On the Geometry of Homogeneous Spaces

1 Under the microscope 1.1 Osculating spaces of homogeneous varieties Let G be a simply connected complex semisimple Lie group, g its Lie algebra, and g its tensor algebra. The universal envelopping algebra of U(g) is the quotient of this tensor algebra by the ideal generated by the elements xy ? yx ? x; y], x; y 2 g. This quotient algebra inherits a ltration from the natural grading of the tensor algebra. Forming the associated graded algebra, we have an isomorphism grad k U(g) = U(g) k =U(g) k?1 = S k g; where S g denotes the symmetric algebra. Fix a maximal torus T and a Borel subgroup B of G containing T. Our convention is that B is generated by the negative roots, and we write the corresponding root space decomposition of g as g = t M 2 + (g g ?); where + denotes the set of positive roots. Let V be an irreducible G-module with highest weight , and v 2 V a highest weight vector. The induced action of g extends to the universal envelopping algebra, and we get an induced ltration of V whose k-th term is V (k) = U(g) k v : Let x = x be the line of V generated by v , and X = G=P PV its G-orbit. Here P is the stabilizer of x , it is a parabolic subgroup of G. The tangent bundle TX is a homogeneous bundle and we identify T x X with the associated P-module g=p. The osculating spaces and the fundamental forms have a simple representation theoretic interpretation: Proposition 1.1 The cone over the k-th osculating space at x is ^ T k x X = V (k) ; so that N k = V (k) =V (k?1). Moreover, there is a commutative diagram where the bottom horizontal map is the k-th fundamental form at x .