Acta Mathematica Sinica-English Series最新文献

Let W be a real symplectic space and (G, G′) an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let (widetilde {rm{G}}) be the preimage of G in the metaplectic group (widetilde {{rm{Sp}}}({rm{W}})). Given an irreducible unitary representation Π of (widetilde {rm{G}}) that occurs in the restriction of the Weil representation to (widetilde {rm{G}}), let Θ_Π denote its character. We prove that, for a suitable embedding T of (widetilde {{rm{Sp}}}({rm{W}})) in the space of tempered distributions on W, the distribution T(Θ̌_Π) admits an asymptotic limit, and the limit is a nilpotent orbital integral. As an application, we compute the wave front set of Π′, the representation of (widetilde {{G^prime}}) dual to Π, by elementary means.

Let (mathfrak{g}) be a classical complex simple Lie algebra and (mathfrak{q}) be a parabolic subalgebra. Let M be a generalized Verma module induced from a one dimensional representation of (mathfrak{q}). Such M is called a scalar generalized Verma module. In this paper, we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand–Kirillov dimension of the corresponding highest weight modules.

Consider the polynomial differential system of degree m of the form

where μ and ν are real numbers such that ((mu^{2}+nu^{2})(mu+nu(m-2))(a_{1}^{2}+a_{2}^{2})ne 0,m > 2) and Ω_m−1(x,y) is a homogenous polynomial of degree m − 1. A conjecture, stated in J. Differential Equations 2019, suggests that when ν = 1, this differential system has a weak center at the origin if and only if after a convenient linear change of variable (x,y) → (X,Y) the system is invariant under the transformation (X,Y,t) → (−X,Y, −t). For every degree m we prove the extension of this conjecture to any value of ν except for a finite set of values of μ.

In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of G = GL_n(ℂ) possessing the minimal Gelfand–Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of G of type (n − 1,1). We give the transition matrix between the two bases for the corresponding coherent families.

K-UR, K-LUR and K-R are the generalizations of UR, LUR and R respectively, which are of great significance in Banach space theory. While in Orlicz–Lorentz function space (Lambda_{varphi,omega}^{circ}[0,gamma)) equipped with the Orlicz norm, the research methods of K-UR, K-LUR and K-R are much more complicated than those of UR, LUR and R. In this paper we obtain some criteria of K-UR, K-LUR and K-R of (Lambda_{varphi,omega}^{circ}[0,gamma)) by means of the norm of dual space and H_μ property of (Lambda_{varphi,omega}^{circ}[0,gamma)).

In this paper, we characterize weighted composition operators that preserve frames on the weighted Hardy spaces in the unit disk. In particular, we obtain the symbol properties of the bounded invertible weighted composition operators. Moreover, we establish the equivalence between bounded invertible operators and frame-preserving operators. Furthermore, we show that weighted composition operator preserves frames if and only if it preserves the Riesz bases property. Additionally, we investigate the weighted composition operators that preserve tight or normalized tight frames on the Dirichlet space.

In this paper, we prove that in a hyperconvex domain Ω in ℍⁿ, if a non-negative Borel measure is dominated by a quaternionic Monge–Ampère measure, then it is a quaternionic Monge–Ampère measure of a function in the class ({cal E}(Omega )).

In this paper, we study the projectively Ricci-flat general (α, β)-metrics within to a spray framework and also bring out the rich variety of behaviour displayed by an important projective invariant. Projective Ricci curvature is one of the essential projective invariant in Finsler geometry which has been introduced by Z. Shen. The projective Ricci curvature is defined as Ricci curvature of a projective spray associated with a given spray G on Mⁿ with a volume form dV on Mⁿ.

We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a polynomial W, with coefficients in a field of characteristic 2, is a square matrix Q of polynomial entries satisfying Q² = W · Id. We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances. Our main example is the Lagrangian submanifold ℝP² ⊂ ℂP² and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.

We explain how to construct a quantum deformation of a spectral curve associated to a tau-function of the KP hierarchy. This construction is applied to Witten–Kontsevich tau-function to give a natural explanation of some earlier work. We also apply it to higher Weil–Petersson volumes and Witten’s r-spin intersection numbers.