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Real and Complex Structures 7:41 a.m. August 22, 2013 Real and complex structures Bill Casselman University of British Columbia [email protected] This essay explains the principal ways in which real and complex algebraic groups are closely related—base field extension, restriction of scalars, and Galois descent. Contents 1. Vector spaces 2. Complex and real varieties 1. Vector spaces Suppose V to be any vector space over C. Its conjugate V is the same space, but with a new scalar multiplication c • v = cv. A C•linear map from V to V is an R•linear map from V to itself such that f(cv)= cf(v). It is called a conjugation if it is of order two. COMPLEXIFICATION AND DESCENT. If V is a real vector space, its complexification is the tensor product VC = C ⊗R V , with C acting on the left. It has the same dimension over C that V does over R. There exists a canonical copy of V in VC, namely R ⊗R V . 1.1. Lemma. If V is any real vector space, there exists a unique conjugation on VC fixing the canonical copy of V . The copy of V is in fact precisely the subspace of vectors fixed by this conjugation. Proof. It takes c ⊗ v to c ⊗ v. There is a converse to this: 1.2. Lemma. If V is any complex vector space and σ a conjugation of V , then the canonical map from C ⊗ V σ to V is an isomorphism. Proof. The space V isthedirectsumofits ±1•eigenspaces V ± for σ. But multiplicationby i is an isomorphism of one with the other, which implies that V = V σ ⊕ iV σ. 1.3. Proposition. If V is a complex vector space, then any two conjugations of V are conjugate in GL(V ). Proof. Let σ and τ be two conjugations. Then V σ and V τ are both real vector spaces whose complexification is V . There therefore exists a real linear isomorphism ϕ of on with the other. It extends to a complex linear 1 automorphism of V such that ϕσ = τϕ, which means precisely that ϕσϕ− = τ. This has an equivalent formulation: 1.4. Corollary. Suppose that σ is a conjugation of the complex vector space V , ϕ a C•linear automorphism of V . Then ϕϕσ = I if and only if there exists a C•linear automorphism of V such that ff −σ = ϕ. Proof. Left as exercise. These results can be summarized as saying that over R there is exactly one structure of vector space in each dimension. Something similar holds for symplectic forms, and there is hence a suitable version of the previous Proposition for symplectic automorphisms. RESTRICTION OF SCALARS. Any complex vector space is of course also a real vector space. What can we say about its complexification C ⊗R V ? Multiplication ι by i is an R•linear map from V to itself, but it does not possess eigenvectors in V . It does, however, acquire them upon complexification, and C ⊗ V may be Real and complex structures 2 split into the direct sum of eigenspaces for ±i. For example, if V = C itself and we choose as coordinates the real and imaginary components of x + iy then ι has matrix 0 −1 0 1 1 with eigenvectors = + i·ι . 1 0 ∓i 0 0 Following this example, we see that the projections (1 ⊗ v) ∓ (i ⊗ iv) v −→ v± = 2 are those onto the ±i eigenspaces in C ⊗ V . This leads to: 1.5. Proposition. If V is a complex vector space then the map c ⊗ v −→ (cv, cv) is an isomorphism of C ⊗R V with V ⊕ V . The subspace V in C ⊗ V is that fixed by swaps. For this last, note that v → v from V to V is conjugate•linear. The last assertion can be reformulated by saying that the following diagram is commutative: C ⊗R V V ⊕ V c⊗v −→c¯⊗v u⊗v −→v⊗u C ⊗R V V ⊕ V Correspondingly, the C•linear dual of C ⊗ V is the direct sum of linear and conjugate•linear maps from V to C. This is the starting point in Hodge theory. LIE ALGEBRAS. All these results apply to Lie algebras. If g is a real Lie algebra, then gC is a complex Lie algebra. If σ is a conjugation of g that is also an automorphism of Lie algebra structure, gσ is a real Lie algebra whose complexification is g. If g is a complex Lie algebra then its conjugate g has conjugate structure constants, and C ⊗R g = g ⊕ g . All descents of a complex Lie algebra to R are isomorphic as vector spaces, but not necessarily as Lie algebras. The following is elementary: 1.6. Proposition. Two conjugations of the complex Lie algebra g give isomorphic real Lie algebras if and only if they are conjugate in Aut(g). Thus the analogue of Proposition 1.3 is that the conjugation classes of conjugations of g are in bijection with isomorphism classes of real Lie algebras whose complexification is isomorphic to g. Example. let sl2 be the real Lie algebra of SL2(R). It may be identified with the vector space of real matrices with trace 0. It has dimension three, with basis t, e, f and Lie bracket [t,e]= 2e [t,f]= −2f [e,f]= t . It corresponds to the usual conjugation X → X of sl2(C). Second, let su2 be the Lie algebra of 2 × 2 skew•Hermitian complex matrices of trace 0, those satisfying −tX = X—in other words it corresponds to Real and complex structures 3 the conjugation X → −tX of sl2(C). I leave it as exercise to see that up to conjugacy these are all the Lie algebra conjugations of sl2(C). Example. Similarly, suppose g to be any split real Lie algebra, containing t as Cartan subalgebra and Borel subalgebra b containing t. Let θ be an opposition involution of g taking b to its opposite and acting as −1 on t. Then because θ is defined over R it commutes with conjugation, and X −→ θ(X) is the conjugation defining the Lie algebra of a real group whose rational points are compact. 2. Complex and real varieties Suppose F to be either R or C. An affine scheme defined over F is specified by its affine ring AF [V ], which can be any ring finitely generated over F . The F •rational points of the variety are in bijection with maximal ideals m of AF [V ] for which AF [V ]/m = F . The complexification of a variety defined over R has affine ring AC[V ] = C ⊗R AR[V ]. If V is a complex variety, the conjugate variety has the same affine ring as V but is assigned the conjugate C•structure, which is to say that AC[V ] is AC[V ] with the conjugate structure of vector space. A real variety whose complexification is isomorphic to V is completely determined by an automorphism of the ring AC[V ] that is conjugation•linear and of order two. If V has etxra structure—for example, if it is an algebraic group—then descents inherit this structure if the conjugation is an automorphism of this structure. Restriction of scalars starts with a variety over C, and defines a variety RC/RV over R in such a way that the real points on it may be naturally identified with the complex points of V . The cleanest way to define it is to first define the complex variety W = V × V , and then define its descent to R as that associated to the conjugation defined by swap. As with Lie algebras, any reductive group over C has at least two descents, one a group that is split over R, and the other whose group of real points is compact. As before, to see this one starts with the split group overR, extends to C, then defines the new conjugation to be θ(g), where θ is the canonical involution determined by a choice of Borel subgroup B, torus T ⊆ B, and elements {eα} spanning simple root spaces t −1 gα. For classical groups suitably defined, θ(X)= X . For example, U(n) is the group of all n × n complex −1 matrices X such that tX = X..
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