Induced orbit data and induced unitary representations for complex groups

We use induced orbit covers to define induced orbit data.By studying the space of regular functions on orbit cover,we know that the induced representation has close connection with the induced orbit datum under the meaning of Vogan’s conjecture.Therefore,when verifying Vogan’s conjecture,many cases can be reduced to the case of rigid orbit data.

Dirac cohomology of unitary representations of equal rank exceptional groups

In this paper, we consider the unitary representations of equal rank exceptional groups of type E with a regular lambda-lowest K-type and classify those unitary representations with the nonzero Dirac cohomology.

Special Unipotent Representations of Simple Linear Lie Groups of Type A

Let G be a special linear group over the real,the complex or the quaternion,or a special unitary group.In this note,we determine all special unipotent representations of G in the sense of Arthur and Barbasch-Vogan,and show in particular that all of them are unitarizable.

Kac Determinant Formula for the q-deformed Virasoro Algebra of Hom-type

We investigate the highest weight representations of the q-deformed Virasoro algebra of Hom-type. In order to determine its unitarity and irreducible highest weight representations, we present its Kac determinant formula when q is nonzero and non-root of unity.

CONTINUOUS FRAME WAVELETS

Let π be a unitary representation of a locally compact topological group G on a separable Hilbert space H. A vector ψ∈ H is called a continuous frame wavelet if there exist A, B 〉 0 such that A｜｜Ф｜｜2≤fG｜〈π（G）ψ,Ф｜2dg≤B｜｜Ф｜｜2（Ф∈H）,in which dg is the left Haar measure of G. Similar to the study of wavelets, an essential problem in the study of continuous frame wavelets is how to characterize them under the given unitary representation. Moreover, we investigate a relation between admissible vectors of π and its components.

Braid Group Representations from Twisted Tensor Products of Algebras

We unify and generalize several approaches to constructing braid group representa-tions from finite groups,using iterated twisted tensor products.We provide some general characterizations and classification of these representations,focusing on the size of their images,which are typically finite groups.The well-studied Gaussian representations associated with metaplectic modular categories can be understood in this framework,and we give some new examples to illustrate their ubiquity.Our results suggest a relationship between the braiding on the G-gaugings of a pointed modular category C(A,Q)and that of C(A,Q)itself.

Theta-lifting and geometric quantization for GL（n, C）

In this paper, we verify Vogan＇s conjecture on quantization in the representation theory for G = GL（n, C）. Also we get some relationship between the induction of orbits and Howe＇s θ-lifting of unitary representations

Tensor products of complementary series of rank one Lie groups

We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.

Operator valued frames and structured quantum channels

Operator-valued frames are natural generalization of frames that have been used in many applied areas such as quantum computing, packets encoding and sensor networks. We focus on developing the theory about operator-valued frame generators for projective unitary representations of finite or countable groups which can be viewed as the theory of quantum channels with group structures. We present new results for operator-valued frames concerning (general and structured) dilation property, orthogonal frames, frame representation and dual frames. Our results are complementary to some of the recent work of Kaftal, Larson and Zhang, and in some cases our treatment is more elementary and transparent.

Estimate of the L^p-Fourier Transform Norm on Strong *-Regular Exponential Solvable Lie Groups

We study the L^p-Fourier transform for a special class of exponential Lie groups, the strong ＊-regular exponential Lie groups. Moreover, we provide an estimate of its norm using the orbit method.

New method of applying conformal group to quantum fields

Most of previous work on applying the conformal group to quantum fields has emphasized its invariant aspects, whereas in this paper we find that the conformal group can give us running quantum fields, with some constants, vertex and Green functions running, compatible with the scaling properties of renormalization group method （RGM）. We start with the renormalization group equation （RGE）, in which the differential operator happens to be a generator of the conformal group, named dilatation operator. In addition we link the operator/spatial representation and unitary/spinor representation of the conformal group by inquiring a conformal-invariant interaction vertex mimicking the similar process of Lorentz transformation applied to Dirac equation. By this kind of application, we find out that quite a few interaction vertices are separately invaxiant under certain transformations （generators） of the conformal group. The significance of these transformations and vertices is explained. Using a particular generator of the conformal group, we suggest a new equation analogous to RGE which may lead a system to evolve from asymptotic regime to nonperturbative regime, in contrast to the effect of the conventional RGE from nonperturbative regime to asymptotic regime.

Global geometrical optics method for vector-valued Schrodinger problems

We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valued Schrodinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin （WKB） analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level SchrSdinger system and the other is the periodic SchrSdinger equation. Numerical tests validate the theoretical results.

On the Cohomology of Some Complex Hyperbolic Arithmetic 3-Manifolds

Given an arithmetic lattice of the unitary group U(3,1)arising from a hermitian form over a CM-field,we show that all unitary representations of U(3,1)with nonzero cohomology contribute to the cohomology of the attached arithmetic complex 3-manifold,at least when we pass to a finite-index subgroup of the given arithmetic lattice.