We establish the sharp upper bounds on the indexes for most of the twisted flag varieties under the spin groups ${\operatorname {\mathrm {Spin}}(n)}$ .

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Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem. * Supported in part by the Simons Foundation

Let G/B be a flag variety over C, where G is a simple algebraic group with a simply laced Dynkin diagram, and B is a Borel subgroup. We say that the product of classes of Schubert divisors in the Chow ring is multiplicity free if it is possible to multiply it by a Schubert class (not necessarily of a divisor) and get the class of a point. In the present paper we find the maximal possible degree (in the Chow ring) of a multiplicity free product of classes of Schubert divisors. PreliminariesWe denote the subset of positive roots by Φ + , and the set of simple roots by Π. We enumerate simple roots as in [1]. Denote the fundamental weight corresponding to a simple root α i by ̟ i . We choose the scalar multiplication on Φ so that the scalar square of each simple root is 2. The scalar product of two roots α and β is denoted by (α, β). Note that with this choice of scalar multiplication, we can use a simple formula for reflection: usually, we writeBut with our choice of scalar product, we can writeMore generally, for two arbitrary vectors v, w in the ambient space of Φ, w = 0, denoteThen, with our choice of the scalar product, we have (•, α) = •, α for each α ∈ Φ. We use the following Pieri formula:

Let G be a semisimple algebraic group whose decomposition into a product of simple components does not contain simple groups of type A, and P ⊆ G be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples (G, P, n) such that (a) there exists an open G-orbit on the multiple flag variety G/P × G/P × . . . × G/P (n factors), (b) the number of G-orbits on the multiple flag variety is finite.