In this paper we give a geometric description in terms of the Grassmann manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the vector kconstrained KP hierarchy. We also show in a geometric way that these hierarchies are equivalent to Krichever's general rational reductions of the KP hierarchy.
Abstract. Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, σ an involution of G defined over k, H a kopen subgroup of the fixed point group of σ, G k (resp. H k ) the set of krational points of G (resp. H) and G k /H k the corresponding symmetric kvariety. A representation induced from a parabolic k-subgroup of G generically contributes to the Plancherel decomposition of L 2 (G k /H k ) if and only if the parabolic k-subgroup is σ-split. So for a study of these induced representations a detailed description of the H k -conjucagy classes of these σ-split parabolic ksubgroups is needed.In this paper we give a description of these conjugacy classes for general symmetric k-varieties. This description can be refined to give a more detailed description in a number of cases. These results are of importance for studying representations for real and p-adic symmetric k-varieties.
We study the continuous, desingularized Newton method for Weierstrass' }-functions. This leads to a family of autonomous differential equations in the plane, which depends on two complex parameters ! 1 and ! 2 . For the associated flows there are, up to conjugacy, precisely three possibilities. These are determined by the form of the parallelogram spanned by ! 1 and ! 2 : square, rectangular but not square, and non-rectangular.
Commutative subalgebras of the complex k × k-matrices are known to generate both matrix and Toda-type hierarchies. In this paper a certain class of infinite chains of closed subspaces of a separable Hilbert space H will be introduced. To each such a flag one associates a sequence of solutions of the matrix hierarchy related to this subalgebra. They compose to a solution of the lower triangular Toda hierarchy corresponding to the transposed algebra. Both solutions can be expressed in determinants of suitable Fredholm operators, the so-called τ -functions. These last functions also have a geometric interpretation in terms of line bundles over the flagvariety. They measure the failure of equivariance w.r.t. to the commuting flows of certain global sections.
Mathematics Subject Classification (2000)17B80 · 37K10 · 58B25 · 81R12
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.