The authors define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties such as the Fourier inversion formula, and give some applications. The definition of the...The authors define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties such as the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of K-admissible measures. The authors prove that K-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of K-admissible measures.展开更多
In this paper,we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps.In particular,a few classical results of Steinberg and Deligne&Lusztig on complex represe...In this paper,we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps.In particular,a few classical results of Steinberg and Deligne&Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.展开更多
This is the third part of a pedagogical introduction to the theory of buildings of Jacques Tits.We describe the construction and properties of the Bruhat-Tits building of a reductive group over a local field.
In this article,we give a survey of some progress of the complex geometry,mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years.In particular,we e...In this article,we give a survey of some progress of the complex geometry,mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years.In particular,we explore some works in the special area in Di erential Geometry,Lie Group and Complex Homogeneous Space.Together with the special area in nonlinear analysis on complex manifolds,they are the two major aspects of my research interests.展开更多
We study solutions of the nonlinear Schroedinger equation （NLSE） and higher-order nonlinear Sehroedinger equation （HONLSE） with variable coefficients. By considering all the higher-order effect of HONLSE as a new ...We study solutions of the nonlinear Schroedinger equation （NLSE） and higher-order nonlinear Sehroedinger equation （HONLSE） with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method （CLGRM）, the abundant solutions of NLSE and HONLSE are obtained.展开更多
In this article,we give a further survey of some progress of the applications of group actions in the complex geometry after my earlier survey around 2020,mostly related to my own interests.
Let G be a connected reductive group defined over F_q, the finite field with q elements. Let B be a Borel subgroup defined over F_q. In this paper, we completely determine the composition factors of the induced module...Let G be a connected reductive group defined over F_q, the finite field with q elements. Let B be a Borel subgroup defined over F_q. In this paper, we completely determine the composition factors of the induced module M(tr) = kG ■tr(where tr is the trivial B-module) for any field k.展开更多
Let G -- Spec A be an affine K-group scheme and A = （w ∈ A＊ ： dimK A＊-w 〈 ∞, dimK w. A＊ 〈 ∞}. Let （-, -） ： A＊ × A → K, （w, w） ：= tr（ww）, be the trace form. We prove that G is linearly reductiv...Let G -- Spec A be an affine K-group scheme and A = （w ∈ A＊ ： dimK A＊-w 〈 ∞, dimK w. A＊ 〈 ∞}. Let （-, -） ： A＊ × A → K, （w, w） ：= tr（ww）, be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A＊.展开更多
Existing groupwise dimension reduction requires given group structure to be non-overlapped. This confines its application scope. We aim at groupwise dimension reduction with overlapped group structure or even unknown ...Existing groupwise dimension reduction requires given group structure to be non-overlapped. This confines its application scope. We aim at groupwise dimension reduction with overlapped group structure or even unknown group structure. To this end, existing groupwise dimension reduction concept is extended to be compatible with overlapped group structure. Then, the envelope method is ameliorated to deal with overlapped groupwise dimension reduction. As an application, Gaussian graphic model is employed to estimate the structure between predictors when the group structure is not given, and the amended envelope method is used for groupwise dimension reduction with graphic structure. Furthermore, the rationale of the proposed estimation procedure is explained at the population level and the estimation consistency is proved at the sample level. Finally, the finite sample performance of the proposed methods is examined via numerical simulations and a body fat data analysis.展开更多
These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varie...These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.展开更多
基金supported by the 973 Project Foundation of China (#TG1999075102)
文摘The authors define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties such as the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of K-admissible measures. The authors prove that K-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of K-admissible measures.
基金supported by National Natural Science Foundation of China (Grant No. 11321101)
文摘In this paper,we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps.In particular,a few classical results of Steinberg and Deligne&Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.
文摘This is the third part of a pedagogical introduction to the theory of buildings of Jacques Tits.We describe the construction and properties of the Bruhat-Tits building of a reductive group over a local field.
基金the Natural Science Foundation of Henan University。
文摘In this article,we give a survey of some progress of the complex geometry,mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years.In particular,we explore some works in the special area in Di erential Geometry,Lie Group and Complex Homogeneous Space.Together with the special area in nonlinear analysis on complex manifolds,they are the two major aspects of my research interests.
基金National Natural Science Foundation of China under Grant No.10675065
文摘We study solutions of the nonlinear Schroedinger equation （NLSE） and higher-order nonlinear Sehroedinger equation （HONLSE） with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method （CLGRM）, the abundant solutions of NLSE and HONLSE are obtained.
文摘In this article,we give a further survey of some progress of the applications of group actions in the complex geometry after my earlier survey around 2020,mostly related to my own interests.
基金supported by National Natural Science Foundation of China (Grant Nos.11501546 and 11671297)。
文摘Let G be a connected reductive group defined over F_q, the finite field with q elements. Let B be a Borel subgroup defined over F_q. In this paper, we completely determine the composition factors of the induced module M(tr) = kG ■tr(where tr is the trivial B-module) for any field k.
文摘Let G -- Spec A be an affine K-group scheme and A = （w ∈ A＊ ： dimK A＊-w 〈 ∞, dimK w. A＊ 〈 ∞}. Let （-, -） ： A＊ × A → K, （w, w） ：= tr（ww）, be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A＊.
基金supported by a grant from the University Grant Council of Hong Kong of ChinaNational Natural Science Foundation of China (Grant No. 11371013)Tian Yuan Foundation for Mathematics
文摘Existing groupwise dimension reduction requires given group structure to be non-overlapped. This confines its application scope. We aim at groupwise dimension reduction with overlapped group structure or even unknown group structure. To this end, existing groupwise dimension reduction concept is extended to be compatible with overlapped group structure. Then, the envelope method is ameliorated to deal with overlapped groupwise dimension reduction. As an application, Gaussian graphic model is employed to estimate the structure between predictors when the group structure is not given, and the amended envelope method is used for groupwise dimension reduction with graphic structure. Furthermore, the rationale of the proposed estimation procedure is explained at the population level and the estimation consistency is proved at the sample level. Finally, the finite sample performance of the proposed methods is examined via numerical simulations and a body fat data analysis.
基金partially supported by a National Science Foundation Grant(Grant No.ID:DMS-1601303)partially supported by a National Science Foundation Grant(Grant No.ID:DMS-1500966)+1 种基金Simons Foundation Collaboration Grant for MathematiciansSimons Fellowship
文摘This is a survey of some recent results on spherical tropical geometry.
文摘These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.
