix,
0_{s,r} = zero s × r − matrix,
I_{p,q} := ( \begin{matrix}1_{p} & 0 \\ 0 & −1_{q} \end{matrix} )

O_{2n}(ℂ) is the orthogonal group for the symmetric bilinear form ( \begin{matrix}0 & 1_{n} \\ 1_{n} & 0 \end{matrix} ). O_{2p,2q}(ℝ) is the orthogonal group for the symmetric bilinear form ( \begin{matrix}I_{p,q} & 0 \\ 0 & I_{p,q} \end{matrix} ).

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 C(GL_{2n}(ℝ)), C_{r}(GL_{n}(ℂ)), C(O_{2n}(ℝ)) and C_{0}(Sp_{2n}(ℝ)).

Proof. (of Theorem 1 b)): Suppose first G = GL_{2n+1}(ℝ) or O_{2n+1}(ℂ). The determinant of any J ∈ C(G) is ±1. On the other hand the eigenvalues of J are ±i, so det J = i^{r} · (−i)^{2n+1−r} = (−1)^{n+r+1}i for some r = 0, …, 2n + 1, a contradiction. Moreover for G = O_{p,q}(ℝ) one uses the fact that every J ∈ C(O_{p,q}(ℝ)) defines a nondegenerate hermitian form H = I_{p,q} + iI_{p,q}J on the complex vector space V_{J} = (ℝ^{p+q}, J). But then 2 ind_{ℂ}H = ind_{ℝ}Re H = ind_{ℝ}I_{p,q} = q, so q is even. Since p + q = 2 dim_{ℂ} V_{J} is even, p is also even.

Proof. (of Theorem 1, case C(GL_{n}(ℍ))): This is a consequence of results of Louise Wolf. Recall the complex representation ρ : GL_{n}(ℍ) ↪ GL_{2n}(ℂ), A+Bj ↦ ( \begin{matrix}A & B \\ −\overline{B} & \overline{A} \end{matrix} ). 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 : GL_{n}(ℍ) ↪ GL_{2n}(ℂ)} appear as complex conjugate pairs. This implies that every complex structure J ∈ C(GL_{n}(ℍ)) is conjugate to i1_{n} ∈ GL_{n}(ℍ). Finally M^{−1}(i1_{n})M = i1_{n} in GL_{n}(ℍ) if and only if M commutes with i, or equivalently if M ∈ GL_{n}(ℂ) ⊂ GL_{n}(ℍ). This implies the assertion for C(GL_{n}(ℍ)).

The proofs of the remaining cases follow more or less the same pattern. Therefore we present here only the proofs of the most complicated cases G = U_{p,q}(ℂ) and αU_{m}(ℍ). The other proofs are just variations of these.

Proof. (of Theorem 2 c)): Denote n = p + q. By definition one has

C(U_{p,q}(ℂ)) = ⋃_{r=0}^{n} ⋃_{k=0}^{n} C_{k,r}(U_{p,q}(ℂ)).

The index k meets the following requirements:

Step I: C_{k,r}(U_{p,q}(ℂ)) = Ø unless k ≡ r − q (mod 2) Suppose J ∈ C_{k,r}(U_{p,q}(ℂ)). By definition J is conjugate to iI_{r,s} where s = n − r, i.e.

J = N^{−1}iI_{r,s}N

with some N ∈ GL_{n}(ℂ). On the other hand J being unitary of type (p, q) implies that ^{t}\overline{N}^{−1}I_{p,q}N^{−1} commutes with iI_{r,s}. This means that ^{t}\overline{N}^{−1}I_{p,q}N^{−1} is a block matrix of the form

t\overline{N}^{−1}I_{p,q}N^{−1} = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )

with α ∈ GL_{r}(ℂ) and β ∈ GL_{s}(ℂ). In particular α and β are hermitian and

ind_{ℂ}α + ind_{ℂ}β = ind_{ℂ} I_{p,q} = q.  (1)

By definition we have

k = ind_{ℂ}(iI_{p,q}J) = ind_{ℂ}(iI_{p,q}N^{−1}(iI_{r,s})N)
= ind_{ℂ}(−^{t}\overline{N}^{−1}I_{p,q}N^{−1}I_{r,s}) = ind_{ℂ}(−( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )I_{r,s})
= ind_{ℂ}( \begin{matrix}−α & 0 \\ 0 & β \end{matrix} ) = ind_{ℂ}(−α) + ind_{ℂ}β
= r − ind_{ℂ}α + ind_{ℂ}β

(2)

Combining (1) and (2) gives

k = r − q + 2 ind_{ℂ} β ≡ r − q(mod 2).

Step II: The homomorphism

φ : U_{p,q}(ℂ) → C_{k,r}(U_{p,q}(ℂ)), M ↦ M^{−1}( \begin{matrix}i1_{r} & 0 \\ 0 & −i1_{s} \end{matrix} )M

is surjective.

According to Step I, we may assume that k = r − q + 2m for some integer m. Moreover, we may assume that U_{p,q}(ℂ) is the unitary group for the symmetric

bilinear form I := ( \begin{matrix}1_{r−q+m} \\ & −1_{q} \\ & & 1_{s−m} \end{matrix} ) on ℂ^{p+q}. Suppose J ∈ C_{k,r}(U_{p,q}(ℂ)).

Then as in Step I, replacing I_{p,q} by I there is an N ∈ GL_{n}(ℂ) with J = N^{−1}iI_{r,s}N and ^{t}\overline{N}^{−1}I N^{−1} = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) with hermitian matrices α ∈ GL_{r}(ℂ) and β ∈ GL_{s}(ℂ). Equations (1) and (2) imply that ind_{ℂ}β = m and ind_{ℂ}α = q − m. Hence there are matrices A ∈ GL_{r}(ℂ) and B ∈ GL_{s}(ℂ) such that

t\overline{A}αA = ( \begin{matrix}1_{r−q+m} & 0 \\ 0 & −1_{q−m} \end{matrix} ) , ^{t}\overline{B}βB = ( \begin{matrix}−1_{m} & 0 \\ 0 & 1_{s−m} \end{matrix} ) .

An immediate computation shows that the matrix M := ( \begin{matrix}A & 0 \\ 0 & B \end{matrix} )^{−1}N is an element of U_{p,q}(ℂ) and that φ(M) = M^{−1}iI_{r,s}M = J. Step III: φ^{−1}(iI_{r,s}) = U_{r−q+m,q−m}(ℂ) × U_{s−m,m}(ℂ). The stabilizer of iI_{r,s} in U_{p,q}(ℂ) is the group

\begin{aligned}φ^{−1}(iI_{r,s}) & = & \{M ∈ U_{p,q}(ℂ) | M I_{r,s} = I_{r,s}M\} \\ & = & \{M = ( \begin{matrix}M_{1} & 0 \\ 0 & M_{2} \end{matrix} ) ∈ U_{p,q}(ℂ)| \begin{matrix}M_{1} ∈ GL_{r}(ℂ) \\ M_{2} ∈ GL_{s}(ℂ) \end{matrix} \} \end{aligned}

Since we still assume that U_{p,q}(ℂ) is the unitary group for the hermitian form I = ( \begin{matrix}I_{r−q+m,q−m} & 0 \\ 0 & −I_{m,s−m} \end{matrix} ), this implies the assertion of Step III.

Combining Steps I to III implies the assertion of Theorem 2 c).

Proof. (of Theorem 2 d)): Obviously

C(αU_{n}(ℍ)) = ⋃_{k=0}^{n} C_{k}(αU_{n}(ℍ)).

Notice that J_{0} = −iI_{n−k,k} is an element of C_{k}(αU_{n}(ℍ)).

Step I: φ : αU_{n}(ℍ)) → C_{k}(αU_{n}(ℍ)), M ↦ M^{−1}J_{0}M is surjective.

Suppose J ∈ C_{k}(αU_{n}(ℍ)). Obviously C_{k}(αU_{n}(ℍ)) is a subset of C(GL_{n}(ℍ)). Hence J is conjugate to i1_{n}, i.e.,

J = N^{−1}(i1_{n})N for some N ∈ GL_{n}(ℍ).

Now J being an element of αU_{n}(ℍ) = \{M ∈ GL_{n}(ℍ)|M(i1_{n})^{t}\overline{M} = i1_{n}\} implies that N(i1_{n})^{t}\overline{N} commutes with i1. So necessarily N(i1_{n})^{t}\overline{N} is a matrix with complex entries:

N(i1_{n})^{t}\overline{N} ∈ GL_{n}(ℂ).

Moreover, by definition, N(i1_{n})^{t}\overline{N} is skew hermitian. So AN(i1)^{t}\overline{N} ^{t}\overline{A} = −iI_{n−a,a} for some A ∈ GL_{n}(ℂ) and an integer a, 0 ≤ a ≤ n. Using the fact that A commutes with i one sees that

k = ind_{ℍ}(Ji) = ind_{ℍ}(N^{−1}(i1)N(i1))
= ind_{ℍ}((i1)N(i1)^{t}\overline{N})
= ind_{ℍ}((i1)AN(i1)^{t}\overline{N} ^{t}\overline{A})
= ind_{ℍ}I_{n−a,a} = a,

i.e. k = a. The matrix

M := ( \begin{matrix}j1_{n−k} & 0 \\ 0 & 1_{k} \end{matrix} )AN

is an element of αU_{n}(ℍ), because

\begin{aligned}M(i1)^{t}\overline{M} & = & ( \begin{matrix}j1_{n−k} & 0 \\ 0 & 1_{k} \end{matrix} )AN(i1)^{t}\overline{N} ^{t}\overline{A}( \begin{matrix}−j1_{n−k} & 0 \\ 0 & 1_{k} \end{matrix} ) \\ & = & ( \begin{matrix}jij1_{n−k} & 0 \\ 0 & i1_{a} \end{matrix} ) = i1_{n}. \end{aligned}

Moreover,

\begin{aligned}φ(M) = M^{−1}J_{0}M & = & N^{−1}A^{−1}( \begin{matrix}−j1_{n−k} \\ & 1_{k} \end{matrix} )( \begin{matrix}−i1_{n−k} & 0 \\ 0 & i1_{k} \end{matrix} )( \begin{matrix}j1_{n−k} \\ & 1_{k} \end{matrix} )AN \\ & = & N^{−1}(i1)N = J. \end{aligned}

This implies the assertion. Step II: φ^{−1}(J_{0}) ≃ U_{n−k,k}(ℂ). Note that

φ^{−1}(J_{0}) = {M ∈ αU_{n}(ℍ)|J_{0}M = MJ_{0}} = αU_{n}(ℍ) ∩ U_{n−k,k}(ℍ).

But αU_{n}(ℍ) ∩ U_{n−k,k}(ℍ) is isomorphic to U_{n−k,k}(ℂ). To see this write U ∈

U_{n−k,k}(ℂ) in the form U = ( \begin{matrix}a & b \\ c & d \end{matrix} ) with a ∈ M_{n−k}(ℂ), b and ^{t}c ∈ M((n − k) × k, ℂ)

and d ∈ M_{k}(ℂ). Then the assignment

U = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ ( \begin{matrix}\overline{a} & −jb \\ j\overline{c} & d \end{matrix} )

defines an isomorphism U_{n−k,k}(ℂ) → αU_{n}(ℍ) ∩ U_{n−k,k}(ℍ).

Remark 1.1. In the proof of Theorem 2 d) above it is shown that C_{k}(αU_{n}(ℍ)) is isomorphic to the symmetric space αU_{n}(ℍ)/U_{n−k,k}(ℂ) where U_{n−k,k}(ℂ) embeds

into αU_{n}(ℍ) by ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ ( \begin{matrix}\overline{a} & −jb \\ j\overline{c} & d \end{matrix} ) with a ∈ M_{n−k}(ℂ), b, ^{t}c ∈ M((n − k) × k, ℂ)

and d ∈ M_{k}(ℂ). Here the respective embeddings for the other cases are collected

C(GL_{2n}(ℝ)) : GL_{n}(ℂ) ↪ GL_{2n}(ℝ), A + iB ↦ ( \begin{matrix}A & −B \\ B & A \end{matrix} )
C(GL_{n}(ℍ)) : GL_{n}(ℂ) ↪ GL_{n}(ℍ) induced by ℂ ↪ ℍ
C(O_{2p,2q}(ℝ)) : U_{p,q}(ℂ) ↪ O_{2p,2q}(ℝ), A + iB ↦ ( \begin{matrix}A & −BI_{p,q} \\ I_{p,q}B & I_{p,q}AI_{p,q} \end{matrix} ),

where O_{2p,2q}(ℝ) is the group of isometries for the symmetric bilinear form ( \begin{matrix}I_{p,q} & 0 \\ 0 & I_{p,q} \end{matrix} ).

C(O_{2n}(ℂ)) : GL_{n}(ℂ) ↪ O_{2n}(ℂ), α ↦ ( \begin{matrix}α & 0 \\ 0 & ^{t}α^{−1} \end{matrix} )

C(U_{p,q}(ℍ)) : U_{p,q}(ℂ) ↪ U_{p,q}(ℍ), ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ↦ ( \begin{matrix}\overline{α} & −jβ \\ j\overline{γ} & δ \end{matrix} )

(the same embedding as for αU_{n}(ℍ)).

C(Sp_{2n}(ℂ)) : GL_{n}(ℂ) ↪ Sp_{2n}(ℂ), α ↦ ( \begin{matrix}α & 0 \\ 0 & ^{t}α^{−1} \end{matrix} )
C_{k}(Sp_{2n}(ℝ)) : U_{n−k,k}(ℂ) ↪ Sp_{2n}(ℝ), A + iB ↦ ( \begin{matrix}A & −BI_{n−k,k} \\ I_{n−k,k}B & I_{n−k,k}AI_{n−k,k} \end{matrix} )
C_{r}(GL_{n}(ℂ)) : GL_{r}(ℂ) × GL_{s}(ℂ) ↪ GL_{n}(ℂ)

the natural embedding

C_{k,r}(U_{p,q}(ℂ)) : U_{r−q+m,q−m}(ℂ) × U_{s−m,m}(ℂ) ↪ U_{p,q}(ℂ)

the natural embedding.

2. Applications

An immediate consequence of Theorems 1 and 2 is the fact that there exist normal forms for complex structures contained in classical groups. To be more precise, for every space C_{∗}(G) there exists a normal form J_{0} ∈ C_{∗}(G) such that any other complex structure in C_{∗}(G) is conjugate to J_{0} in G. Some normal forms J_{0} are listed in the following table

\begin{matrix}C_{∗}(G) & J_{0} & C_{∗}(G) & J_{0} \\ C(GL_{2n}(ℝ)) & ( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ) & C_{k,r}(U_{p,q}(ℂ)) & iI_{r,p+q−r} \\ C_{r}(GL_{n}(ℂ)) & iI_{r,n−r} & C(U_{p,q}(ℍ)) & iI_{p,q} \\ C(GL_{n}(ℍ)) & i1_{n} & C_{k}(Sp_{2n}(ℝ)) & ( \begin{matrix}0 & I_{n−k,k} \\ −I_{n−k,k} & 0 \end{matrix} ) \\ C(O_{2p,2q}(ℝ)) & ( \begin{matrix}0 & I_{p,q} \\ −I_{p,q} & 0 \end{matrix} ) & C(Sp_{2n}(ℂ)) & iI_{n,n} \\ C(O_{2n}(ℂ)) & iI_{n,n} & C_{k}(αU_{n}(ℂ)) & iI_{k,n−k} \end{matrix}

As mentioned already in the introduction, Theorems 1 and 2 admit also applications of purely linear algebraic nature. Here we present only the following two applications, various other results could be deduced in a similar way.

Corollary 2.1. For M ∈ O_{p+q}(ℂ) we have

det Im (M ( \begin{matrix}i1_{p} & 0 \\ 0 & 1_{q} \end{matrix} )) = \{ \begin{matrix}0 & q ≡ 1 (mod 2) \\ ≠ 0 & q ≡ 0 (mod 2). \end{matrix}

Corollary 2.2. Let τ ∈ M_{p+q}(ℂ) such that I_{p,q}τ is symmetric. Then I_{p,q} + tτI_{p,q}\overline{τ} is a hermitian form with at most p positive (respectively q negative) eigenvalues, i.e.

I_{p,q} + ^{t}τI_{p,q}\overline{τ} = ^{t}αI_{p,q}\overline{α} for some α ∈ M_{p+q}(ℂ).

The significance of these corollaries lies in the fact that the spaces C_{∗}(G) can be interpreted as open subspaces of certain Grassmannian varieties. In these terms the statements of the above corollaries are in fact equivalent to the results of Theorem 1 for the groups O_{p,q}(ℝ) and U_{p,q}(ℍ). It seems difficult to prove these corollaries directly.

Proof. (of Corollary 2.1): Write n = p + q. Consider the following subvariety of the Grassmannian Gr_{n}(ℂ^{2n}) of n-dimensional subvector spaces of ℂ^{2n}:

S = \{V ∈ Gr_{n}(ℂ^{2n}) | \begin{matrix}V is isotropic w.r.t. \\ the symmetric form & ( & \begin{matrix}1_{n} & 0 \\ 0 & I_{p,q} \end{matrix} & ) \end{matrix} \} .

Every V ∈ Gr_{n}(ℂ^{2n}) is of the form V = π(ℂ^{n}) with π ∈ M(2n × n, ℂ). Denote \overline{V} := \overline{π}(ℂ^{n}) the complex conjugate subspace. This definition does not depend on the choice of the matrix π. Let S^{0} denote the subset

S^{0} = \{V ∈ S | V ∩ \overline{V} = \{0\} \}.

Suppose V = π(ℂ^{n}) ∈ S^{0}. Then V ⊕ \overline{V} ≃ ℂ^{2n} and the complex structure of V induces a complex structure J_{V} on ℂ^{2n} by :

V ⊕ \overline{V} ⟶^{(π,\overline{π})} ℂ^{2n}
( \begin{matrix}i1_{n} & 0 \\ 0 & −i1_{n} \end{matrix} ) ↓  ↓ J_{V}
V ⊕ \overline{V} ⟶^{(π,\overline{π})} ℂ^{2n}.

The following computation shows that J_{V} is a real matrix.

J_{V} = (π, \overline{π})( \begin{matrix}i1_{n} & 0 \\ 0 & −i1_{n} \end{matrix} )(π, \overline{π})^{−1}
= (π, \overline{π})( \begin{matrix}1_{n} & −i1_{n} \\ 1_{n} & i1_{n} \end{matrix} )( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} )( \begin{matrix}1_{n} & −i1_{n} \\ 1_{n} & i1_{n} \end{matrix} )^{−1}(π, \overline{π})^{−1}
= (Re π, Im π)( \begin{matrix}0 & 1_{n} \\ −1_{n} & o \end{matrix} )(Re π, Im π)^{−1}.

On the other hand, using the fact that V is isotropic with respect to ( \begin{matrix}1_{n} & 0 \\ 0 & I_{p,q} \end{matrix} ), i.e. ^{t}π( \begin{matrix}1_{n} & 0 \\ 0 & I_{p,q} \end{matrix} )π = 0, it is easy to see that J_{V} is orthogonal with respect to the symmetric bilinear form ( \begin{matrix}1_{n} & 0 \\ 0 & I_{p,q} \end{matrix} ). So J_{V} ∈ C(O_{2n−q,q}(ℝ)). The assignment V → J_{V} defines a map S^{0} → C(O_{2n−q,q}(ℝ)) which is equivariant with respect to the natural action of O_{2n−q,q}(ℝ) on both spaces. Obviously the map S^{0} → C(O_{2n−q,q}(ℝ)) is injective. Hence Theorem 1 says that

S^{0} ≃ \{ \begin{matrix}C(O_{2n−q,q}(ℝ)) & & q ≡ 0(mod2) \\ & if \\ Ø & & q ≡ 1(mod2). \end{matrix}

Now suppose M ∈ O_{n}(ℂ). Then M defines an element V_{M} ∈ S by

V_{M} := π_{M}(ℂ^{n}) with π_{M} := ( \begin{matrix}M( \begin{matrix}i1_{p} & 0 \\ 0 & 1_{q} \end{matrix} ) \\ 1_{n} \end{matrix} ) .

It is easy to see that V_{M} ∈ S^{0} if and only if (π_{M}, \overline{π_{M}}) or equivalently if the matrix (Re π_{M}, Im π_{M}) is invertible. But

(Re π_{M}, Im π_{M}) = ( \begin{matrix}Re (M( \begin{matrix}i1_{p} & 0 \\ 0 & 1_{q} \end{matrix} )) & Im (M( \begin{matrix}i1_{p} & 0 \\ 0 & 1_{q} \end{matrix} )) \\ 1_{n} & 0 \end{matrix} ) .

So V_{M} ∈ S^{0} if and only if det Im(M( \begin{matrix}i1_{p} & 0 \\ 0 & 1_{q} \end{matrix} )) ≠ 0. This implies the assertion.

Proof. (of Corollary 2.2). Write n = p + q. Consider the following subvariety of the Grassmannian Gr_{n}(ℂ^{2n}):

T := \{V ∈ Gr_{n}(ℂ^{2n}) | \begin{matrix}V is isotropic w. r. t. \\ the alternating form & ( & \begin{matrix}0 & I_{p,q} \\ −I_{p,q} & 0 \end{matrix} & ) \end{matrix} \} .

Suppose V = π(ℂ^{n}) ∈ Gr_{n}(ℂ^{2n}), with π ∈ M(2n × n, ℂ). Define V^{0} ∈ Gr_{n}(ℂ^{2n}) by V^{0} := ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )\overline{π}(ℂ^{n}). Note that this definition does not depend on the choice of π. The set

T^{0} := {V ∈ T | V ∩ V^{0} = {0} }

is an open subset of T.

Step I. T^{0} ≃ C(U_{p,q}(ℍ)).

Suppose V = π(ℂ^{n}) ∈ T^{0} and write π = ( \begin{matrix}u \\ v \end{matrix} ). Since V ⊕ V^{0} ≃ ℂ^{2n}, the matrix (π, ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )\overline{π}) = ( \begin{matrix}u & −\overline{v} \\ v & \overline{u} \end{matrix} ) is invertible. Recall the natural complex representation of quaternion matrices

ρ : GL_{n}(ℍ) ↪ GL_{2n}(ℂ), A + Bj ↦ ( \begin{matrix}A & B \\ −\overline{B} & \overline{A} \end{matrix} ).

This shows that u + vj = ^{t}(^{t}u + ^{t}vj) is an element of GL_{n}(ℍ). Define J_{V} := (u + vj)(i1)(u + vj)^{−1}. The fact that V is isotropic with respect to ( \begin{matrix}0 & I_{p,q} \\ −I_{p,q} & 0 \end{matrix} ) means in terms of u and v that ^{t}uI_{p,q}v − ^{t}vI_{p,q}u = 0. Using this one easily sees that ^{t}J_{V} I_{p,q}\overline{J}_{V} = I_{p,q}, i.e., J ∈ U_{p,q}(ℍ). The assignment V ↦ J_{V} defines a map T^{0} → C(U_{p,q}(ℍ)). Note that

ρ(U_{p,q}(ℍ)) = \{M = ( \begin{matrix}A & B \\ −\overline{B} & \overline{A} \end{matrix} ) ∈ SL_{2n}(ℂ)|M( \begin{matrix}0 & I_{p,q} \\ −I_{p,q} & 0 \end{matrix} )^{t}M = ( \begin{matrix}0 & I_{p,q} \\ −I_{p,q} & 0 \end{matrix} )\}

So U_{p,q}(ℍ) acts on T via the representation ρ. By definition the map T^{0} → C(U_{p,q}(ℍ)) is equivariant with respect to the action of U_{p,q}(ℍ) on both spaces. Moreover it is easy to see that T^{0} → C(U_{p,q}(ℍ)) is injective. Since by Theorem 1 the action of U_{p,q}(ℍ) on C(U_{p,q}(ℍ)) is transitive, T^{0} → C(U_{p,q}(ℍ)) is also surjective.

Step II: Consider the hermitian form H := ( \begin{matrix}I_{p,q} & 0 \\ 0 & I_{p,q} \end{matrix} ) on ℂ^{2n}. For every V = π(ℂ^{n}) ∈ T^{0} the pull back of H to V ⊕ V^{0} is nondegenerate implying that H|V = ^{t}πH\overline{π} is nondegenerate. By Step I the set T^{0} is a connected homogeneous space, so the map T^{0} → ℤ, V ↦ ind(H|V) is constant. Note that V_{0} = ( \begin{matrix}1 \\ 0 \end{matrix} )ℂ^{n} = ℂ^{n} × {0}^{n} is an element of T^{0} and that H|V_{0} is given by the matrix I_{p,q}. Hence ind(H|V) = q for all V ∈ T_{0}. Suppose τ ∈ M_{n}(ℂ) with τI_{p,q} symmetric. Then V_{τ} = ( \begin{matrix}1 \\ τ \end{matrix} )ℂ^{n} is an element of T and

H|V_{τ} = I_{p,q} + ^{t}τI_{p,q}\overline{τ}.

Since T^{0} is an open subset of T this implies the assertion.

3. Flag Domains

Let G_{ℂ} be a connected reductive complex Lie group. Given a parabolic subgroup P of G_{ℂ} the projective variety G_{ℂ}/P is called complex flag manifold. Any real form G_{ℝ} of G_{ℂ} acts on G_{ℂ}/P in a natural way. There exist only finitely many G_{ℝ}-orbits in G_{ℂ}/P. The connected components of the open G_{ℝ}-orbits are called flag domains (see [3]). Flag domains are used in Representation Theory in order to describe certain representations of G_{ℝ}. To be more precise, the Dolbeault cohomology groups of certain homogeneous line bundles on flag domains of G_{ℝ} yield the discrete series of representations of G_{ℝ}. The geometric properties of flag domains were studied by J. A. Wolf, W. Schmid and others. For example flag domains are always simply connected. Moreover a flag domain X = G_{ℝ}/H is (s + 1)-complete in the sense of Andreotti and Grauert, where s is the maximum of the dimensions of all compact complex subvarieties of X. This is shown in [2] in the case of compact H, if H is reductive this is proved in [4]. In [5] J. A. Wolf presents a proof of the general case. The (s+1)-completeness implies in particular that H^{q}(X, F) = 0 for all q > s and every coherent sheaf F on X.

The classical groups G of above are not necessarily connected. They consist of one (for G = GL_{n}(ℂ), GL_{n}(ℍ), U_{p,q}(ℂ), U_{p,q}(ℍ), Sp_{2n}(ℝ), Sp_{2n}(ℂ) or αU_{n}(ℍ)), two (for G = GL_{n}(ℝ), O_{n}(ℝ), O_{n}(ℂ)) respectively four (for G = O_{p,q}(ℝ)) components. Denote by n_{G} the number of connected components of G. The group G can be considered as a real form of a connected reductive complex Lie group G_{ℂ}. Using Theorems 1 and 2 we show

Proposition 3.1. The spaces C_{∗}(G) of Theorems 1 and 2 are disjoint unions of n_{G} flag domains for the real Lie group G.

Proof. Proof for G = GL_{2n}(ℝ): GL_{2n}(ℝ) consists of two components. Accordingly the symmetric space C(GL_{2n}(ℝ)) has two connected components, both isomorphic to SL_{2n}(ℝ)/GL_{n}(ℂ)∩SL_{2n}(ℝ). The group GL_{2n}(ℝ) is a real form of the connected reductive complex Lie group GL_{2n}(ℂ). Consider the parabolic subgroup P = \{( \begin{matrix}A & B \\ 0_{n} & D \end{matrix} ) ∈ GL_{2n}(ℂ)\}. An immediate computation shows that

the embedding GL_{2n}(ℝ) ↪ GL_{2n}(ℂ), M ↦ ( \begin{matrix}1_{n} & 0 \\ i1_{n} & 1_{n} \end{matrix} )M( \begin{matrix}1_{n} & 0 \\ i1_{n} & 1_{n} \end{matrix} )^{−1} induces an

embedding of GL_{2n}(ℝ)/GL_{n}(ℂ) into GL_{2n}(ℂ)/P. Since both manifolds are of dimension n^{2}, the image of GL_{2n}(ℝ)/GL_{n}(ℂ) in GL_{2n}(ℂ)/P is an open GL_{2n}(ℝ)- orbit, i.e., a flag domain.

As for the proofs of the other cases we only give the following list of corresponding embeddings G ↪ G_{ℂ}, flag manifolds, and flag domains:

\underline{C_{r}(GL_{n}(ℂ)) :}

GL_{n}(ℂ) ↪ GL_{n}(ℂ) × GL_{n}(ℂ), M ↦ (M, ^{t}\overline{M}^{−1}),

Flag manifold: GL_{n}(ℂ)^{2}/\{( \begin{matrix}A & B \\ 0_{s,r} & D \end{matrix} ) ∈ GL_{n}(ℂ)\}^{2},

Flag domain: C_{r}(GL_{n}(ℂ)) ≃ GL_{n}(ℂ)/(GL_{r}(ℂ) × GL_{s}(ℂ)).

\underline{C(GL_{n}(ℍ)) :}

GL_{n}(ℍ) ↪ GL_{2n}(ℂ), A + Bj ↦ ( \begin{matrix}A & B \\ −\overline{B} & \overline{A} \end{matrix} ),

Flag manifold: GL_{2n}(ℂ)/\{( \begin{matrix}A & B \\ 0_{n} & D \end{matrix} ) ∈ GL_{2n}(ℂ)\},

Flag domain: C(GL_{n}(ℍ)) ≃ GL_{n}(ℍ)/GL_{n}(ℂ).

\underline{C(O_{2p,2q}(ℝ)) :} O_{2p,2q}(ℝ) ↪ O_{2(p+q)}(ℂ), M ↦ M_{0}^{−1}MM_{0} with

M_{0} := ( \begin{matrix}\frac{1}{2} X & X \\ \frac{i}{2} \overline{X} & −i\overline{X} \end{matrix} ) , and X := ( \begin{matrix}i1_{p} & 0 \\ 0 & 1_{q} \end{matrix} ),

Flag manifold: O_{2(p+q)}(ℂ)/ \{( \begin{matrix}A & B \\ 0_{p+q} & D \end{matrix} ) ∈ O_{2(p+q)}(ℂ)\} ,

Flag domains: the two (resp. four) components of C(O_{2p,2q}(ℝ)).

\underline{C(O_{2n}(ℂ)) :}

O_{2n}(ℂ) ↪ O_{2n}(ℂ) × O_{2n}(ℂ), M ↦ (M, ^{t}\overline{M}^{−1}),

Flag manifold: O_{2n}(ℂ)^{2}/ \{( \begin{matrix}A & B \\ 0_{n} & D \end{matrix} ) ∈ O_{2n}(ℂ)\}^{2} ,

Flag domains: the two components of C(O_{2n}(ℂ)).

\underline{C_{k,r}(U_{p,q}(ℂ)) :} U_{p,q}(ℂ) ↪ GL_{p+q}(ℂ), M ↦ M,

Flag manifold: GL_{p+q}(ℂ)/ \{( \begin{matrix}A & B \\ 0_{s,r} & D \end{matrix} ) ∈ GL_{p+q}(ℂ)\} ,

Flag domain: C_{k,r}(U_{p,q}(ℂ)) ≃ U_{p,q}(ℂ)/(U_{r−q+m,q−m}(ℂ) × U_{s−m,m}(ℂ)).

\underline{C(U_{p,q}(ℍ)) :}

U_{p,q}(ℍ) ↪ Sp_{2(p+q)}(ℂ), A + Bj ↦ P^{−1}( \begin{matrix}A & BI_{p,q} \\ −I_{p,q}\overline{B} & I_{p,q}\overline{A}I_{p,q} \end{matrix} ) P with

P = ( \begin{matrix}0 & 0 & 1_{p} & 0 \\ 0 & 1_{q} & 0 & 0 \\ 1_{p} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1_{q} \end{matrix} )

Flag manifold: Sp_{2(p+q)}(ℂ)/ \{( \begin{matrix}A & B \\ 0_{p+q} & D \end{matrix} ) ∈ Sp_{2(p+q)}(ℂ)\} ,

Flag domain: C(U_{p,q}(ℍ)) ≃ U_{p,q}(ℍ)/U_{p,q}(ℂ).

\underline{C_{k}(Sp_{2n}(ℝ)) :}

Sp_{2n}(ℝ) ↪ Sp_{2n}(ℂ), M ↦ ( \begin{matrix}1_{n} & 0 \\ iI_{p,q} & 1_{n} \end{matrix} )M( \begin{matrix}1_{n} & 0 \\ iI_{p,q} & 1_{n} \end{matrix} )^{−1},

Flag manifold: Sp_{2n}(ℂ)/ \{( \begin{matrix}A & B \\ 0_{n} & D \end{matrix} ) ∈ Sp_{2n}(ℂ)\} ,

Flag domain: C_{k}(Sp_{2n}(ℝ)) ≃ Sp_{2n}(ℝ)/U_{n−k,k}(ℂ).

\underline{C(Sp_{2n}(ℂ)) :}

Sp_{2n}(ℂ) ↪ Sp_{2n}(ℂ) × Sp_{2n}(ℂ), M ↦ (M, ^{t}\overline{M}^{−1}),

Flag manifold: Sp_{2n}(ℂ)^{2}/ \{( \begin{matrix}A & B \\ 0_{n} & D \end{matrix} ) ∈ Sp_{2n}(ℂ)\}^{2} .

Flag domain: C(Sp_{2n}(ℂ)) ≃ Sp_{2n}(ℂ)/GL_{n}(ℂ). \underline{C_{k}(αU_{n}(ℍ)) :}

αU_{n}(ℍ) ↪ O_{2n}(ℂ), A + Bj ↦ P^{−1}( \begin{matrix}A & B \\ −\overline{B} & \overline{A} \end{matrix} )P with

P = ( \begin{matrix}0 & 0 & 1_{k} & 0 \\ 0 & 1_{n−k} & 0 & 0 \\ 1_{k} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1_{n−k} \end{matrix} )

Flag manifold: O_{2n}(ℂ)/ \{( \begin{matrix}A & B \\ 0_{n} & D \end{matrix} ) ∈ O_{2n}(ℂ)\} .

Flag domain: C_{k}(αU_{n}(ℍ)) = αU_{n}(ℍ)/U_{n−k,k}(ℂ).

4. Families of Nondegenerate Complex Tori Associated to Spaces of Complex Structures

A polarized abelian variety of dimension n is a pair (X, H) with a complex torus X = ℂ^{n}/Λ (Λ is a lattice in ℂ^{n}) and a positive definite hermitian form H on ℂ^{n} such that Im H(Λ × Λ) ⊆ ℤ. More generally a nondegenerate complex torus of index k is a pair (X, H) with a complex torus X = ℂ^{n}/Λ and a nondegenerate hermitian form H on ℂ^{n} with Im H(Λ×Λ) ⊆ ℤ such that ind_{ℂ}(H) = k (for more details see [1]). It is well-known that the space C_{0}(Sp_{2n}(ℝ)) of complex structures of index 0 in the real symplectic group parametrizes polarized abelian varieties of dimension n: For J ∈ C_{0}(Sp_{2n}(ℝ)) the pair V_{J} := (ℝ^{2n}, J) is a complex vector space of dimension n and X_{J} := V_{J}/ℤ^{2n} is a complex torus. The matrix ^{t}J( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ) + i( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ) defines a positive definite hermitian form H_{J} on V_{J} such that Im H_{J} is integral valued on the lattice ℤ^{2g}. Thus (X_{J}, H_{J}) is a polarized abelian variety. Obviously the same construction applied to J ∈ C_{k}(Sp_{2n}(ℝ)) gives a nondegenerate complex torus (X_{J}, H_{J}) of index k. It can be shown that conversely every nondegenerate complex torus of index k and dimension n is of the form (X_{J}, H_{J}) (see [1]). In a similar way any of the spaces C_{∗}(G) parametrizes nondegenerate complex tori (X, H) with some extra condition on the endomorphism algebra of (X, H). We exhibit this construction for G = Sp_{2n}(ℂ) and U_{p,q}(ℍ). A similar but in the notation considerably more complicated construction applies to every space C_{∗}(G).

\underline{C(Sp_{2n}(ℂ)):} Suppose J ∈ C(Sp_{2n}(ℂ)). Considering ℂ^{2n} as a real vector space the pair V_{J} := (ℂ^{2n}, J) is again a complex vector space of dimension 2n. Let H_{J} denote the ℝ-bilinear form

H_{J} : \{ \begin{matrix}ℂ^{2n} × ℂ^{2n} → ℂ, \\ (v, w) ↦ tr_{ℂ/ℝ}(^{t}v^{t}J( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} )w) + i tr_{ℂ/ℝ}(^{t}v( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} )w). \end{matrix}

It is easy to see that H_{J} is in fact a nondegenerate hermitian form on the complex vector space V_{J}. According to Theorem 1 the space C(Sp_{2n}(ℂ)) is connected and thus the index of H_{J} equals the index of H_{J_{0}} , where J_{0} = iI_{n,n} (see Section 2). Using^{6} this shows that ind_{ℂ}H_{J} = ind_{ℂ}H_{J_{0}} = \frac{1}{2}ind_{ℝ}Re H_{J_{0}} = n. Suppose F is an imaginary quadratic field and let M ⊂ F^{2n} be a ℤ-module of

rank 4n with tr_{F/ℚ}(^{t}a( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} )b) ∈ ℤ for all a, b ∈ M. Identify F ⊗_{ℚ} ℝ with ℂ^{2},

then M ⊂ (F ⊗_{ℚ} ℝ)^{2n} = ℂ^{2n} is a lattice and

(X_{J} = V_{J}/M, H_{J})

is a nondegenerate complex torus of dimension 2n and index n. In particular by construction the endomorphism algebra End(X_{J}) ⊗_{ℤ} ℚ contains F, i.e., (X_{J}, H_{J}) admits multiplication in F (for the definition of endomorphism algebra and multiplication in F see [1].) □

\underline{C(U_{p,q}(ℍ)):} Suppose J ∈ C(U_{p,q}(ℍ)). Considering ℍ^{n} = ℍ^{p+q} as a real vector space the pair V_{J} := (ℍ^{n}, J) is a complex vector space of dimension 2n. Consider the anti-involution q ↦ q′ = i\overline{q}i^{−1} on ℍ. Then the ℝ-bilinear form

H_{J} : ℍ^{n} × ℍ^{n} → ℂ, H_{J}(v, w) := tr_{ℍ/ℝ}(^{t}v(^{t}JiI_{p,q})w′) + itr_{ℍ/ℝ}(^{t}v(iI_{p,q})w′)

defines a nondegenerate hermitian form on the complex vector space V_{J}. According to Theorem 1 the space C(U_{p,q}(ℍ)) is connected and thus the index of H_{J} equals the index of H_{J_{0}} with normal form J_{0} = iI_{p,q}. Now an immediate computation shows that ind_{ℂ}H_{J_{0}} = n. Let F be a definite quaternion algebra over ℚ with anti-involution ′. It is easy to see that F can be embedded into ℍ in such a way that the anti-involution ′ extends either to quaternion conjugation on ℍ or the anti-involution q′ = i\overline{q}i^{−1} of above. Let us assume the latter case. Let M ⊂ F^{n} be a free ℤ-submodule of rank 4n and T ∈ M_{n}(F) a nondegenerate matrix satisfying ^{t}T′ = −T such that tr_{F/ℚ}(^{t}vT w′) ∈ ℤ for all v, w ∈ M. For simplicity we assume here that T = iI_{p,q}. Then M ⊂ F^{n} ⊂ ℍ^{n} is a lattice and

(X_{J} := V_{J}/M, H_{J})

is a nondegenerate complex torus of dimension 2n and index n. Moreover by construction the endomorphism algebra End(X_{J}) ⊗_{ℤ} ℚ contains F such that the Rosati-involution (see [1]) restricts to the anti-involution ′ on F, i.e., (X_{J}, H_{J}) admits multiplication in (F, ′). □

References

[1]  Birkenhake, Ch., and H. Lange, “Complex Tori,” forthcoming Monograph.

[2] Schmid, W., and J. A. Wolf, A Vanishing Theorem for Open Orbits On Complex Flag Manifolds, Proc. Amer. Math. Soc., 92 (1984), 461–464.

2^{t}( \begin{matrix}v_{1} \\ v_{2} \end{matrix} )( \begin{matrix}A & −B \\ −B & −A \end{matrix} )( \begin{matrix}w_{1} \\ w_{2} \end{matrix} ).

6For M = A + iB ∈ M_{n}(ℂ) and v = v_{1} + iv_{2}, w = w_{1} + iw_{2} ∈ ℂ^{n} one has tr_{ℂ/ℝ}(^{t}vMw) =

[3] Wolf, J. A., The action of a real semisimple Lie group on a complex flag manifold, I: Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237.

[4] —, The Stein condition for cycle spaces of open orbits on complex flag manifolds, Annals of Math. 136 (1991), 541–555.

[5] —, Exhaustion functions and cohomology vanishing theorems for open orbits on complex flag manifolds, Math. Research Letters 2 (1995), 179– 191.

[6] Wolf, L., Similarity of matrices in which the elements are real quaternions, Bull. Amer. Math. Soc. 42 (1936), 737–743.

Christina Birkenhake Mathematisches Institut Bismarckstr. 1 1/2 D-91054 Erlangen birkenhake@mi.uni-erlangen.de

Received February 20, 1998 and in final form October 22, 1998