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The Complex Dual to the Real Sphere? Simon Gindikin

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The Complex Dual to the Real Sphere? Simon Gindikin WHATIS... ? the Complex Dual to the Real Sphere? Simon Gindikin The observation of this note is connected with dot-product corresponds to the same quadratic some modern considerations in integral geome- form. try.Atthesametimeitreturnsusbacktothe eraof For the real sphere S = Sn great projective geometry of Poncelet-Plücker and x2 + x2 + · · · + x2 = 1, to the understanding that some phenomena of real 1 2 n+1 geometry need a language of complex geometry. we consider its complexification CS, This era started with Poncelet’s discovery that cir- z2 + z2 + · · · + z2 = 1, z = x + iy ∈ Cn+1 cles can be defined as ellipses passing through two 1 2 n+1 universal imaginary points at infinity—the cyclic and complex horospheres E(ζ)—intersections of points. We consider here a canonical object dual CS by the hyperplanes to the real sphere S = Sn that was not considered ζ · z = ζ z + · · · + ζ z = 1, ζ · ζ = 0, ζ 6= 0. earlier, probably because it is complex. 1 1 n+1 n+1 Spherical and hyperbolic geometry are real So complex horospheres are parameterized by forms of the same complex geometry, but in many points of the complex cone C ⊂ Cn+1 without the respects hyperbolic geometry is richer than spheri- vertex. The crucial moment in such constructions cal geometry. In hyperbolic geometry, horospheres comes when one selects from all horospheres some (“spheres of infinite radius”) play an important that have a special relation with the real sphere. role, but they have no analogues in spherical ge- We suggest considering horospheres E(ζ) that do ometry. Our initial point is that it makes perfect not intersect the real sphere S and interpreting sense to consider complex horospheres on the real the manifold Sˆ ⊂ C of their parameters ζ as the sphere. dual object for the sphere S. Direct computation Let us start from the hyperbolic picture. We shows that the domain Sˆ on the cone C is defined realize hyperbolic space as the hyperboloid H = by the condition Hn ⊂ Rn+1, ξ2 + · · · + ξ2 < 1, ξ = ℜζ. 2 2 2 1 n+1 x1 − x2 − · · · − xn+1 = 1, This domain is invariant relative to the orthogonal relative to the action of the pseudoorthogonal group O(n+1) (but, of course, is inhomogeneous). ˆ group O(1; n). The dual object is the cone H, To support this interpretation we will state one 2 2 2 ξ1 − ξ2 − · · · − ξn+1 = 0, ξ 6= 0, analytic fact. Analytic dualities as consequences of without the vertex, where the group O(1; n) al- geometric dualities are important components of such considerations (the Radon transform and pro- so acts transitively. Points ξ ∈ Hˆ parameterize the horospheres, which are intersections of H by jective duality is the classic example). Let Hyp(S) ⊂ C the (isotropic) hyperplanes ξ · x = 1. Here the be the space of hyperfunctions on S S— functionals on the space O(S) of holomorphic C Simon Gindikin is Board of Governors Professor at Rut- functions on S in some neighborhoods of S— gers University. His email address is gindikin@math. and let O(S)ˆ be the space of holomorphic functions rutgers.edu. in Sˆ. 942 Notices of the AMS Volume 56, Number 8 Theorem 1. There is an O(n)-isomorphism be- interpret fˆ as boundary values of f . The operator tween Hyp(S) and O(S)ˆ . f → fˆ is compatible with the isomorphism in The- orem 2 above. Let C be the intersection of C by The operators, which establish the isomorphism z the hyperplane ζ · z = 1 and ω be a holomorphic in both directions, are explicit. If f ∈ Hyp(S) and (n−1)-form such that d(ζ ·z) ∧ω = µ. Let γ ⊂ C ζ ∈ Sˆ, then the evaluation of this functional on the z be any cycle homological to the sphere Sn−1. function ϕζ = 1/(1 − ζ · z), which is holomorphic in a neighborhood of S, gives fˆ (ζ) ∈ O(S)ˆ . To Theorem 3. We have construct the inverse operator we need an ana- fˆ ω = c(m, n)f (z), m = deg fˆ . logue of the Cauchy-Fantappie integral formula Zγ on CS, which makes it possible to extend func- In this formula we reconstruct the extensions tionals from the functions of the form ϕ to all ζ of spherical polynomials on the whole space. We holomorphic functions in neighborhoods of S. do not give the explicit value of the constant All regular functions and distributions are c. To make this formula similar to the Poisson contained in Hyp(S). Let µ be the invariant formula on H we need to use the homogeneity form of maximal degree on S: µ ∧ d(z · z) = dz. of fˆ to replace the integration in C by the inte- Then for any function ψ(z), z ∈ S, we con- z gration in a fixed section of C. Doing so will add sider the hyperfunction-functional (f [ψ], φ) = to the integrand a factor—a Poisson kernel. The ψ(z)φ(z)µ, φ ∈ O(S). We identify fˆ [ψ](ζ) = S new essential moment comes when we integrate Rˆ ψ(ζ). If ψ is holomorphic in some neighborhood not on the whole complex boundary but on any ˆ of S, then in the integral defining ψ(ζ) we can cycle there. Let us mention that the connections ˆ by deforming S extend ψ(ζ) holomorphically between spherical polynomials on S and homoge- ˆ outside of the domain S. If ψ is holomorphic on neous polynomials on the complex cone C were C S then ψˆ holomorphically extends to the whole discovered by Maxwell although he considered a of C. different isomorphism. Theorem 2. There is an O(n, C)-isomorphism be- There are interesting complex constructions tween O(CS) and O(C) that identifies the spaces of connecting with the hyperbolic geometry as well. polynomials on these manifolds. Here is one example. Let H+ be one sheet of the hy- perboloid H (x1 > 0). Let us consider its complex This isomorphism is surprising since complex neighborhood Crown(H) = {z = x + iy ∈ Cn+1, homogeneous manifolds CS and C are not iso- 2 2 2 z · z = 1, x1 − x2 − · · · − xn+1, x1 > 0}, which we morphic as homogeneous manifolds, nor are they will call the complex crown of H. It is biholomor- isomorphic as complex ones. There are some phically equivalent to the future tube. intermediate isomorphisms for spaces of holomor- phic functions on horospherically convex domains Theorem 4. All spherical functions on H+ admit D ⊂ CS (their complements are unions of horo- holomorphic extensions on Crown(H), anditisthe spheres). This situation is similar to the complex maximal joint holomorphy domain for these func- linear convexity of Martineau. It is essential that tions. the sphere S is horospherically convex compact. It All these constructions can be generalized to would be interesting to investigate horospherical- arbitrary compact symmetric spaces. ly convex compacts inside S as an example of the influence of complex geometry on real geometry. Further Reading In the isomorphism of Theorem 2, homoge- [1] S. Gindikin, Complex horospherical transform on neous polynomials on C correspond to spherical real sphere, Geometric Analysis of PDE and Sever- polynomials on S. Spherical polynomials are eigen- al Complex Variables, Contemporary Mathematics, functions of the Laplace-Beltrami operator on the number 368, Amer. Math. Soc., 2005, pp. 227–232. sphere. Similarly, we can consider spherical func- [2] , Horospherical Cauchy-Radon transform on tions on the hyperbolic space H. In the latter case compact symmetric spaces, Mosk. Math. J., 6(2) there is the Poisson integral reconstructing spher- (2006), 299–305. ical functions through their boundary values (we [3] , Harmonic analysis on symmetric mani- transfer to the bounded model in the intersection folds from the point of view of complex analysis, Japanese J. Math., 1 (2006), 87–105. of H by the hyperplane x1 = 1). Is there an analogue of the Poisson integral for spherical polynomials? Of course, S has no real boundary, but we can consider the complex boundary of CS, which we will identify with the projectivization B of the cone C. We extend spher- ical polynomials f (x) on Cn+1 and take restrictions fˆ to the cone C. They are homogeneous polyno- mials on C—sections of line bundles on B. We September 2009 Notices of the AMS 943.
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