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Deforming commutative algebras in the lower triangular (ℤ×ℤ)-matrices yields lower triangular Toda hierarchies and their associated nonlinear equations. Like for their counterpart in the ring of pseudodifferential operators, the KP-hierarchy, one also has for these hierarchies a geometric picture: certain infinite chains of subspaces in an separable Hilbert space provide solutions of lower triangular Toda hierarchies. The KP-hierarchy and its multi-component version contain many interesting subsystems, like e.g. the nth Gelfand–Dickey hierarchy and the AKNS-hierarchy. In this paper one considers analogues of these two subsystems in the context of the lower triangular Toda hierarchies and a geometric description of solutions to both type reductions is given.

Commutative subalgebras of the complex -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 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.

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 k-open subgroup of the fixed point group of σ, Gk (resp. Hk) the set of k-rational points of G (resp. H) and Gk/Hk the corresponding symmetric k-variety. A representation induced from a parabolic k-subgroup of G generically contributes to the Plancherel decomposition of L2(Gk/Hk) if and only if the parabolic k-subgroup is σ-split. So for a study of these induced representations a detailed description of the Hk-conjucagy classes of these σ-split parabolic k-subgroups is needed. In this paper we give a description of these conjugacy classes for general symmetric kvarieties. 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.

In this paper, we present several instances where infinite-dimensional flag varieties and their holomorphic line bundles play a role in integrable systems. As such, we give the correspondence between flag varieties and Darboux transformations for the KP hierarchy and the nth KdV hierarchy. We construct solutions of the nth MKdV hierarchy from the space of periodic flags and we treat the geometric interpretation of the Miura transform. Finally, we show how the group extension connected with these line bundles shows up at integrable deformations of linear systems on ℙ1(ℂ).

In this paper one considers the problem of finding solutions to a number of Toda-type hierarchies. All of them are associated with a commutative subalgebra of the k×k-matrices. The first one is formulated in terms of upper triangular ℤ×ℤ-matrices, the second one in terms of lower triangular ones and the third is a combination of the two foregoing types. It is shown that in an appropriate group setting solutions of the linearization of these Lax equations can be constructed by using a Birkhoff-type decomposition in the relevant group.

To a compact Riemann surface, a point on the surface, a local coordinate around this point, and a holomorphic transfer function of degree m is associated a point in the Grassmann manifold of l2(Z)m. This leads to solutions of the nonlinear hierarchies corresponding to maximal Abelian subalgebras of End(Cm).