Shimura Varieties

Shimura varieties

Spring 2015

These are my notes from the working group on Shimura varieties http://www.math.u-bordeaux1.fr/~jtong/GAGA.html Please send any remarks to [email protected]

Contents

1 Linear algebraic groups (review of results)1 1.1 Basic definitions and examples...... 1 1.2 Tori...... 2 1.3 K /k-forms...... 3 1.4 Structure of algebraic groups...... 3 1.5 Root systems...... 5 1.6 Classification over an algebraically closed field...... 11 1.7 Classification over a perfect field...... 11

2 Hermitian symmetric spaces: examples and basic properties 12 2.1 Almost complex manifolds...... 12 2.2 Hermitian manifolds...... 13 2.3 Symmetric spaces...... 14 2.4 Bergman metric...... 15 2.5 A higher-dimensional example: Siegel upper half-space...... 17 2.6 Isometries of Hermitian symmetric domains...... 18

3 Classiﬁcation of symmetric Hermitian domains 20 3.1 Preliminaries: Cartan involutions...... 20

8 January 2015 (Alan Hertgen)

1 Linear algebraic groups (review of results)

The used reference for this is Satake, Classiﬁcation theory of semi-simple algebraic groups. A modern reference for algebraic groups is http://jmilne.org/math/CourseNotes/ala.html

1.1 Basic definitions and examples Let G be an algebraic group over k, by which we mean a smooth group scheme over k. (REMARK: usually we also assume that G is of finite type.) If the underlying scheme is affine, then we say that G is an affine algebraic group.

Example 1.1. • Any ﬁnite group may be seen as an algebraic group (take the constant functor of points).

1 • The additive group G corresponds to the functor R (R, ), which is represented by Speck[T ]. a + 1 • The additive group G corresponds to the functor R (R×, ), which is represented by Speck[T,T − ] m · = Speck[T ,T ]/(T T 1). 1 2 1 2 − • The special linear group SLn corresponds to the functor R SLn(R), which is represented by

Speck[T ,...,T ]/(det(T ,...,T ) 1), 11 nn 11 nn −

where det(T11,...,Tnn) is the determinant X sgn(σ)T T . 1,σ1 ··· n,σn σ Sn ∈

• The general linear group GLn corresponds to the functor R GLn(R), which is represented by

Speck[T ,...,T ,T ]/(det(T ,...,T )T 1). 11 nn 11 nn −

More generally, if V is a ﬁnite dimensional vector space over k, then we may deﬁne GLV as a representable functor R Aut(V R). ⊗k

• Similarly, we have SOn and Spn (for chark , 2). N

Definition 1.2. An algebraic subgroup of GLV is called a linear algebraic group. In particular, a linear algebraic group is affine. Conversely, every affine algebraic group is isomorphic to a linear algebraic group. A theorem of Chevalley asserts that an algebraic group G is built up from a linear algebraic group and an abelian variety. Precisely, we have

Theorem 1.3 (Chevalley). If G is a connected algebraic group over a perfect ﬁeld k, then there exists a unique short exact sequence of algebraic groups 1 H G A 1 → → → → where H is a linear algebraic group and A is an abelian variety.

Proposition 1.4. Let G be an algebraic group over k. There is a unique short exact sequence

1 G◦ G π (G) 1 → → → 0 → where G◦ is connected, and π0(G) is ﬁnite étale over k.

1.2 Tori

d Deﬁnition 1.5. An algebraic group is called a split torus (tore déployé) if it is isomorphic to Gm. An algebraic group T is called a torus if Tk is a split torus. Example 1.6. The algebraic group D GL formed by diagonal matrices is a split torus. n ⊂ n   ∗    ∗   .   ..  ∗

In fact, any subtorus T in GLn /k is split iff it lies in Dn after some basis change; that is, iff there is a matrix P 1 ∈ GL (k) such that PTP − D . N n ⊂ n

2 Deﬁnition 1.7. The group of characters of a torus T is given by

X ∗(T ) : Hom(T ,(G ) ). = k m k

It is a free Z-module of ﬁnite rank, which is the dimension of T . Further, X ∗(T ) is equipped with an action of Gal(k/k).

Theorem 1.8. There is an equivalence of categories

tori over k free modules of ﬁnite rank with Gal(k/k)-action, ↔ T X ∗(T ).

1.3 K /k-forms Let G be an algebraic group over k and let K /k be a Galois extension.

Deﬁnition 1.9. A K /k-form of G is an algebraic group H/k such that there exists an isomorphism f : H G . K → K For K k we will simply say that H/k is a k-form of G. = Let Γ : Gal(K /k). For each σ Γ we have f σ : H G , and we put = ∈ K → K σ 1 φ : f f − Aut(G ). σ = ◦ ∈ K

Such maps φσ satisfy the cocycle condition 1 φ− φ φ . σ ◦ τ = στ One shows that two K /k-forms are isomorphic iff the corresponding cocycles are cohomological:

Theorem 1.10. There is a natural bijection

{K /k-forms of G}/ H 1(Γ,Aut(G )). isomorphism ↔ K d Example 1.11. A torus of dimension d is by deﬁnition a k-form of Gm.

1 Gd 1 Z Z H (Γ,Aut(( m)k )) H (Γ,GLn( )) with the trivial action Homcont(Γ,GLn( ))/conjugation N

1.4 Structure of algebraic groups

Deﬁnition 1.12. Let G be an algebraic group over k. An element u G(k) is called unipotent if for an embedding ∈ G , (GL ) its image in GL (k) is unipotent, i.e. (u 1)k 0 for some k 1,2,3,.... k → n,k n − = = An algebraic group G is called unipotent if all elements of G(k) are unipotent.

Example 1.13. Consider the algebraic group U GL consisting of upper triangular matrices with 1 on the diag- n ⊂ n onal 1  ∗ ∗ ··· ∗  1     ∗ ··· ∗  1   ··· ∗.   .. .   . .  1

Un is unipotent. N

Deﬁnition 1.14. We say that an algebraic group G is solvable if G(k) is solvable.

3 Proposition 1.15. Assume that k k and let G be a connected solvable algebraic group over k. =

1. By the Lie–Kolchin theorem G is isomorphic to an algebraic subgroup of the group Tn of upper triangular matrices.

2. There is a unique maximal unipotent algebraic subgroup Gu. It is a connected normal subgroup with k-points

G (k) {u G(k) u is unipotent}. u = ∈ |

3. Let T be the maximal subtorus of G. Then G G o T. = u 4. The maximal tori are conjugate by an element of G(k).

Deﬁnition 1.16. Let G be an algebraic group over k. G has a unique maximal connected solvable normal subgroup R(G), called the radical of G. G has a unique maximal connected unipotent normal subgroup Ru(G), called the unipotent radical. We say that G is semisimple if R(G ) 0 and that G is reductive if R (G ) 0. k = u k = G Example 1.17. SLn is semisimple. GLn is reductive, with radical R((GLn)k ) ( m)k . N We say that G is simple if it has no normal algebraic subgroups of dimension 0. Recall that a morphism of > algebraic groups f : G H is called an isogeny if it is surjective and has ﬁnite kernel. → Proposition 1.18. 1. A semisimple algebraic group G is isogenous to a direct product of simple algebraic groups.

2. A reductive algebraic group G is isogenous to a direct product of a semisimple group and a torus.

Deﬁnition 1.19. A Borel subgroup in G is a maximal solvable connected algebraic subgroup.

Proposition 1.20. Assume k k. Let G be a connected algebraic group over k. = 1. The Borel subgroups in G are conjugate.

2. If B is a Borel subgroup, then G/B is projective.

3. Any Borel subgroup equals its normalizer.

S 1 4.G g G(k) g − B g. = ∈ The proof is based on the Borel ﬁxed-point theorem.

Proposition 1.21 (Borel). Let G be a connected solvable algebraic group acting regularly on a non-empty, complete algebraic variety W over k k. Then there exists a fixed point. = Borel’s result generalizes Lie–Kolchin theorem. If G GL is a connected solvable algebraic group, then G acts ⊂ V on the complete flags of V 0 V ( V ( ( V V = 0 1 ··· n = and such flags form a projective variety, so by the Borel’s theorem there is a fixed flag. The corresponding choice of basis gives an embedding G , T . → n Definition 1.22. The rank of G is the dimension of a maximal torus in G. The k-rank of G is the dimension of a maximal split torus. We say that G is anisotropic if its k-rank is 0.

4 1.5 Root systems Let G be a connected semisimple algebraic group over k and let T be a maximal subtorus.

Deﬁnition 1.23. We say that a character α X ∗(T ) is a root if there exists an isomorphism u : G U to a sub- ∈ a −→ α group U G such that for all t T and s G α ⊂ ∈ ∈ a 1 t u(s)t − u(α(t)s). = The set V of roots is called the root system (with respect to T ).

The subgroup Uα is uniquely determined by α, and V is a ﬁnite set. We denote by N(T ) the normalizer of T in G and by Z (T ) the centralizer of T in G, so that Z (T ) N(T )◦ T . = = Deﬁnition 1.24. The group W : N(T )/T is called the Weyl group. =

W is a finite étale group. It may be seen as a group of automorphisms of X ∗(T ). For each s N(T ) we have an ∈ automorphism ws : X X defined by → 1 w (χ)(t) : χ(s− t s), s = where χ X ∗(T ) and t T (??? cf. Satake, p. 44). ∈ ∈ So we have a triple (X ,V ,W ), where X X ∗(T ) is a free Z-module of finite rank d dimT , the root system V = = is a finite subset of X , and W is a group of automorphisms of X . It satisfies the following properties:

(i)0 V , and if α V , then α V . ∉ ∈ − ∈ (i)∗ If α V and cα V for some c Q, then c 1. ∈ ∈ ∈ = ±

(ii) For each α V there is wα W acting by ∈ ∈

w (χ) χ α∗(χ)α, α = −

where α∗ X ∨. Moreover, wα maps V into itself: wα(V ) V . ∈ ⊆ (iii) The vector space XQ : X Z Q is spanned by V . = ⊗

(iv) W is generated by {wα α V }. | ∈

Deﬁnition 1.25. In general, (X ,V ) satisfying (i), (i)∗, (ii), (iii) is called an abstract root system, and we say that its Weyl group is the group W generated by wα.

Example 1.26. Consider the algebraic group SL of 2 2 matrices with determinant 1: 2 × µa b¶ { ad bc 1}. c d | − =

A maximal subtorus is given by diagonal matrices with determinant 1:

µa 0 ¶ T { 1 } = 0 a−

µa 0 ¶ This is isomorphic to Gm via a 1 , and we see that the group of characters is 7→ 0 a−

X ∗(T ) : Hom(T,Gm) Z, = µa 0 ¶ which is generated by χ: 1 a. We have a subgroup in SL2 which is isomorphic to Ga: 0 a− 7→

5 µ1 b¶ u : G SL , a −→ 0 1 ⊂ 2 µ1 s¶ s 7→ 0 1

µa 0 ¶ For t 1 T and s Ga we get = 0 a− ∈ ∈ µ ¶ µ ¶ µ 1 ¶ µ 2 ¶ 1 a 0 1 s a− 0 1 a s 2 t u(s)t − 1 u(χ(t) s). = 0 a− · 0 1 · 0 a = 0 1 =

This shows that α 2χ is a root. Moreover, α is also a root. Since XQ is a one-dimensional vector space, V = − = {α, α} is the whole root system. The Weyl group W is generated by reﬂection w . − α wα −α α

Similarly, for SL3 there are six roots as follows:

The Weyl group for SL3 is isomorphic to the symmetric group S3.

In general, SLd 1 will give a root system of type Ad . It may be seen in (d 1)-dimensional space as d (d 1) + + + vectors ei e j with 1 i d 1 and 1 j d 1, where i , j. − ≤ ≤ + ≤ ≤ +

Since W is a ﬁnite group, we may endow XQ with a W -invariant scalar product ( , ) (positive deﬁnite symmetric · · bilinear form). Now from (ii) we have

(w (χ),w (χ)) (χ,χ) α α = (χ α∗(χ)α, χ α∗(χ)α) = − − 2 (χ,χ) 2α∗(χ)(α,χ) α∗(χ) (α,α), = − + and hence (α,χ) α∗(χ) 2 . = (α,α) We see that (α,χ) wα(χ) χ 2 α = − (α,α) is the reﬂection with respect to the hyperplane perpendicular to α:

H : {χ (α,χ) 0}, α = | = i.e. w (α) α, and w leaves H pointwise ﬁxed. In particular, it is an involution, i.e. w2 1. α = − α α α =

6 NOTE: compare this to Bourbaki’s definition (Groupes et algèbres de Lie, Chapitre VI, n◦ 1.1). Let V be a Eu- clidean space with product ( , ). For a vector α V the reflection with respect to the hyperplane H perpendicular · · ∈ α to α is given by (χ,α) sα(χ) χ 2 α. = − (α,α) This map fixes pointwise H and sends α to α.A root system R V satisfies the following axioms: α − ⊂ (SR ) R is finite, 0 R, and spans V . I ∉ (SR ) For each α R the reflection s leaves R invariant: s (R) R. II ∈ α α ⊆ ( , ) (SR ) For all α,β R the number c : s (β) β 2 α β α is an integer multiple of α. III ∈ αβ = α − = (α,α) β | | (Geometric meaning: each root has half integral projection on another root, because cαβ 2 α cos(φ).) = | | If for each α R the only multiples are α, then the root system is called reduced. The Weyl group W (R) is the ∈ ± group of automorphisms of V generated by s for α R. α ∈ For example; there are three irreducible reduced root systems of rank 2, namely A2,B2,G2.

A2 B2 G2

Deﬁnition 1.27. For a root system the Cartan integers are given by

(α,β) cαβ : 2 Z, = (α,α) ∈

where α,β V . ∈

15 January 2015 (Alan Hertgen)

Proposition 1.28. For a root system V there exists a subset of roots ∆ such that ∆ is a basis of the vector space XQ, and each root β V may be written as a combination of elements of ∆ with integer coefﬁcients of the same sign, i.e. ∈ X β mα α, = α ∆ ∈ with either m 0 for all α (and then we say that β is a positive root with respect to ∆) or m 0 for all α (and then α ≥ α ≤ we say that β is a negative root with respect to ∆). Such a set ∆ is called a base1 of V and its elements are called simple roots.

1“Base” in Bourbaki’s terminology and “fundamental system” in Satake’s book; in these notes I will use word “base”

7 Bases arise in the following way. For a root α V we may consider the hyperplane Hα orthogonal to α. The S ∈ connected components of XR \ α V Hα are called Weyl chambers. For a vector γ lying in some Weyl chamber we may consider the set of positive roots∈ with respect to γ:

V +(γ) : {α V (γ,α) 0}, = ∈ | > i.e. these are the roots lying on the “positive” side of the hyperplane Hγ. Among these roots V +(γ) we pick indecomposable roots, i.e. those that cannot be written as α1 α2 with α1,α2 V +(γ). Then indecomposable roots in + ∈ V +(γ) form a base. This way each Weyl chamber gives a base. Example 1.29. For the root system A we may pick a base ∆ {α,β} as in the picture (the shaded part denotes the 2 = Weyl chamber giving rise to ∆):

α + ββ

−α α

−α − β −β

N Proposition 1.30. • A base ∆ consists of d roots, where d is the rank of the Z-module X .

• Each root α may be written as α W W W (α ). αir αir 1 αi1 i0 = − ··· • W is generated by {W α ∆}. α | ∈ • W acts transitively and freely on the set of bases (= on the set of Weyl chambers).

Deﬁnition 1.31. Let V be a root system in X and let V1 V be a nonempty subset such that V1 {V1}Z V . The ⊂ = ∩ latter means that V1 is a root system in X1 : {V1}Q X . We say that V1 is a subsystem in V . = ∩ A root system V is called irreducible if it cannot be written as V V1 V2 for two subsystems V1 and V2 with = ∪ V1 V2 (i.e. (α,β) 0 whenever α V1 and β V2). ⊥ = ∈ ∈ Each root system may be decomposed into irreducible root systems:

V V1 Vs , = ∪ ··· ∪ where Vi are orthogonal, and a base ∆ has the corresponding decomposition ∆ ∆ ∆ , = 1 ∪ ··· ∪ s where ∆i : ∆ Vi . A base ∆ is called irreducible if it is a base of an irreducible root system. = ∩ Let now ∆ {α ,...,α } be an irreducible base. Then the vectors α ,...,α are linearly independent, ∆ does = 1 d 1 d not decompose into orthogonal subsets, and the Cartan integers

(αi ,αj ) 2 (αi ,αi ) are 0 for all i , j. Such ∆ may be classiﬁed by Dynkin diagrams. Namely, we consider a graph with vertices ≤ 2 corresponding to αi ∆. Between αi and αj with i , j we put ∈ (αi ,αj ) (αi ,αj ) cαi αj cαj αi 2 2 · = (αi ,αi ) · (αj ,αj )

2There was a little mistake at this point at the seminar

8 edges (and one deduces from basic properties of root systems that this number is 0,1,2,3). Further, if αi and αj have different length, we put an arrow that goes from the longer vector to the shorter. The resulting graph is connected thanks to irreducibility of ∆.

Example 1.32. Consider the root system G . We can choose a base ∆ {α,β} as the picture below shows. 2 = 3 α + 2 β

β α + β 2 α + β 3 α + β

−α α

−3 α − β −2 α − β −α − β −β

−3 α − 2 β

We calculate that c 1 and c 3, so we put three edges between α and β. Moreover, β is longer than α, αβ = − βα = − so we should show it by an arrow:

α β

9 All Dynkin diagrams may be classiﬁed, and the complete list is the following.

α α α αd−2 αd−1 α Ad 1 2 3 d (d≥1)

α α α αd−2 αd−1 α Bd 1 2 3 d (d≥2)

α α α αd−2 αd−1 α Cd 1 2 3 d (d≥3)

αd−1

α α α αd−3 αd−2 Dd 1 2 3 (d≥4)

αd

The series Ad , Bd , Cd , Dd arise from the following algebraic groups:

Ad Bd Cd Dd SLd 1 SO2d 1 Spd SO2d + + chark , 2?

SL (R) : {M GL (R) detM 1}, n = ∈ n | = O (R) : {M GL (R) MM > M > M I }, n = ∈ n | = = n µ ¶ µ ¶ 0 In 0 In Sp (R) : {M GL (R) M > M } 2n = ∈ 2n | I 0 = I 0 − n − n

The diagrams E6,E7,E8,F4,G2 arise from root systems of certain exceptional algebraic groups.

10 1.6 Classiﬁcation over an algebraically closed ﬁeld

Let G and G0 be connected semisimple algebraic groups over k. Let φ: G G0 be an isogeny. This induces a → t morphism of maximal subtori φ: T T 0, and hence a map between groups of characters φ: X ∗(T 0) X ∗(T ). → → The latter is injective and has a finite cokernel. Let V X : X ∗(T ) and V 0 X 0 : X ∗(T 0) be the corresponding root systems. For each α V there exists a ⊂ = ⊂ = t ∈ unique root α0 V 0 such that φ(Uα) U 0 , and φ(α0) qα α, where qα is a power of chark. In fact, one has ∈ = α = t Y degφ [X : φ(X 0)] q . = α α V ∈ In particular, degφ 0 and φ is an isomorphism when chark 0. = = Definition 1.33. An isogeny of two root systems (X ,V ) and (X 0,V 0) is an injective morphism ρ : X 0 X with finite → cokernel such that there exists a bijection f : V V 0 with (ρ f )(α) qα α for all α V . → ◦ = ∈ Theorem 1.34 (Uniqueness; Chevalley). Let G and G0 be connected semisimple algebraic groups over k. Let T and T 0 be respectively a maximal subtorus in G and G0, with corresponding character groups X and X 0. If there exists an t isogeny of root systems φ: X 0 X , then there exists an isogeny of groups φ: G G0 such that ρ φ. Moreover, φ is → → = unique up to conjugation by an element of T (k).

Theorem 1.35 (Existence; Chevalley). Let k0 be a prime field. Let X be a Z-free module of finite rank and let (X ,V ) be a root system. Then there exists a connected semisimple algebraic group, defined over k0, called a Chevalley group, such that (X ,V ) is its root system. Moreover, the corresponding torus T may be chosen to be split. The uniqueness and existence theorems give a classification of semisimple algebraic groups over an algebraically closed field.

1.7 Classification over a perfect field Let k be a perfect field. Let G be a connected semisimple algebraic group over k. Then there exists a Chevalley

group G0 such that GK GK0 for some ﬁnite Galois extension K /k (that is, G is a K /k form of G0). Let T be a maximal torus in G, let X be the corresponding character group and V the root system. Denote Γ : Gal(K /k). The group X has a Γ-module structure, and we consider = X σ X0 : {χ X χ 0}. = ∈ | σ Γ = ∈

It is the annihilator of a maximal split torus A in T . The character group of A is X ∗(A) X /X . We take the set of = 0 roots V0 : X0 V , and it is a root subsystem in V . Let ∆ be a base of V . Then ∆0 : ∆ V0 is a base of V0. We = ∩ = ∩ denote by W the Weyl group of V and by W0 the Weyl group of V0 (the latter is generated by wα for α V0). ∈ We ﬁx some Γ-linear order on X , i.e. such that

χ 0 for χ X χσ 0 for all σ Γ. > ∉ 0 ⇒ > ∈ Then ∆σ : {ασ α ∆} = | ∈ is a base corresponding to the Γ-linear order, and there is a unique w W such that ∆σ w ∆. σ ∈ 0 = σ This allows to deﬁne an action of Γ on X by

[σ] 1 σ χ : w− χ . = σ

Deﬁnition 1.36. The data (X ,∆,∆0,[σ]) is called a Satake diagram. Here (X ,∆) corresponds to a Dynkin diagram, ∆ ∆ is the subset of base that annihilate A, and [σ] is the action of Γ on ∆. 0 ⊂ One depicts a Satake diagram as a Dynkin diagram, with vertices from ∆0 being colored and the action of Γ being shown by arrows.

11 Example 1.37. Here is a Satake diagram:

N Deﬁnition 1.38. An anisotropic kernel of G is the group [Z (A), Z (A)] where A is a maximal split torus. It is a semisimple anisotropic group. Theorem 1.39. The anisotropic kernel and the Satake diagram of a semisimple algebraic group G determine it up to isomorphism. So in order to classify semisimple algebraic groups over a perfect ﬁeld, one should classify (1) semisimple anisotropic groups and (2) Satake diagrams that arise from semisimple algebraic groups.

29 January 2015 (Alexey Beshenov)

2 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.

2.1 Almost complex manifolds

Definition 2.1. 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 2.2. 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 2.3. 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

12 Example 2.4. 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

2.2 Hermitian manifolds Deﬁnition 2.5. 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 2.6. 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 2.7. 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 2.8. 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 · · =

13 2.3 Symmetric spaces

Deﬁnition 2.9. 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 2.10. 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 2.11. 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 2.12. 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

14 2.4 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 2.13. 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 kernel3.

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 2.14. 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

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

15 Proposition 2.15. 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 2.16. 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 2.17. 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 2.18. 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 ← + [

16 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 2.19. 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.

2.5 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). =

17 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 ∈

2.6 Isometries of Hermitian symmetric domains Theorem 2.20. 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 ⊂

18 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 2.21. 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 2.22. 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 2.23. 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 2.24. 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)

19 Example 2.25. 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

3 Classiﬁcation of symmetric Hermitian domains

3.1 Preliminaries: Cartan involutions Deﬁnition-theorem 3.1. 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 3.2. Trivially, if G is a compact group, then the identity map θ : G G is a Cartan involution. N → Example 3.3. 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

20 Example 3.4. Let G be a reductive algebraic group over R and let G , GL(V ) be a faithful representation over R. → A group G GL(V ) is reductive iff there exists some basis of V such that G is stable under the transpose X X >. ⊂ 1 7→ We pick such a transpose, and then the restriction of X (X >)− to G is a Cartan involution. In fact, all Cartan 7→ involutions arise this way. N

µa b¶ µ d b¶ Example 3.5. 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