By using a hodograph transformation, we calculate a part of the Chern-Moser invariants on the boundary of tube domains. As an application of this result, we give another proof of a result of Yang on the characterization of tube domains with a spherical boundary.
We introduce a new measure on partitions. We assign to each partition λ a probability Sλ(x; t)sλ(y)/Zt where sλ is the Schur function, Sλ(x; t) is a generalization of the Schur function defined by Macdonald (Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford, 1995) and Zt is a normalization constant. This measure, which we call the t-Schur measure, is a generalization of the Schur measure (A. Okounkov. Infinite wedge and random partitions. Selecta Math. (N.S.) 7 (2001), 57-81) and is closely related to the shifted Schur measure studied by Tracy and Widom (A limit theorem for shifted Schur measures. Preprint (2002), math.PR/0210255) for a combinatorial viewpoint. We prove that a limit distribution of the length of the first row of a partition with respect to t-Schur measures is given by the Tracy-Widom distribution, i.e. the limit distribution of the largest eigenvalue suitably centered and normalized in the Gaussian unitary ensemble.
We study the special values at s = 2 and 3 of the spectral zeta function ζQ(s) of the non-commutative harmonic oscillator Q(x, Dx) introduced in A. Parmeggiani and M. Wakayama (Proc. Natl Acad. Sci. USA 98 (2001), 26-31; Forum Math. 14 (2002), 539-604). It is shown that the series defining ζQ(s) converges absolutely for Re s > 1 and further the respective values ζQ(2) and ζQ(3) are represented essentially by contour integrals of the solutions, respectively, of a singly confluent Heun ordinary differential equation and of exactly the same but an inhomogeneous equation. As a by-product of these results, we obtain integral representations of the solutions of these equations by rational functions.
Let Pn be the real n-dimensional projective space. We determine the group structure of the self-homotopy set of the double suspension of Pn where n is 3, 4, 5 and 6 using the ideas and methods of the second author (The suspension order of the real even dimensional projective space, J. Math. Kyoto Univ. 43(4) (2003), 755-769).
We evaluate the integral appearing in the solution of the Knizhnik-Zamolodchikov equation associated with the vector representation of the simple Lie algebra g of type Bn, Cn or Dn. This integral, which can be considered as a generalization of the Beta integral, is expressed in terms of an alternating product of the Gamma function.
We study the homotopy types of the spaces of free algebraic loops on real projective spaces. In particular, we show that the Morse theoretic principle holds for these spaces in the stable homotopy category.
We study, for a model class of classical pseudodifferential operators with symplectic characteristics of multiplicity k, necessary and sufficient conditions for the hypoellipticity with loss of r + k/2 derivatives (r > 0).