xn x2 x2 x2 1 1 n the klein model

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Unformatted text preview: n = {(x0 , . . . , xn ) ∈ Rn+1 , −x2 + x2 + . . . + x2 = −1, x0 > 0} 0 1 n with Iso(X ) equal to the subgroup O(n, 1)+ of index 2 of the orthogonal group O(n, 1) which consists of transformations of spinor norm 1, that is, transformations that can be written as a product of reflections in vectors with positive norm. The Riemannian metric is induced by the hyperbolic metric in Rn+1 , ds2 = −dx2 + dx2 + . . . + dx2 . 0 1 n We will be using a projective model of H n , considering H n as the image of the subset C = {(x0 , . . . , xn ) ∈ Rn+1 , −x2 + x2 + . . . + x2 < 0} 0 1 n REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 7 in the projective space Pn (R). The isometry group of the projective model is naturally identified with the group PO(n, 1). By choosing a representative of a point from H n with x0 = 1, we can identify H n with the real ball K n : {(x1 , . . . , xn ) : |x|2 = x2 + . . . + x2 < 1} 1 n (the Klein model ). The metric is given by ds2 = 1 1 − |x|2 n n dx2 + i i=1 1 ( xi dxi )2 . (1 − |x|2 )2 i=1 The closure of H n in Pn (R) is equal to the image of the set ¯ C = {(x0 , . . . , xn ) ∈ Rn+1 , −x2 + x2 + . . . + x2 ≤ 0} 0 1 n in Pn (R). The boundary is called the absolute. One defines the notion of a hyperplane in a space of constant curvature. If X = E n , a hyperplane is an affine hyperplane. If X = S n , a hyperplane is the intersection of S n with a linear hyperplane in Rn+1 (a great circle when n = 2). If X = H n , a hyperplane is the nonempty intersection of H n with a projective hyperplane in Pn (R). Each hyperplane H in E n is a translate a + L = {x + a, x ∈ L} of a unique ˜ linear hyperplane H in the corresponding standard euclidean space V = Rn . If X n = S n or H n , then a hyperplane H is uniquely defined by a linear hyperplane ˜ H in V = Rn+1 equipped with the standard symmetric bilinear form of Sylvester signature (t+ , t− ) = (n + 1, 0) or (n, 1). Any point x ∈ V can be written uniquely in the form x = h + v, ˜ where h ∈ H and v ∈ V is orthogonal to H . We define a reflection with mirror hyperplane H by the formula rH (x) = h − v. One can also give a uniform definition of a hyperplane in a space of constant curvature as a totally geodesic hypersurface and define a reflection in such a space as an isometric involution whose set of fixed points is a hyperplane. Let H be a hyperplane in X n . Its complement X n \ H consists of two connected components. The closure of a component is called a halfspace. A reflection rH permutes the two halfspaces. One can distinguish the two halfspaces by a choice of one of the two unit vectors in V orthogonal to the corresponding linear hyperplane ˜ H . We choose it so that it belongs to the corresponding halfspace. For any vector v perpendicular to a hyperplane in H n we have (v, v ) > 0 (otherwise the intersection of the hyperplane with H n is empty). + + Let H1 , H2 be two halfspaces, and e1 , e2 be the corresponding unit vectors. If + + − − n X = H ,...
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