SHIMURA DATUM HAO (BILLY) LEE Abstract. These are notes I created for a seminar talk, following the paper of “Introduction to Shimura Varities” by Milne. I claim no credit to the originality of the contents of these notes. Nor do I claim that they are without errors, nor readable. 1. Review Notations: Let U1 be the complex unit circle Definition 1.1. Let M be a complex manifold. It is homogeneous if its automorphism group (automorphism can be Riemannian, Hermitian, ...) acts transitively (for all p; q there exists an automorphism sending on to the other). It is symmetric if it is homogeneous, and for some p (hence for all) there exists an involution with p as an isolated fixed point (some nbhd st. p is the only fixed point). A connected symmetric Hermitian manifold is called a Hermitian symmetric space. It is a Hermitian symmetric domain if it is a hermitian symmetric space of non-compact type (non-positive, but not identically zero sectional curvature). Proposition 1.2. Let (M; g) be a Hermitian symmetric domain and h be the Lie algebra of Hol(M)+ (this is a Lie Group, in fact, so is Is (M; g)). There is a unique connected algebraic subgroup G of GL(h) such that + + G(R) = Hol(M) inside GL(h): For such a G, + G(R) = G(R) \ Hol (M) inside GL(h) and so G(R)+ is the stabilizer of G(R) of M. This group G is adjoint (in particular semisimple) and G(R) is not compact. Definition 1.3. G is adjoint if Ad : G ! Aut(g) is faithful (non-trivial center). It’s semi-simple. Proposition 1.4. Let D be a HSD, for each p 2 D, there exists a unique homomorphism up : U1 ! Hol(D) = G(R) such that up(z) fixes p and acts on TpD as multiplication by z. Remark 1.5. These homomorphisms are algebraic, because the representations of U1 have the same description when regarded as a Lie group or as an algebraic group Theorem 1.6. Let D be a Hermitian symmetric domain, G the associated real adjoint algebraic group. The homomor- phisms up : U1 ! G attached to each p 2 D satisfies: −1 (1) Only the characters z; 1; z occur in the representation of U1 on Lie(G)C defined by Ad ◦ up (2) ad (up(−1)) is a Cartan Involution (3) up(−1) does not project to 1 in any simple factor of G Conversely, let G be a real adjoint algebraic group, and let u : U1 ! G satisfy the above. Then the set D of conjugates of u by elements of G(R)+ has a natural structure of a hermitian symmetric domain, for which G(R)+ = Hol(D)+ and u(−1) is the symmetry at u (viewed as a point of D). Lemma 1.7. Let H be an adjoint real Lie group (semisimple with trivial center), and let u : U1 ! H be a homomorphism satisfying SU1 and SU2. Then TFAE: (1) u(−1) = 1 (2) u is trivial (u(z) = 1 for all z) 1 SHIMURA DATUM 2 (3) H is compact 2 ±1 Proof. First is because u factors through U1 ! U1 so z can’t occur in the representation of U1. The second part is H compact (H(R)) iff identity map on G is a Cartan involution, so adu(−1) = idG. Hence, u(−1) 2 Z(H) and since Z(H) = 1, u(−1) = 1. This means that the third condition of the theorem, is really just saying that G has no compact simple factors. (some non-compactness condition). 2. Connected Shimura Datum Note. • Since extensions and quotients of solvable algebraic groups are solvable (G ⊇ G1 ⊇ ::: ⊇ e normal, where each quotient is commutative), G contains a maximal connected solvable (commutative) normal subgroup called the radical R(G) of G. G defined over an algebraically closed field is semisimple if R(G) = e. Else, say it’s semisimple if it is after base extension. (for us, always connected) • Simple algebraic or Lie group is a semi-simple group whose only proper normal subgroups are finite (almost-simple). • Group G is unipotent if every representation of G has a non-zero fixed vector. Also fixed under extensions and quotients. Therefore, every connected group variety has a maximal connected unipotent normal subgroup variety. Define reductive similarly • Gad is the image of G under adjoitn representation • Gder is [G; G], the intersection of all normal subgroups N of G such that G=N is commutative. It’s the smallest subgroup such that G=Gder is commutative. (s.s.). Definition 2.1. A connected Shimura Datum is a pair (G; D) of a semi-simple algebraic group G over Q and a Gad(R)+-conjugacy class D of homomorphisms u : U ! Gad satisfying: 1 R −1 ad • SU1: for all u 2 D, only the characters z; 1; z occur in the representation of U1 on Lie G defined by C ad ad Ad ◦ u : U1 ! G ! Aut Lie G . C • SU2: for all u 2 D, ad (u(−1)) is a Cartan involution on Gad. R • SU3: Gad has no Q-factor H such that H(R) is compact Remark 2.2. We can also talk about pairs (G; u) and if the u satisfies the conditions SU1, and SU2, so will any conjugate of it. Therefore, it determines a connected Shimura datum, but this is better, to avoid having a distinguished point. × Let S be the algebraic trosu over R obtains from Gm over C by restriction of scalars (Deligne torus). That is, S(R) = C and SC = Gm × Gm. We have an exact sequence of tori ! 0 ! Gm ! S ! U1 ! 0 where in the real values gives −1 z7! z × r7!r × z¯ 0 ! R ! C ! U1 ! 0: z When H is a s.s. real alg group with trivial center, u : U1 ! H gives a homomorphism h : S ! H by the rule h(z) = u( z¯). −1 z z¯ ad Then acting by characters z; 1; z is the same as S acting via z¯; 1; z . Converse also holds. Change, ... and adh(i), G has no Q-factor on which the projection of h is trivial. For G a connected algebraic group, there exists a Cartan involution iff G is reductive (any two are conjugate by G(R)). Proof: Satake. ! 2 a b Example 2.3. u : U1 ! P GL2(R) be the homomorphism sending z = (a + bi) to mod ± I. Let D be the −b a ! a b −1 set of conjugates (so A A mod ± I) for A 2 SL2(R). Hence, (SL2;D) is a Shimura datum. −b a This is really just up from above when p = i 2 H. SHIMURA DATUM 3 Definition 2.4. A semi-simple algebraic group G=Q is of compact type if G(R) is compact. It’s of non-compact type if it does not contain a non-trivial normal subgroup. G semi-simple over Q, then there exists isogeny G1 × ::: × Gr ! G; (fact in Milne) where Gi’s are simple. All Gi’s are compact (non-compact) iff all Gi’s are non-compact. If it’s simply connected or adjoint, this is non-compact iff no simple factor is of compact type. Proposition 2.5. To give a connected Shimura Datum is the same as to give • a semisimple algebraic group G over Q of non-compact type • a Hermitian symmetric domain D • an action of G(R)+ on D defined by a surjective homomorphism Gad(R)+ ! Hol(D)+with compact kernel Proof. Let (G; D) be a connected Shimura datum, and let u 2 D. Decompose Gad into a product of its simple factos R H1 × ::: × Hs, and write u = (u1; :::; us) each projection onto the i-th factor. If Hi is compact, then ui = 1 (by the lemma). Otherwise, Hi is not compact, and so ui satisfies the conditions of the 0 + 0 + 0 real version, so we get an irreducible HSD Di such that Hi(R) = Hol (Di) and Di is in natural 1-1 correspondence + with the set Di of Hi(R) -conjugates of ui. The 0 Y 0 D = Di + ad + + is an HSD on which Hi(R) acts via a surjective homomorphism G (R) ! Hol(D) with compact kernel (the Hi’s 0 Q where ui = 1). There’s a natural identification D with D = Di. 3. Connected Shimura Variety Proposition 3.1. Let D be a hermitian symmetric domain, and Γ a discrete subgroup of Hol(D)+. If Γ is torsion free, then Γ acts on D freely. Additionally, there is a unique complex structure on ΓnD that makes the quotient map a local isomorphism. Remark 3.2. Want BIG groups Γ so that we have finite co-volume. Definition 3.3. Let G be an algebraic group over Q. A subgroup Γ of G(Q) is arithmetic if it is commensurable (equivalence relation) with G(Q) \ GLn(Z) (finite index in both) for some embedding G,! GLn (then true for every embedding into GLn0 ). Let H be a connected real Lie group. A subgroup Γ of H is arithmetric, if there is an algebraic group G over Q, a + + surjective homomorphism G(R) ! Hol(D) with compact kernel, and an arithmetric subgroup Γ0 of G(Q) such that + Γ0 \ G(R) maps onto Γ. Proposition 3.4. Let H be a s.s. real Lie group which admits a faithful finite-dimensional rep. Every arithmetic subgroup Γ of H is discrete of finite covolume, and contains a torsion-free subgroup of finite index.