Masaki Kashiwara

TRANSFORMATION GROUPS FOR SOLITON EQUATIONS

Abstract A new approach to soliton equations, based on τ functions (or Hirotas dependent variables), vertex operators and the Clifford algebra of free fermions, is applied to study a new hierarchy of Kadomtsev-Petviashvili type equations (the BKP hierarchy). The infinite-dimensional orthogonal group acts on the space of BKP τ-functions. The Sawada-Kotera equation is obtained as a reduction of BKP. Its infinitesimal transformations constitute the Euclidean Lie Algebra A2(2).

Crystalizing the

For an irreducible representation of theq-analogue of a universal enveloping algebra, one can find a canonical base atq=0, named crystal base (conjectured in a general case and proven forAn, Bn, Cn andDn). The crystal base has a structure of a colored oriented graph, named crystal graph. The crystal base of the tensor product (respectively the direct sum) is the tensor product (respectively the union) of the crystal base. The crystal graph of the tensor product is also explicitly described. This gives a combinatorial description of the decomposition of the tensor product into irreducible components.

q

In [7], D. Kazhdan and G. Lusztig gave a conjecture on the multiplicity of simple modules which appear in a Jordan-H61der series of the Verma modules. This multiplicity is described in the terms of Coxeter groups and also by the geometry of Schubert cells in the flag manifold (see [8]). The purpose of this paper is to give the proof of their conjecture. The method employed here is to associate holonomic systems of linear differential equations with R.S. on the flag manifold with Verma modules and to use the correspondance of holonomic systems and constructible sheaves. Let G be a semi-simple Lie group defined over • and g its Lie algebra. We take a pair (B,B-) of opposed Borel subgroups of G and let T=B~Bbe a maximal torus and W the Weyl group. Let b, b and f the corresponding Lie algebras and 9l the nilpotent radical of b. Let us denote by Jg the category of holonomic systems with R.S. on X=G/B whose characteristic varieties are contained in the union of the conormal bundles of Xw=BWB/B (we W). On the other hand, let (9 denote the category of finitely-generated U(g)-modules which are Tl-finite. By (gtrlv we denote the category of the modules in (9 with the trivial central character. We shall prove that J / / a n d (~trlv are equivalent by the correspondances S0l ~--*F(X;gJI) and M~--~,~| Here ~ is the sheaf of differential operators on X. Let us denote by M w the Verma module with highest weight -w(p)-p and let ~Jl w be the dual g -module of ~codimXwt/~ ~ Then, ~ w and Mw ~ [X~] ~,~X] correspond by the above correspondence. For any 9 J l e ~ , we can calculate the character of F(X; 93l) by the formula

-analogue of universal enveloping algebras

The hierarchy of Kadomtsev-Petvisahvili (KP) equation is studied on the basis of free fermion operators. Particular emphasis is laid on relating the operator approach to the Grassmann formulation of M. and Y. Sato. A new bilinear identity for wave functions is derived, and is shown to generate the series of Hirota bilinear equations for the KP hierarchy. Extension to the multicomponent case is also discussed.

Kazhdan-Lusztig Conjecture and Holonomic Systems.

We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample to the conjecture of Kazhdan--Lusztig on the irreducibility of the characteristic variety of the intersection cohomology sheaves associated with the Schubert cells of type A and also to the similar problem asked by Lusztig on the characteristic variety of the perverse sheaves corresponding to canonical bases.

Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–

We study the properties of level zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by Akasaka and the present author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremal weight. The universal extremal weight module with level zero fundamental weight as an extremal weight is irreducible, and isomorphic to the affinization of an irreducible finite-dimensional module.

Geometric construction of crystal bases

The theory of ordinary differential equations with regular singularities has been well studied and has become one of the most fundamental fields of analysis. But the regular singularities of differential equations with several variables have not been studied in satisfactory form, although they appear in several important fields of mathematics. An example is the Laplacian of symmetric spaces of noncompact type. The typical one is the Laplacian

On level-zero representation of quantized affine algebras

A series of new hierarchies of soliton equations are presented on the basis of the Kadomtsev-Petviashvili (KP) hierarchy. In contrast to the KP case, which admits GL( ∞) as its transformation group, these new hierarchies are shown to correspond to O( ∞) or Sp( ∞). Soliton, rational and quasi-periodic solutions are constructed.

Systems of differential equations with regular singularities and their boundary value problems

On etudie la theorie geometrique des representations du groupe de Weyl du point de vue des systemes holonomes sur des algebres de Lie

KP Hierarchies of Orthogonal and Symplectic Type : Transformation Groups for Soliton Equations VI

A decomposition of the level-oneq-deformed Fock space representations ofUq(sln) is given. It is found that the action ofU′q(sln) on these Fock spaces is centralized by a Heisenberg algebra, which arises from the center of the affine Hecke algebraĤN in the limitN → ∞. Theq-deformed Fock space is shown to be isomorphic as aU′q(sln)-Heisenberg-bimodule to the tensor product of a level-one irreducible highest weight representation ofU′q(sln) and the Fock representation of the Heisenberg algebra. The isomorphism is used to decompose theq-wedging operators, which are intertwiners between theq-deformed Fock spaces, into constituents coming fromU′q(sln) and from the Heisenberg algebra.