We consider the graded space R of syzygies for the coordinate algebra A of projective variety X = G/P embedded into projective space as an orbit of the highest weight vector of an irreducible representation of semisimple complex Lie group G. We show that R is isomorphic to the Lie algebra cohomology H = H • (L 2 , C), where L 2 is graded Lie subalgebra of the graded Lie s-algebra L = L1 ⊕ L 2 Koszul dual to A. We prove that the isomorphism identifies the natural associative algebra structures on R and H coming from their Koszul and Chevalley DGA resolutions respectively. For subcanonically embedded X a Frobenius algebra structure on the syzygies is constructed. We illustrate the results by several examples including the computation of syzygies for the Plücker embeddings of grassmannians Gr(2, N ). Lie algebra cohomology. the first author thanks scientific school supporting grant NSh-8004.2006.2 and the second author -RFBR grant 04-02-16538 for partial support. 1 see [1], [2] 2 see [9], [32], [33], [34], [35], etc. 3 in the sense of [17] 4In fact, besides the grassmannians, Berkovits and others consider also a singular quadratic cone over Griso(5, 11), which is especially interesting for physicists because it serves some model of gravity. But the mathematical statements here are rather unclear to us and most likely require appropriate A∞ enchantment of the smooth case. In these notes we restrict ourselves to the smooth homogeneous spaces only. 1 ̺ p = 1 2 α∈∆p α . Besides B, we will sometimes consider its opposite Borel subgroup B ′ . If B = T ⋉ U , where U is the unipotent part of B, then B ′ = T ⋉ U ′ and U ′ T U is a dense open subset in G. Similarly, we will write ̺ ′ = −̺ for the half sum of all the negative roots. 1.1.2. Vector bundles on G/P . With any representation E of P is associated a vector bundle E = G × P E over X = G/P with the fiber E. Its total space consists of pairs (g, e) ∈ G × E modulo the equivalence (gp, e) ∼ (g, pe) for p ∈ P . Global sections X ⊂ s f -E are naturally identified with functions G f -E such that (1.5) pf (g) = f (gp −1 ) for all p ∈ P . 7 recall that (̺, αi) = 1 for any simple root αi Proposition 1.2.2. If X = G/P is a d-dimensional SHW-orbit with ω X = O X (−N ), then all the non zero cohomologies H q (X, O X (k)) are only H 0 (X, O X (m)) and H d (X, O X (−N − m)) , where m 0 in the both cases. Proof. Since λ is a highest weight, −mλ ∈ C low lies in the lowest weights chamber for all m 0. So, for all O X (m) = O X −mλ we have H 0 (X, O X (m)) = 0 and H q (X, O X (m)) = 0 when q > 0. By the Serre duality, this implies H d (X, O X (−N − m)) = H 0 (X, O X (m)) * = 0 and vanishing of all H q (X, O X (−N − m)) with q = d.11 the formulation most commonly used in the representation theory (see, for example, [21]) actually describes G-modules H q (O µ ) in terms highest weights but the underlying homogeneous space is always taken to be G/B ′ , that is the lowest weight vector orbit 12 here the length should be defined w. r. t. the reflections by the walls of the lowest chamber C low
