Hongyu He
We study square integrable functions on the metaplectic group and functions on the space of unitary symmetric matrices. We relate them using the oscillator representations.
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For all the irreducible dual pairs of type I $(G,G')$, we analyze the restriction of the oscillator representation as a $(\mathfrak{g}', K')$-module, when $G'$ is the smaller group. For all $(G,G')$ in the stable range, as well as one more case, the modules obtained are projective. We use the duality correspondence introduced by Howe to analyze these restrictions.
Orthogonal Oscillator Representations, Laplace Equations and Intersections of Determinantal Varieties
该研究证明了由正交振子表示经傅里叶变换得到的特殊线性李代数无限维不可约表示的相伴簇是显式行列式簇的交，建立了表示论、偏微分方程与代数几何间的显式关联。
The n-dimensional projective group gives rise to a one-parameter family of inhomogeneous first-order differential operator representations of sl(n+1). By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of sl(n+1). Letting these differential operators act on the corresponding polynomial algebra and the space of exponential-polynomial functions, we construct new multi-parameter families of explicit infinite-dimensional irreducible representations for s(n+1) and sp(2m+2) when n=2m+1. Our results can be viewed as …
We describe a new technique to obtain representations of the braid group B_n from the R-matrix of a quantum deformed algebra of the one dimensional harmonic oscillator. We consider the action of the R-matrix not on the tensor product of representations of the algebra, that in the harmonic oscillator case are infinite dimensional, but on the subspace of the tensor product corresponding to the lowest weight vectors.
Differential operator approach to $\imath$quantum groups and their oscillator representations
本文通过修正的$q$-Weyl代数建立到$\u0131$量子群的代数同态，给出其微分算子实现，并构造振子表示及其不可约子表示的晶体基。
Using the projective oscillator representation of sl(n+1) and Shen's mixed product for Witt algebras, Zhao and the second author (2011) constructed a new functor from sl(n)-Mod to sl(n+1)-Mod. In this paper, we start from n = 2 and use the functor successively to obtain a full projective oscillator realization of any finite-dimensional irreducible representation of sl(n+1). The representation formulas of all the root vectors of sl(n+1) are given in terms of first-order differential operators in n(n+1)/2 variables. One can use the result to study tensor decompositions of finite-dimensional irre…
We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible sl(n, F)-modules that appeared in the Z^2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand-Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.
An oscillator group $G$ is a semidirect product of a Heisenberg group with a one-parameter group. In this article we construct Olshanski semigroups for infinite-dimensional oscillator groups. These are complex involutive semigroups which have a polar decomposition. The main application will be for representations $π$ of $G$ which are semibounded, i.e., there exists a non-empty open subset $U$ of the Lie algebra $\mathfrak{g}$ such that the operators $idπ(x)$ from the derived representation are uniformly bounded from above for $x\in U$. More precisely we show that every semibounded representati…
We introduce a category $\widehat{\mathcal{O}}_{\rm osc}$ of $q$-oscillator representations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_n)$. We show that $\widehat{\mathcal{O}}_{\rm osc}$ has a family of irreducible representations, which naturally corresponds to finite-dimensional irreducible representations of quantum affine algebra of untwisted affine type $A$. It is done by constructing a category of $q$-oscillator representations of the quantum affine superalgebra of type $A$, which interpolates these two family of irreducible representations. The category $\widehat{\mathca…
Oscillator representations of quantum affine orthosymplectic superalgebras
本文引入D型量子仿射超代数的q-振子表示范畴并构造新族不可约表示，基于超对偶理论证明其插值两类不可约表示，建立了Howe对偶对应的量子仿射类比。
We explore a model for the one-dimensional quantum oscillator based upon the Lie superalgebra sl(2|1). For this purpose, a class of discrete series representations of sl(2|1) is constructed, each representation characterized by a real number beta>0. In this model, the position and momentum operators of the oscillator are odd elements of sl(2|1) and their expressions involve an arbitrary parameter gamma. In each representation, the spectrum of the Hamiltonian is the same as that of the canonical oscillator. The spectrum of the momentum operator can be continuous or infinite discrete, depending …
We develop a functorial theory of spinor and oscillator representations parallel to the theory of Schur functors for general linear groups. This continues our work on developing orthogonal and symplectic analogues of Schur functors. As such, there are a few main points in common. We define a category of representations of what might be thought of as the infinite rank pin and metaplectic groups, and give three models of this category in terms of: multilinear algebra, diagram categories, and twisted Lie algebras. We also define specialization functors to the finite rank groups and calculate the …
We study the unitary representations of the non-compact real forms of the complex Lie superalgebra sl(n|m). Among them, only the real form su(p,q|m) (p+q=n) admits nontrivial unitary representations, and all such representations are of the highest-weight type (or the lowest-weight type). We extend the standard oscillator construction of the unitary representations of non-compact Lie superalgebras over standard Fock spaces to generalised Fock spaces which allows us to define the action of oscillator determinants raised to non-integer powers. We prove that the proposed construction yields all th…
An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis is given. Particular attention is paid to the so-called star type I representations (``unitary representations''), and to a simple class of representations V(p), with p any positive integer. Then, the notion of Wigner Quantum Oscillators (WQOs) is recalled. In these quantum oscillator models, the unitary representations of gl(1|DN) are physical state spaces of the N-particle D-dimensional oscillator. So far, physical properties of gl(1|DN) WQOs were describe…
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