Contents 1 Hermitian Symmetric Spaces: Examples and Basic

Contents

1 Hermitian symmetric spaces: examples and basic properties1 1.1 Almost complex manifolds...... 1 1.2 Hermitian manifolds...... 2 1.3 Symmetric spaces...... 3 1.4 Bergman metric...... 4 1.5 A higher-dimensional example: Siegel upper half-space...... 6 1.6 Isometries of Hermitian symmetric domains...... 7

2 Classiﬁcation of symmetric Hermitian domains9 2.1 Preliminaries: Cartan involutions...... 9

1 Hermitian symmetric spaces: examples and basic properties

The next goal is to explain classiﬁcation of Hermitian symmetric domains. We begin with some preliminary expla- nations of what these are, along with toy examples.

1.1 Almost complex manifolds

Definition 1.1. Let M be a smooth real manifold (of class C ∞). We say that it is an almost complex manifold if we are given a smooth tensor field J on M of type (1,1) such that for each vector field X on M holds J(J(X )) X . = − In other words, we ask that to each point p M we associate (in a smooth manner) an endomorphism of the real ∈ tangent space J : T M T M such that J 2 Id. p p → p p = − n Example 1.2. Let M be a complex manifold of dimension n, i.e. a ringed space (M,OM ) locally isomorphic to C with the sheaf of holomorphic functions. Then M may be viewed as a smooth real manifold of dimension 2n. The tangent spaces T M carry naturally an action of i C, and M is an almost complex manifold. N p ∈ Example 1.3. Let V be a real vector space. To specify an action of C on V means to pick an R-algebra representation 2 ρ : R[J]/(J 1) EndR(V ). + → Since the algebra R[J]/(J 2 1) is generated by J, everything amounts to picking an endomorphism J : V V satis- + → fying J 2 Id. Then this gives an action of the whole C by = − (x J y) v x v y J(v). + · = + Note that V becomes a C-vector space, so we conclude that dimR V must be an even number, otherwise V does not admit an almost complex structure. Vice versa, if dimR V 2n, then we can pick a basis (x , y ,...,x , y ) and define an action of C by thinking of = 1 1 n n these as of coordinates z x i y : k = k + k

J(x ) : y , k = k J(y ) : x . k = − k However, there is nothing canonical about this! N

Example 1.4. We see that a real manifold M does not admit an almost complex structure if dimR M is odd. If dimR M is even, then each tangent space Tp M carries an almost complex structure Jp , but they might not glue to a smooth vector ﬁeld J on M such that J 2 Id. For instance, among the spheres S2n, only S2 and S6 admit = − an almost complex structure (S2 simply because it may be viewed as the Riemann sphere; for S6 it is actually not known whether it has a complex structure, it’s a famous open problem). See e.g. N. Steenrod, The Topology of Fibre Bundles, §41, and J. P.May, A Concise Course in Algebraic Topology, §24.4. N

1 1.2 Hermitian manifolds Deﬁnition 1.5. Let M be a smooth manifold. We say that it is Hermitian if

• M carries an almost complex structure J.

• M carries a Riemannian structure, i.e. a smooth tensor ﬁeld g of type (2,0), such that

– g(X ,Y ) g(Y , X ) for all vector fields X ,Y on M, = – g : T M T M R is a positive definite nondegenerate bilinear form for each p M. p p × p → ∈ • The Riemannian structure is invariant under J, i.e. for all vector fields X ,Y on M holds

g(JX , JY ) g(X ,Y ). =

Example 1.6. Consider the complex plane C with real coordinates Z X i Y . The standard Riemannian metric = + is given by µ ∂ ∂ ¶ µ ∂ ∂ ¶ µ ∂ ∂ ¶ gp (p), (p) 0, gp (p), (p) 1, gp (p), (p) 1. ∂X ∂Y = ∂X ∂X = ∂Y ∂Y = We recall that usually this is written as “ds2 dx2 d y2”. The action by i is ∂ ∂ , ∂ ∂ , and so clearly = + ∂X 7→ ∂Y ∂Y 7→ − ∂X g(i X ,i Y ) g(X ,Y ), and C is Hermitian. N · · = Example 1.7. Consider the complex upper half-plane

H : {z C Imz 0}. = ∈ | > It carries naturally a complex structure, and real coordinates Z X i Y . We consider a Riemannian metric given 2 (d X )2 (dY )2 = + by ds + , i.e. = y2 µ ∂ ∂ ¶ µ ∂ ∂ ¶ 1 µ ∂ ∂ ¶ 1 gp (p), (p) 0, gp (p), (p) , gp (p), (p) , ∂X ∂Y = ∂X ∂X = y2 ∂Y ∂Y = y2

and we see immediately that g(i X ,i Y ) g(X ,Y ), so H is a Hermitian manifold. N · · = Example 1.8. Consider P1(C) C { } as the Riemann sphere S2 , R3, which comes with a natural complex and = ∪ ∞ → Riemannian structure. z

∞

x 0 C y

Geometrically, the action of i rotates C counterclockwise, and this satisﬁes g(i X ,i Y ) g(X ,Y ). N · · =

2 1.3 Symmetric spaces

Deﬁnition 1.9. Let M be a connected complex manifold. Then its underlying smooth manifold M ∞ has a canonical almost complex structure. Assume that M ∞ also carries a Riemannian structure which is compatible with the almost complex structure, in the sense that M ∞ is a Hermitian manifold. We say that M is a Hermitian symmetric space if for each point p M there exists a symmetry s : M M ∈ p → that satisﬁes the following properties:

1) s is an involution, i.e. s2 Id, p p =

2) sp is holomorphic with respect to the complex structure on M,

3) s is an isometry with respect to the speciﬁed Riemannian structure on M ∞, i.e. g(s (X ),s (Y )) g(X ,Y ), p p p =

4) sp has p as its isolated ﬁxed point.

Example 1.10. As we observed, the complex plane C is naturally a Hermitian manifold. To see that it is also symmetric, note that the group of plane translations acts transitively on C, and these actions are holomorphic isometries. Hence we may check the conditions 1)–4) for one point, for example 0. Then a symmetry is given by z z: 7→ − it is an involution, a holomorphic isometry, and p is its only ﬁxed point. Similarly, we have Hermitian symmetric spaces C/Λ, where Λ ω Z ω Z is a lattice. N = 1 ⊕ 2 Example 1.11. Recall that the group

µa b¶ SL (R) : {γ M (R) detγ ad bc 1} 2 = = c d ∈ 2 | = − =

acts on H by Möbius transformations az b γ z : + . · = cz d + 2 (d X )2 (dY )2 The maps z γ z are holomorphic, and they are also isometries. Indeed, the Riemannian metric ds + 7→ · = y2 may be written in coordinates z x i y and z x i y as = + = − dz dz ds2 4 . = − (z z)2 − Using the formula d( z) det dz dz , we see that γ γ (cz d)2 (cz d)2 · = + = + d(γ z)d(γ z) dz dz 4 · · 4 . − (γ z γ z)2 = − (z z)2 · − · − µ py x/py¶ The action is transitive, since x i y i for any x i y H . Hence we again may consider one + = 0 1/py · + ∈ particular point, e.g. i H . Now z 1/z is an involutive holomorphic isometry, having i as the only ﬁxed point. ∈ 7→ − We conclude that H is a Hermitian symmetric space. N

Example 1.12. Consider P1(C) with the natural Riemannian structure on the Riemann sphere

C { } , R3. ∪ ∞ → Any rotation of the sphere is a holomorphic isometry, and the group of rotations acts transitively. Rotation by π along the Z -axis (that is, z z on C and ) is an involution, leaving ﬁxed the poles 0 and . So P1(C) is a 7→ − ∞ 7→ ∞ ∞ Hermitian symmetric space. N

3 1.4 Bergman metric Let Ω Cn be a bounded domain, i.e. a nonempty bounded open connected subset. It turns out that there is a ⊂ canonical way to assign a Riemannian metric to Ω, so that every biholomorphic map Ω Ω is an isometry with → respect to it. We sketch the construction following S.G. Krantz, Function Theory of Several Complex Variables, referred as [Krantz] below. As always, for functions f ,g : Ω C we have an inner product → Z f ,g : f (z)g(z)dz, 〈 〉 = Ω which is correctly deﬁned on the space of square-integrable functions

L2(Ω) : {f : Ω C f , f }. = → | 〈 〉 < ∞ We have the subspace H(Ω) L2(Ω) of holomorphic square integrable functions, and in fact it is a closed subspace. ⊂ This means that H(Ω) is a Hilbert space, just as L2(Ω)—namely, it is complete with respect to the associated norm p f : f , f . k k = 〈 〉 Definition-theorem 1.13. There exists a unique map K : Ω Ω C, such that × → 1) for fixed w Ω the function z K (z,w) lies in H(Ω), ∈ 7→ 2)K (z,w) K (w,z), = 3) it satisfies the reproducing property Z f (z) K (z,w) f (w)dw = Ω for all f H(Ω). ∈ We call K (z,w) the Bergman kernel1.

The construction is roughly the following. Let (φk )k N be an orthonormal basis of H(Ω). We set ∈ X K (z,w) : φk (z)φk (w). = k

This converges, and in fact satisfies 1) [Krantz, Proposition 1.4.7]. It is clear that 2) is satisfied as well. Finally, we check 3): X Z f f ,φk φk K ( ,w) f (w)dw. = k 〈 〉 = Ω − The uniqueness of K (z,w), and hence the fact that it does not depend on the choice of an orthonormal basis of H(Ω), is due to 2) and 3). Indeed, suppose K 0(z,w) is another Bergman kernel. Then Z Z K (z,w) K (w,z) K 0(z,t)K (w,t)dt K (w,t)K 0(z,t)dt K 0(z,w) K 0(z,w). = = Ω = Ω = = Example 1.14. As one can imagine, explicitly calculating the Bergman kernel is difficult. It is possible in some easy cases, e.g. for the unit open disk D : {z C z 1}. = ∈ | | | < The Bergman kernel for D is given by 1 1 K (z,w) = π (1 z w)2 − · —see the calculation in [Krantz, Theorem 1.4.22]. N

1 STEFAN BERGMAN (1895–1977), a Polish-born American mathematician

4 Proposition 1.15. Let Ω Cn be a bounded domain and let K (z,w) be the corresponding Bergman kernel. Then ⊂ K (z,z) 0 for all z Ω. > ∈ Proof. We have from the construction of the Bergman kernel

X X 2 K (z,z) φk (z)φk (z) φk (z) , = k = k | |

and there are k such that φ (z) , 0, since there are nonzero functions in H(Ω). | k | ■ Deﬁnition-theorem 1.16. Let Ω Cn be a bounded domain. Then there exists a canonical Hermitian metric on Ω ⊂ given by 2 X i j ∂ g gi j dz dz , gi j (z) logK (z,z). = = ∂zi ∂z j This is called the Bergman metric on Ω.

(Checking that g (z) is positive deﬁnite for all z Ω requires some thought; see [Krantz, Chapter 1, Exercise 39].) i j ∈ Here is a truly remarkable property of the Bergman metric:

Theorem 1.17. Let Ω ,Ω Cn be two bounded domains, and let f : Ω Ω be a biholomorphic map. Then f is 1 2 ⊂ 1 → 2 an isometry with respect to the Bergman metric.

(This is [Krantz, Proposition 1.4.15].)

Example 1.18. For the open unit disk we have K (z,z) 1 1 , and a little calculation (e.g. in real coordinates D π (1 z 2)2 ³ ´ = ³−| | ´ x i y z, keeping in mind that ∂ 1 ∂ i ∂ and ∂ 1 ∂ i ∂ ) gives the Bergman metric + = ∂z = 2 ∂x − ∂y ∂z = 2 ∂x + ∂y

∂2 2 g(z) logK (z,z) . = ∂z ∂z = (1 z 2)2 − | | In fact, this is the so-called Poincaré metric. N We recall that H and D are two different models of the hyperbolic plane, and the upper half plane H corresponds to the open disk D via the Cayley transform

H D, → i z z − 7→ i z 1 z + i − z. 1 z ← + [

5 i

A little calculation shows that the canonical Bergman metric on D corresponds to the metric on H that we considered above (up to scaling the latter by an appropriate constant). Indeed, let z,z denote the coordinates on H , and let w,w be the coordinates on D.

µ i z ¶ 2i dz dw d − − , = i z = (i z)2 + + µ i z ¶ 2i dz dw d + , = i z = (i z)2 − −

2i dz 2i dz 2 − 2dw dw · (i z)2 · (i z)2 8 dz d z 8 dz d z 2 dz d z 2 + − 2 2 2 2 . (1 w w) = Ã i z i z! = ((i z)(i z) (i z)(i z)) = (2i (z z)) = − (z z) − 1 − + + − − − + − − − i z · i z + − Remark 1.19. Any Hermitian symmetric domain D may be embedded in some Cn as a bounded symmetric domain Ω. This means that D has a unique Hermitian metric that maps to the Bergman metric on Ω.

1.5 A higher-dimensional example: Siegel upper half-space

We have the transitive action of SL2(R) on the upper half-plane H by Möbius transformations. We see that the stabilizer of i H identiﬁes with ∈ µ a b¶ SO (R) { a,b R, a2 b2 1}, 2 = b a | ∈ + = − which is a maximal compact subgroup, and we have

H SL2(R)/SO2(R). =

6 This has a higher-dimensional generalization, the Siegel upper half-space, consisting of symmetric complex n n matrices with positive deﬁnite imaginary part: ×

Hn : {Z X i Y X ,Y Mn(R), X > X , Y > Y , Y 0} = = + | ∈ = = > n (n 1)/2 (in particular, H1 H is the upper half-plane). We may see this as an open subset of C + by sending a matrix = n (n 1)/2 (zi j ) to the point (zi j )j i C + , so there is a natural complex structure. ≥ ∈ We have the symplectic group µ ¶ µ ¶ µ ¶ AB 0 In 0 In Sp (R) : {γ M (R) γ> γ } 2n = = CD ∈ 2n | · I 0 · = I 0 − n − n

(in particular, SL2(R) Sp (R)), and Sp (R) acts transitively on Hn by “Möbius transformations” = 2 2n 1 γ Z (AZ B)(CZ D)− . · = + + We have a subgroup of unitary matrices

U {Z M (C) Z † Z ZZ † I }, n = ∈ n | = = n † where Z Z > denotes the conjugate transpose (in particular, SO2(R) U1). This may be identiﬁed with a sub- = group of Sp2n(R): µ XY ¶ X iY , + 7→ YX − and this is a maximal compact subgroup of Sp2n(R). In fact,

Hn Sp (R)/Un. = 2n Now consider the set of matrices

† Dn : {Z Mn(C) Z > Z and In Z Z 0}, = ∈ | = − > n (n 1)/2 which may be identiﬁed with a bounded domain in C + (in particular, for n 1 it is just the open unit disk), = so we know that for Dn there is a canonical Hermitian metric—the Bergman metric. There is a “Cayley transform”, identifying Hn and Dn:

Hn Dn, → 1 Z (i I Z )(i I Z )− 7→ n − n + 1 i (In Z )(In Z )− Z . − + ←[ This allows us to conclude that Hn also carries an invariant Hermitian metric. Further, it is a Hermitian symmetric space (and hence Dn). For this observe that the point i In Hn is the only ﬁxed point of the involution µ ¶ ∈ 0 In − Sp2n(R). In 0 ∈

1.6 Isometries of Hermitian symmetric domains Theorem 1.20. Let (M,g) be a symmetric space, with group of orientation preserving isometries Isom(M,g). Denote by Isom(M,g)+ the connected component of the identity (in the analytic topology). For a point p M denote by ∈ K Isom(M,g)+ be the subgroup leaving p ﬁxed. p ⊂

7 For example,

(M,g) isometries orientation-preserving p Kp H PGL2(R) PSL2(R) o Z/2 PSL2(R) i SO2(R)/{ 1} ± P1(C)O (R) SO (R) 0 U SO (R) 3 3 1 = 2

Then Kp is compact, and we have an isomorphism of smooth manifolds

Isom(M,g)+/K M, p → γ K γ(p); · p 7→ in particular, the action of Isom(M,g)+ on M is transitive.

For a proof we refer to [Helgason, Differential geometry, Lie Groups, and Symmetric Spaces, II.4.3].

Theorem 1.21. Let (M,g) be a Hermitian symmetric space. Consider the following groups:

• Isom(M ∞,g) = isometries of the underlying real manifold M ∞.

• Hol(M) = automorphisms M M as a complex manifold. →

• Isom(M,g) = Isom(M ∞,g) Hol(M) = holomorphic isometries. ∩ Then in fact Isom(M ∞,g)+ Hol(M)+ Isom(M,g)+, = = in particular Hol(M)+/Kp M ∞.

— this is claimed in Milne’s notes.

Theorem 1.22. Let (M,g) be a Hermitian symmetric domain. Consider the real Lie group H : Hol(M)+ and its Lie = algebra h : Lie(H). There is a unique connected algebraic subgroup G of GL(h) such that inside GL(h) =

G(R)+ Hol(M)+. = Moreover, G(R)+ G(R) Hol(M). = ∩ Example 1.23. For the upper half-plane H every isometry is either

• holomorphic (and orientation-preserving), or

1 • differs from a holomorphic map by the orientation change z z− , which is antiholomorphic. 7→

We have G PGL , and PGL (R)+ PSL (R). N = 2 2 = 2 Theorem 1.24. Let M be a Hermitian symmetric domain. For each point p M there exists a unique homomor- ∈ phism u : U Hol(M) p 1 → (where U : {z C z 1} is the circle group), such that u (z) ﬁxes p and acts on T M as multiplication by z. 1 = ∈ | | | = p p (A proof of this is sketched in Milne’s notes)

8 Example 1.25. Consider M H . Then Hol(M) PSL2(R) : SL2(R)/{ 1}. We want morphisms that ﬁx i, and we = = ± keep in mind that Stab R (i) SO (R). In fact, U may be viewed as SO (R) via SL2( ) = 2 1 2

h : U SL (R), 1 → 2 µ a b¶ z a ib . = + 7→ b a − 2 h(z) ﬁxes the point i, and we compute that it acts on Ti H as multiplication by z :

µµ ¶ ¶¯ µ ¶¯ d a b ¯ d aX b ¯ f X ¯ f + ¯ d X b a · ¯ = d X bX a ¯ − X i X i = − +¯ = ¯ d aX b ¯ d ¯ + ¯ f (X )¯ = d X bX a ¯X i · d X ¯X i − + = = 2 2 ¯ a b d ¯ + 2 f (X )¯ = (a bi) · d X ¯X i − ¯ = d ¯ (a bi)2 f (X )¯ . = + · d X ¯X i = Since we want action by multiplication by z, we deﬁne

u : U PSL (R) : SL (R)/{ 1}, 1 → 2 = 2 ± z h( pz) (mod 1). 7→ ±

The choice of pz does not change the class modulo 1, and so u is well-deﬁned and u(z) acts on Ti H by ± multiplication by z. N

2 Classiﬁcation of symmetric Hermitian domains

2.1 Preliminaries: Cartan involutions Deﬁnition-theorem 2.1. Let G be a connected reductive algebraic group over R. Then there exists an automorphism θ : G G (as an algebraic group over R) with θ2 Id, such that the group → = G(θ) : {g G(C) g θ(g)} = ∈ | = is compact (by g we denote complex conjugation). Such θ is called a Cartan involution, and it is unique up to conjugation by an element of G(R).

Example 2.2. Let V be a ﬁnite dimensional real vector space. Consider the group GL(V ). A choice of some basis deﬁnes an isomorphism GL(V ) GLn, and on GLn we have the usual matrix transpose X X >. The map 7→ 1 θ : X (X >)− 7→ is a Cartan involution: (θ) 1 GL (R) {Z GL (C) Z (Z >)− } U n = ∈ n | = = n is compact. A different choice of basis of GL(V ) differs from θ by conjugation by an element of GLn(R). N Example 2.3. If G is a reductive algebraic group over R and G , GL(V ) is a faithful representation over R, then → there exists some basis for V for which G is stable under transpose X X > (exactly because G is assumed to be 1 7→ reductive!). The restriction of X (X >)− to G is a Cartan involution. In fact, all Cartan involutions arise this 7→ way. N

9 µa b¶ µ d b¶ Example 2.4. For instance, take SL GL . For a matrix M the inverse is given by 1 − , so we 2 ⊂ 2 = c d detM · c a − take a Cartan involution µa b¶ µ d c¶ θ : − . c d 7→ b a − µ 0 1¶ Note that it is the same as conjugation by : 1 0 − 1 µ 0 1¶ µa b¶ µ 0 1¶− µ d c¶ − . 1 0 · c d · 1 0 = b a − − − Now we see that µ a b¶ SL(θ) { a 2 b2 1} SU , 2 = b a | | | + | | = = 2 − which is a closed bounded subset in C2, hence a compact subgroup. N