Abstract. Let Hg and Dg be the Siegel upper half plane and the gen-eralized unit disk of degree g... more Abstract. Let Hg and Dg be the Siegel upper half plane and the gen-eralized unit disk of degree g respectively. Let C(h,g) be the Euclidean space of all h × g complex matrices. We present a partial Cayley trans-form of the Siegel–Jacobi disk Dg × C(h,g) onto the Siegel–Jacobi space Hg × C(h,g) which gives a partial bounded realization of Hg × C(h,g) by Dg × C(h,g). We prove that the natural actions of the Jacobi group on Dg×C(h,g) and Hg×C(h,g) are compatible via a partial Cayley transform. A partial Cayley transform plays an important role in computing differ-ential operators on the Siegel–Jacobi disk Dg×C(h,g) invariant under the natural action of the Jacobi group on Dg × C(h,g) explicitly.
The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometricall... more The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space. To the memory of my teacher, Professor Shoshichi Kobayashi Table of Contents 1. Introduction 2. Invariant Metrics and Laplacians on the Siegel-Jacobi Space 3. Invariant Differential Operators on the Siegel-Jacobi Space 4. The Partial Cayley Transform 5. Invariant Metrics and Laplacians on the Siegel-Jacobi Disk 6. A Fundamental Domain for the Siegel-Jacobi Space 7. Jacobi Forms 8. Singular Jacobi Forms 9. The Siegel-Jacobi Operator 10. Construction of Vector-Valued Modular Forms from Jacobi Forms 11. Maass-Jacobi Forms 12 The Schrödinger-Weil Representation 13. Final Remarks and Open Problems
In this paper, we introduce the concept of stable automorphic forms on semisimple algebraic group... more In this paper, we introduce the concept of stable automorphic forms on semisimple algebraic groups. We use the stability of automorphic forms to study infinite dimensional arithmetic varieties.
For two positive integers m and n, we let P_n be the open convex cone in R^n(n+1)/2 consisting of... more For two positive integers m and n, we let P_n be the open convex cone in R^n(n+1)/2 consisting of positive definite n x n real symmetric matrices and let R^(m,n) be the set of all m x n real matrices. In this article, we investigate differential operators on the non-reductive manifold P_n × R^(m,n) that are invariant under the natural action of the semidirect product group GL(n,R) R^(m,n) on the Minkowski-Euclid space P_n × R^(m,n). These invariant differential operators play an important role in the theory of automorphic forms on GL(n,R) R^(m,n) generlaizing that of automorphic forms on GL(n,R).
In this paper, we discuss the theory of the Siegel modular variety in the aspects of arithmetic a... more In this paper, we discuss the theory of the Siegel modular variety in the aspects of arithmetic and geometry. This article covers the theory of Siegel modular forms, the Hecke theory, a lifting of elliptic cusp forms, geometric properties of the Siegel modular variety, (hypothetical) motives attached to Siegel modular forms and a cohomology of the Siegel modular variety.
Functoriality conjecture is one of the central subjects of the present-day mathematics. Functoria... more Functoriality conjecture is one of the central subjects of the present-day mathematics. Functoriality is the profound problem formulated by Robert Langlands in the late 1960s in order to establish nonabelian class field theory. In this expository paper, we describe the Langlands-Shahidi method, the local and global Langlands conjectures and the converse theorems which are powerful tools for the establishment of functoriality of some important cases, and survey the interesting results related to functoriality conjecture.
In this article, we prove a decomposition theorem on differential polynomials of theta functions ... more In this article, we prove a decomposition theorem on differential polynomials of theta functions of high level.
In this article, we prove that a lattice representation of the Heisenberg group H_ R^(g,h) associ... more In this article, we prove that a lattice representation of the Heisenberg group H_ R^(g,h) associated to a lattice L and a positive definite symmetric half-integral matrix M of degree h is unitarily equivalent to the direct sum of (det 2M)^g copies of the Schroedinger representation of H_ R^(g,h).
In this paper, we study the right regular representation of a finite group G on the vector space ... more In this paper, we study the right regular representation of a finite group G on the vector space consisting of vector valued functions on Γ G with a subgroup Γ of G and give a trace formula using the work of M.-F. Vigneras.
Let H_g and D_g be the Siegel upper half plane and the generalized unit disk of degree g respecti... more Let H_g and D_g be the Siegel upper half plane and the generalized unit disk of degree g respectively. Let C^(h,g) be the Euclidean space of all h× g complex matrices. We present a partial Cayley transform of the Siegel-Jacobi disk D_g× C^(h,g) onto the Siegel-Jacobi space H_g× C^(h,g) which gives a partial bounded realization of H_g× C^(h,g) by D_g× C^(h,g). We prove that the natural actions of the Jacobi group on D_g× C^(h,g) and H_g× C^(h,g) are compatible via a partial Cayley transform. A partial Cayley transform plays an important role in computing differential operators on the Siegel-Jacobi disk D_g× C^(h,g) invariant under the natural action of the Jacobi group on D_g× C^(h,g) explicitly.
For a fixed positive integer g, we let P_g = {Y∈ R^(g,g) | Y= ^tY>0 } be the open convex cone ... more For a fixed positive integer g, we let P_g = {Y∈ R^(g,g) | Y= ^tY>0 } be the open convex cone in the Euclidean space R^g(g+1)/2. Then the general linear group GL(g, R) acts naturally on P_g by A Y= AY ^tA (A∈ GL(g, R), Y∈ P_g). We introduce a notion of polarized real tori. We show that the open cone P_g parametrizes principally polarized real tori of dimension g and that the Minkowski domain R_g= GL(g, Z) P_g may be regarded as a moduli space of principally polarized real tori of dimension g. We also study smooth line bundles on a polarized real torus by relating them to holomorphic line bundles on its associated polarized real abelian variety.
In this paper, we introduce the concept of stable automorphic forms for semisimple algebraic grou... more In this paper, we introduce the concept of stable automorphic forms for semisimple algebraic groups and use the stability of automorphic forms to study infinite dimensional arithmetic quotients.
Let D_n be the generalized unit disk of degree n. In this paper, we find Riemannian metrics on th... more Let D_n be the generalized unit disk of degree n. In this paper, we find Riemannian metrics on the Siegel-Jacobi disk D_n × C^(m,n) which are invariant under the natural action of the Jacobi group explicitly and also compute the Laplacians of these invariant metrics explicitly. These are expressed in terms of the trace form. We give a brief remark on the theory of harmonic analysis on the Siegel-Jacobi disk D_n × C^(m,n).
For two positive integers m and n, we let H n be the Siegel upper half plane of degree n and let ... more For two positive integers m and n, we let H n be the Siegel upper half plane of degree n and let C (m,n) be the set of all m × n complex matrices. In this article, we study differential operators on the Siegel-Jacobi space H n × C (m,n) that are invariant under the natural action of the Jacobi group Sp(n, R) ⋉ H (n,m) R on H n × C (m,n) , where H (n,m) R denotes the Heisenberg group. We give some explicit invariant differential operators. We present important problems which are natural. We give some partial solutions for these natural problems.
Abstract. This article is a continuation of the paper [21]. In this paper we deal with Maass-Jaco... more Abstract. This article is a continuation of the paper [21]. In this paper we deal with Maass-Jacobi forms on the Siegel-Jacobi space H Cm, where H denotes the Poincaré upper half plane and m is any positive integer. 1.
Abstract. In this paper, we discuss the theory of the Siegel modular variety in the aspects of ar... more Abstract. In this paper, we discuss the theory of the Siegel modular variety in the aspects of arithmetic and geometry. This article covers the theory of Siegel modular forms, the Hecke theory, a lifting of elliptic cusp forms, geometric properties of the Siegel modular variety, (hypothetical) motives attached to Siegel modular forms and a cohomology of the Siegel modular variety. To the memory of my mother
Abstract. Functoriality conjecture is one of the central and influential subjects of the present ... more Abstract. Functoriality conjecture is one of the central and influential subjects of the present day mathematics. Functoriality is the profound lifting problem formulated by Robert Langlands in the late 1960s in order to establish nonabelian class field theory. In this expository article, I describe the Langlands-Shahidi method, the local and global Langlands conjectures and the converse theorems which are powerful tools for the establishment of functoriality of some important cases, and survey the interesting results related to functoriality conjecture.
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Papers by Jae-Hyun Yang