Unipotent representations of real classical groups

Let $\mathbf G$ be a complex orthogonal or complex symplectic group, and let $G$ be a real form of $\mathbf G$, namely $G$ is a real orthogonal group, a real symplectic group, a quaternionic orthogonal group, or a quaternionic symplectic group. For a fixed parity $\mathbb p\in \mathbb Z/2\mathbb Z$, we define a set $\mathrm{Nil}^{\mathbb p}_{\mathbf G}(\mathfrak g)$ of nilpotent $\mathbf G$-orbits in $\mathfrak g$ (the Lie algebra of $\mathbf G$). When $\mathbb p$ is the parity of the dimension of the standard module of $\mathbf G$, this is the set of the stably trivial special nilpotent orbits, which includes all rigid special nilpotent orbits. For each $\mathcal O \in \mathrm{Nil}^{\mathbb p}_{\mathbf G}(\mathfrak g)$, we construct all unipotent representations of $G$ (or its metaplectic cover when $G$ is a real symplectic group and $\mathbb p$ is odd) attached to $\mathcal O$ via the method of theta lifting and show in particular that they are unitary

Bounded-Distortion Metric Learning

Metric learning aims to embed one metric space into another to benefit tasks like classification and clustering. Although a greatly distorted metric space has a high degree of freedom to fit training data, it is prone to overfitting and numerical inaccuracy. This paper presents {\it bounded-distortion metric learning} (BDML), a new metric learning framework which amounts to finding an optimal Mahalanobis metric space with a bounded-distortion constraint. An efficient solver based on the multiplicative weights update method is proposed. Moreover, we generalize BDML to pseudo-metric learning and devise the semidefinite relaxation and a randomized algorithm to approximately solve it. We further provide theoretical analysis to show that distortion is a key ingredient for stability and generalization ability of our BDML algorithm. Extensive experiments on several benchmark datasets yield promising results

A combinatorial characterization of the annihilator varieties of highest weight modules for classical Lie algebras

Let $\mathfrak{g}$ be a classical Lie algebra. Let $L(\lambda)$ be a highest weight module of $\mathfrak{g}$ with highest weight $\lambda-\rho$, where $\rho$ is half the sum of positive roots. In 1985, Joseph proved that the associated variety of a primitive ideal is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. In this paper, we will give some combinatorial characterizations of the annihilator varieties of highest weight modules for classical Lie algebras. In fact, we will give two algorithms, i.e., bipartition algorithm and partition algorithm.Comment: 40page

Fourier-Jacobi models of Deligne-Lusztig characters and depth zero local descent for unitary groups

In this paper, we deduce explicit multiplicity formulas of the Fourier-Jacobi model for Deligne-Lusztig characters of finite symplectic groups, unitary groups, and general linear groups. We then apply these results to deduce the explicit depth zero local descent (\`a la Soudry and Tanay) for $p$-adic unitary groups. The result is a concrete example in the context of non-tempered Gan-Gross-Prasad program.Comment: 31 pages. Comments are welcom

Polymorphisms of CYP1A1 I462V and GSTM1 genotypes and lung cancer susceptibility in Mongolian

Aim: To study the genotype of cytochrome P450 1A1(CYP1A1) I462V and glutathions S-transferase M1( GSTM1) and the relationship of the genetic polymorphism of them with the susceptibility of lung cancer in Mongolia of China. 

Methods: Allele-specific PCR and a multiplex PCR were employed to identify the genotypes of I462V of CYP1A1 and GSTM1 in a case-control study of 210 lung cancer patients with bronchoscopy diagnosis and 210 matched controls free of malignancy.

Results: The frequencies of the variant CYP1A1(Val/Val) genotypes and GSTM1(-) in lung cancer groups were higher than that in control groups (15.24% vs 7.4% and 56.67% vs 40.95% ). The individuals who carried with CYP1A1(Val/Val) or GSTM1(-) genotype had a significantly higher risk of lung cancer, the OR is 2.56 and 1.89 respectively. Stratified histologically the relative risk increased to 2.6 - fold when the patients carried with two valine alleles than the ones carried one valine allele in cases of SCC. GSTM1(-) genotype is the risk factor of SCC (OR=2.39) and AC(OR=2.16). The presence of at least one Val allele of CYP1A1 and GSTM1(-), the risk of lung cancer was increased, the OR was 4.15 for one Val allele and GSTM1(-) and 2.67 for two Val alleles and GSTM1 Considering ages and smoking status, the risk of lung cancer increased when the age less than 50 who carried with CYP1A1 valine (one or two) alleles or the age during the 51 to 65 who carried with GSTM1(-) genotype. The light smokers with CYP1A1 valine alleles and GSTM1(-) have a high risk for lung cancer. No association was found between the light and heavy drinkers with the susceptibility of lung cancer and the genetic polymorphisms of CYP1A1 I462V and GSTM1(-). 

Conclusion: The valine allele of CYP1A1 was the risk factors of lung cancer especially for SCC and GSTM1(-) also was the risk factor of lung cancer and especially for SCC and AC of Mongolian, China. Light smoking has a influence each other with genotype of CYP1A1 I462V and GSTM1(-) and susceptibility of lung cancer. No relationship was found between the susceptibility of lung cancer and drinkers with genetic polymorphisms of CYP1A1 I462V and GSTM1(-). The influence of genotypes on the susceptibility of lung cancer may depend on the ages. There may be a synergetic interaction between CYP1A1 valine allele and GSTM1(-) genotypes on the elevated susceptibility of lung cancer. So do those genotypes with light smokers. Key words polymorphism; genotype; lung cancer; cytochrome P450；glutathione S-transferase Abbreviations: SCC, squamous cell carcinoma; AC, adenocarcinoma; SCLC, small cell lung cancer; LCLC, large cell lung cance

Insights into neutron star equation of state by machine learning

Due to its powerful capability and high efficiency in big data analysis, machine learning has been applied in various fields. We construct a neural network platform to constrain the behaviors of the equation of state of nuclear matter with respect to the properties of nuclear matter at saturation density and the properties of neutron stars. It is found that the neural network is able to give reasonable predictions of parameter space and provide new hints into the constraints of hadron interactions. As a specific example, we take the relativistic mean field approximation in a widely accepted Walecka-type model to illustrate the feasibility and efficiency of the platform. The results show that the neural network can indeed estimate the parameters of the model at a certain precision such that both the properties of nuclear matter around saturation density and global properties of neutron stars can be saturated. The optimization of the present modularly designed neural network and extension to other effective models are straightforward.Comment: 12 pages, 5 figures. Comments are welcom

Associated cycles of local theta lifts of unitary characters and unitary lowest weight modules

In this paper we first construct natural filtrations on the full theta lifts for any real reductive dual pairs. We will use these filtrations to calculate the associated cycles and therefore the associated varieties of Harish-Chandra modules of the indefinite orthogonal groups which are theta lifts of unitary lowest weight modules of the metaplectic double covers of the real symplectic groups. We will show that some of these representations are special unipotent and satisfy a K-type formula in a conjecture of Vogan.Comment: The current version is a major revision of the first draft where we incorporate ideas from a recent paper arXiv:1302.1031. We bypass the K-types and asymptote calculations, and give more geometric and conceptual proof