Dedicated to Jiro Sekiguchi on the occasion of his sixtieth birthday.Abstract. Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair.We take parabolic subgroups P of G and Q of K respectively and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it.In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ⊂ G or Q ⊂ K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.
Single and double asymmetric inductions were achieved by the reduction of pyruvates and benzoylformates with 2,6-dimethyl-3,5-dialkoxycarbonyl-1,4-dihydropyridine in the presence of a mono-ionised zinc species ; differentiation is made between enantiotopic faces of the substrate carbonyl by diastereotopic hydrogen, and between diastereotopic faces by equivalent and diastereotopic hydrogen.ASYMMETRIC reduction of carbonyl compounds by the use of dihydropyridine derivatives is of particular interest from the viewpoint of models of the NAD(P)H reaction as well as for their synthetic utility.
The notion of two-sided cell, which was originally introduced by A. Joseph and reformulated by D. Kazhdan and G. Lusztig, has played an important role in the representation theory. Results concerning them have been obtained by very deep and sometimes ad hoc arguments. Here we introduce certain polynomial invariants for irreducible representations of Weyl groups. Our invariants are easily calculated, and the calculational results show that they almost determine the two-sided cells. Moreover, the factorization pattern of our polynomial invariants seems to be controlled by the natural parameter set MðGÞ of each two-sided cell.wðwÞt l 0 ðwÞ ð0:2Þafter such modification, where w ¼ trace r.Concerning the function (0.2), we can make the following observation by a case study. Let w, w 0 A W 4 , where W 4 is the set of irreducible characters of W. Then the
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