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In this paper we study algebro-geometric aspects of the Christoffel-Darboux kernels for classical orthogonal polynomials with rational coefficients. We find a novel connection between a projective curve defined by theChristoffel-Darboux kernel and a system of Diophantine equations, which was originally designed by Hausdorff (1909) for applications to Waring’s problem, and which is closely related to quadrature formulas in numerical analysis and Gaussian designs in algebraic combinatorics. We prove some nonsolvability results of such Hausdorff-type equations.

In this paper, we give a representation-theoretic interpretation of the specialization E w ∘ λ ( q , ∞ ) E_{w_\circ \lambda } (q,\infty ) of the nonsymmetric Macdonald polynomial E w ∘ λ ( q , t ) E_{w_\circ \lambda }(q,t) at t = ∞ t=\infty in terms of the Demazure submodule V w ∘ − ( λ ) V_{w_\circ }^- (\lambda ) of the level-zero extremal weight module V ( λ ) V(\lambda ) over a quantum affine algebra of an arbitrary untwisted type. Here, λ \lambda is a dominant integral weight, and w ∘ w_\circ denotes the longest element in the finite Weyl group W W . Also, for each x ∈ W x \in W , we obtain a combinatorial formula for the specialization E x λ ( q , ∞ ) E_{x \lambda } (q, \infty ) at t = ∞ t=\infty of the nonsymmetric Macdonald polynomial E x λ ( q , t ) E_{x \lambda } (q,t) and also a combinatorical formula for the graded character gch ⁡ V x − ( λ ) \operatorname {gch} V_{x}^- (\lambda ) of the Demazure submodule V x − ( λ ) V_{x}^- (\lambda ) of V ( λ ) V(\lambda ) . Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape λ \lambda .

Accardi and Cecchini generalized the concept of conditional expectations on von Neumann algebras. In this paper we give some conditions for strong convergence of increasing or decreasing martingales of Accardi and Cecchini's conditional expectations. Introduction. The study of conditional expectations and martingale convergence theorems in the operator algebraic framework was initiated by Umegaki's earlier works [14,15]. Since then, the martingale convergence theorems on von Neumann algebras have been developed by several authors. For example, the almost sure type martingale convergence was obtained by Lance [9] and Dang-Ngoc [4]. The strong martingale convergence was completed in [13] by one of the authors. The conditional expectation of a von Neumann algebra M onto a von Neumann subalgebra N does not generally exist given a faithful normal state (or semifinite weight) p on M. Indeed its existence is equivalent to the invariance of N under the modular automorphism group associated with 'p (see [12]). To recover this drawback of noncommutative conditional expectations, Accardi and Cecchini [1] generalized the concept of conditional expectations on von Neumann algebras by using the Tomita-Takesaki theory [11]. Accardi and Cecchini's conditional expectation E: M -3 N with respect to 'p always exists but is not necessarily a projection onto N and lacks a useful property that c(axb) = ac(x)b for a, b E N and x E M. Araki [2] established the convergence of modular operators and modular conjugations given an increasing net of von Neumann subalgebras and a faithful normal state. By extending the arguments in [2], we have obtained in [8] the martingale type convergence of modular automorphism groups under an increasing or decreasing net of von Neumann subalgebras and a faithful normal semifinite weight. On the same lines, we investigate in this paper the strong convergence of martingales of Accardi and Cecchini's conditional expectations (called here generalized conditional expectations). In ? 1 of this paper we state the definition of the generalized conditional expectation with respect to a faithful normal semifinite weight and give easy technical Received by the editors March 15, 1983 and, in revised form, May 22, 1983. 1980 Mathematics Subject Classification. Primary 46L10.

Let F n F_n be a free group of rank n n and F n N F_n^N the quotient group of F n F_n by a subgroup [ Γ n ( 3 ) , Γ n ( 3 ) ] [ [ Γ n ( 2 ) , Γ n ( 2 ) ] , Γ n ( 2 ) ] [\Gamma _n(3), \Gamma _n(3)][[\Gamma _n(2),\Gamma _n(2)],\Gamma _n(2)] , where Γ n ( k ) \Gamma _n(k) denotes the k k -th subgroup of the lower central series of the free group F n F_n . In this paper, we determine the group structure of the graded quotients of the lower central series of the group F n N F_n^N by using a generalized Chen’s integration in free groups. Then we apply it to the study of the Johnson homomorphisms of the automorphism group of F n F_n . In particular, under taking a reduction of the target of the Johnson homomorphism induced from a quotient map F n → F n N F_n \rightarrow F_n^N , we see that there appear only two irreducible components, the Morita obstruction S k H Q S^k H_{\mathbf {Q}} and the Schur-Weyl module of type H Q [ k − 2 , 1 2 ] H_{\mathbf {Q}}^{[k-2, 1^2]} , in the cokernel of the rational Johnson homomorphism τ k , Q ′ = τ k ′ ⊗ i d Q \tau _{k, \mathbf {Q}}’=\tau _k’ \otimes \mathrm {id}_{\mathbf {Q}} for k ≥ 5 k \geq 5 and n ≥ k + 2 n \geq k+2 .

In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type. This conjecture was proved for closed S in [WZ2,WZ3,FH,M4] and for open S in [M4]. It was proved for the other orbits in [M5] when G_R is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed K_C-orbit when G_R is of Hermitian type. Thus the conjecture is completely solved affirmatively., 15 pages