A representation of the quantum toroidal algebra of type \mathfrak{sl}_N is defined on every integrable irreducible highest weight module of the quantum affine algebra of type \mathfrak{gl}_N. The q-version of the level-rank duality giving the reciprocal decomposition of the q-Fock space with respect to mutually commutative actions of U_q'(\widehat{\mathfrak{gl}}_N) of level L and U_q'(\widehat{\mathfrak{sl}}_L) of level N is described.

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The quantum complex Grassmannian U_q/K_q of rank l is the quotient of the quantum unitary group U_q=U_q(n) by the quantum subgroup K_q=U_q(n−l)×U_q(l). We show that (U_q, K_q) is a quantum Gelfand pair and we express the zonal spherical functions, i.e. K_q-biinvariant matrix coefficients of finite-dimensional irreducible representations of U_q, as multivariable little q-Jacobi polynomials depending on one discrete parameter. Another type of biinvariant matrix coefficients is identified as multivariable big q-Jacobi polynomials. The proof is based on earlier results by Noumi, Sugitani and the first author relating Koornwinder polynomials to a one-parameter family of quantum complex Grassmannians, and certain limit transitions from Koornwinder polynomials to multivariable big and little q-Jacobi polynomials studied by Koornwinder and the second author.

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We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrödinger equations of the following form
(A)   iu_t+u_xx=\mathscr{N}(u, \bar{u}, u_x, \bar{u}_x),   t∈R, x∈R;   u(0, x)=u₀(x),   x∈R,
where
\mathscr{N}(u, \bar{u}, u_x, \bar{u}_x)=\mathscr{K}₁ | u | ²u+i\mathscr{K}₂ | u | ²u_x+i\mathscr{K}₃u²\bar{u}_x+\mathscr{K}₄ | u_x | ²u+\mathscr{K}₅\bar{u}u_x²+i\mathscr{K}₆ | u_x | ²u_x,
\mathscr{K}_j=\mathscr{K}_j(|u|²), \mathscr{K}_j(z)∈C³(R⁺); \mathscr{K}_j(z)=λ_j+O(z), as z→+0, \mathscr{K}₁, \mathscr{K}₆ are real valued functions. Here the parameters λ₁, λ₆∈R, and λ₂, λ₃, λ₄, λ₅∈C, are such that λ₁−λ₃∈R and λ₄−λ₅∈R. If \mathscr{K}₅(z)=\frac{λ₅}{1+μz} and λ₅=μ=±1, \mathscr{K}₁=\mathscr{K}₂=\mathscr{K}₃=\mathscr{K}₄=\mathscr{K}₆=0 equation (A) appears in the classical pseudospin magnet model [9]. We prove that if u₀∈H^{3, 0}∩H^{2, 1} and the norm ||u₀||_{3, 0}+||u₀||_{2, 1}=ε is sufficiently small, then the solution of (A) exists globally in time and satisfies the sharp time decay estimate ||u(t)||_{2, 0, ∞}≤Cε(1+|t|)^−1/2, where ||φ||_{m, s, p}=||(1+x²)^s/2(1−∂_x²)^m/2φ||_L^p, H_p^{m, s}={φ∈S';||φ||_{m, s, p}<∞}. Furthermore we prove existence of modified scattering states and nonexistence of nontrivial scattering states. Our method is based on a certain gauge transformation and an appropriate phase function.

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We study the blowing up π : \bar{X}→X of a 3-dimensional terminal singularity X of index m≥2 such that the exceptional locus of π consists of a prime divisor E with discrepancy 1/m. A complete classification of such blowing ups is given and it is proved that these correspond to weighted blow ups by a certain kind of maximal weights except for the case where X is of type (cD/2). We shall treat the (cD/2) case later. These also give examples of contractions of extremal rays which contract a divisor to a point.

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