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Math 249B. Cartan's Connectedness Theorem 1. Introduction Let G Be A

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Math 249B. Cartan's Connectedness Theorem 1. Introduction Let G Be A Math 249B. Cartan's connectedness theorem 1. Introduction Let G be a connected semisimple group over R. The group G(R) is often disconnected for its analytic topology (in contrast with the situation over C). For example, if G = PGL2m then there is a natural continuous surjection det : G(R) ! R×=(R×)2 = {±1g induced by the determinant on GL2m(R). Likewise, if (V; q) is an indefinite non-degenerate quadratic space over R then SO(q)(R) is generally disconnected (with 2 or 4 connected components). It is a general theorem of Whitney that if X is an R-scheme of finite type then X(R) has only finitely many connected components. (For a proof, see Appendix A of Milnor's book Singular points of complex hypersurfaces, which rests on the Lemma in x1 of \The Lefschetz theorem on hyperplane sections" by Andreotti and Frankel in Annals 69 (1959).) That is overkill for our purposes, and gives very limited information. We shall directly prove that π0(G(R)) is a 2-torsion finite abelian group controlled by the maximal split R-tori in G. This will emerge from our proof of: Theorem 1.1 (E. Cartan). If G is simply connected then G(R) is connected. The original approach of Cartan used Riemannian geometry. Briefly, since G(C) is a connected and topologically simply connected Lie group with G(R) the fixed points of the involution given by complex conjugation, the problem is reduced to showing that any invo- lution of a connected and simply connected Lie group has connected locus of fixed points, a problem Cartan solved by geometric methods. We will take a different approach, due to Borel and Tits (in x4 of their 1972 paper in IHES 41 that is a supplement to their big 1965 paper on reductive groups in IHES 27). This deduces the general case from the anisotropic and split cases by using the relationship between absolute and relative roots. Our presentation of their technique explains some points in more detail and streamlines other aspects (to keep the exposition self-contained relative to this course) by using our prior work with relative root systems. Example 1.2. As a warm-up, let's discuss the anisotropic and split cases. If G is anisotropic then G(R) is compact, as can be proved by adapting Prasad's method from the handout on compactness and anisotropicity over local fields. This is explained more fully as Theorem D.2.4 in the Luminy SGA3 article on reductive group schemes, where it is also shown that G(R) is connected in such cases without any need to assume G is simply connected (in the sense of algebraic groups). The split case is more interesting. For a split maximal R-torus T , the \simply connected" _ _ hypothesis (which amounts to a coroot basis ∆ of Φ(G; T ) being a Z-basis of X∗(T )) ensures that G(R) is generated by its subgroups Ua(R) for a 2 Φ(G; T ) (see Proposition 2.5 in the handout on geometric Bruhat decomposition); for SL2 this is very classical (over any field). But each Ua(R) = R is connected and passes through the identity, so any word in finitely many elements of such subgroups lies in the connected component of the identity (consider the associated \word map" using entire Ua(R)'s). Hence, G(R) is connected! 1 2 The strategy for the general case consists of three steps, using a maximal split R-torus S and the associated relative root system RΦ := Φ(G; S): (I) Show that S(R) ! π0(G(R)) is surjective, so it suffices to prove that S(R) ⊂ G(R)0. (Surjectivity does not use the \simply connected" hypothesis, and gives precise control over #π0(G(R)) for general connected semisimple G by applying Cartan's theorem to the simply connected central cover; see Remark 4.4.) nm (II) Prove that for a basis ∆ of the root system RΦ of non-multipliable relative roots, _ the associated set ∆ of relative coroots is a basis of X∗(S). In particular, S is a direct _ 0 product of copies of GL1 embedded by the coroots in ∆ , so to prove S(R) ⊂ G(R) it suffices to check the result after replacing G with the rank-1 connected semisimple _ nm subgroup D(ZG(Sa)) containing a (GL1) for each a 2 RΦ . (This derived group is simply connected, as for derived groups of torus centralizers in simply connected groups in general; see Corollary 9.5.11 of the course notes.) (III) For a connected semisimple group with relative rank 1 over a field k, construct SL2 as a k-subgroup containing a given maximal split k-torus. This reduces the task from (II) to the connectedness of SL2(R) that is a special case of Example 1.2 (or more concretely, SL2(R) is generated by the R-points of the standard root groups). Step (II) rests on some input from Bourbaki that we will review when needed, and step (III) involves some clever group-theoretic considerations (due to Borel and Tits). 2. Control of π0(G(R)) by S(R) Let G be a connected reductive R-group, and S a maximal split R-torus. We shall show that S(R) meets every connected component of G(R), which is to say S(R)G(R)0 = G(R). r If r = dim S then S(R) = ({±1g × R>0) , so it would follow that π0(G(R)) is 2-torsion r abelian with size at most 2 . (A description of the size of π0(G(R)) is given in Remark 4.4.) To prove S(R) ! π0(G(R)) is surjective, we need: Lemma 2.1. Let P ⊂ G be a minimal parabolic R-subgroup containing S. Then the quotient space G(R)=P (R) = (G=P )(R) is connected. The equality G(R)=P (R) = (G=P )(R) is a special case of a general equality we have proved over any field (with any parabolic subgroup over the ground field), and crucially it is a topological equality since passage to R-points carries smooth morphisms (such as G ! G=P ) to submersions due to the Submersion Theorem. Proof. We have a dynamic description for P , so P = PG(λ) for some λ 2 X∗(S). Thus, for U = UG(−λ) we have a Zariski-dense open immersion U ⊂ G=P . As a variety U is an affine space. Hence, U(R) is connected. Thus, it suffices to show that U(R) is dense in (G=P )(R). Rather generally, if X is a smooth R-scheme and Z ⊂ X is a nowhere-dense locally closed subset then we claim that Z(R) ⊂ X(R) has measure zero and hence empty interior. (Applying this to X = G=P and Z = (G=P ) − U would then give the desired density on R-points.) By stratifying Z we may assume it is smooth, yet its dimension is everywhere strictly smaller than that of X. The map of manifolds Z(R) ! X(R) is therefore nowhere a submersion, so by Sard's Theorem its image has measure 0. 3 If Γ is a locally connected topological group with Γ0 denoting is its (necessarily open) identity component and if H is a subgroup such that Γ=H is connected then H meets every 0 connected component of Γ (so π0(H) ! π0(Γ) is surjective). Indeed, since Γ=Γ is discrete (by openness of Γ0), the quotient space HnΓ=Γ0 is trivial. It follows that H ! Γ=Γ0 is surjective. Setting Γ = G(R) and H = P (R), it follows that π0(P (R)) ! π0(G(R)) is surjective. The Levi decomposition P = ZG(S)oU with U(R) connected then implies that π0(ZG(S)(R)) ! π0(G(R)) is surjective. But the connected reductive R-group ZG(S)=S is anisotropic, so its group (ZG(S)=S)(R) of R-points is compact and connected. The submersive homomorphism of Lie groups ZG(S)(R) ! (ZG(S)=S)(R) has image that is open, hence closed, so by connectedness of the target it is surjective. Thus, ZG(S)(R)=S(R) = (ZG(S)=S)(R), so this quotient modulo S(R) is connected; hence, S(R) meets every connected component of ZG(S)(R). This says S(R) ! π0(ZG(S)(R)) is surjective, so we obtain: Proposition 2.2. For a connected reductive R-group G with maximal split R-torus S, the natural map S(R) ! π0(G(R)) is surjective. nm Now assume G is semisimple. Consider a basis ∆ for RΦ , so the associated relative coroots provide an isogeny Y GL1 ! S a2∆ Q _ _ defined by (ya) 7! a2∆ a (ya). This isogeny is an isomorphism if and only if ∆ is a Z-basis of X∗(S), which is to say that the root datum nm nm _ _ nd (X(S); RΦ ; X∗(S); (RΦ ) = (RΦ ) ) nm _ is \simply connected" in the sense that (RΦ ) generates X∗(S) over Z. We will establish this property when G is simply connected, and that will permit us to reduce the proof of Cartan's theorem to the case of R-rank equal to 1 (which needs some work too!). 3. A result on relative root systems We call a semisimple root datum (X; Φ;X_; Φ_) simply connected when Z · Φ_ = X_, and adjoint type when Z · Φ = X. For a connected semisimple group G over a field k, since the ranks of the absolute and relative root systems can be very different, it is not at all apparent whether the properties of being simply connected or adjoint type for the absolute root datum should be inherited by the relative root datum.
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