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Kayue Daniel Wong

Publications and source records attributed to Kayue Daniel Wong.

25 records · Page 2Linked to original sources
arXiv2020

Dirac series of $GL(n, \mathbb{R})$

The unitary dual of $GL(n, \mathbb{R})$ was classified by Vogan in the 1980s. Focusing on the irreducible unitary representations of $GL(n, \mathbb{R})$ with half-integral infinitesimal characters, we find that Speh representations and the special unipotent representations are building blocks. By looking at the $K$-types of them, and by using a Blattner-type formula, we obtain all the irreducible unitary $(\mathfrak{g}, K)$-modules with non-zero Dirac cohomology of $GL(n, \mathbb{R})$, as well as a formula for (one of) their spin-lowest $K$-types. Moreover, analogous to the $GL(n,\mathbb{C})$ case given in [DW1], we count the number of the FS-scattered representations of $GL(n, \mathbb{R})$.

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arXiv2017

A Casselman-Osborne theorem for rational Cherednik algebras

We define Lie algebra cohomology associated with the half-Dirac operators for representations of rational Cherednik algebras and show that it has property described in the Casselman-Osborne Theorem by establishing a version of the Vogan's conjecture for the half-Dirac operators. Moreover, we study the relationship between Lie algebra cohomology and Dirac cohomology in analogy of the representations for semisimple Lie algebras.

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arXiv2016

On Quantization of a Nilpotent Orbit Closure in $G_2$

Let $G$ be the complex exceptional Lie group of type $G_2$. Among the five nilpotent orbits in its Lie algebra $\mathfrak{g}$, only the 8-dimensional orbit $\mathcal{O}_8$ has non-normal orbit closure $\bar{\mathcal{O}_8}$. In this short note, we will give a quantization model of $\bar{\mathcal{O}_8}$, verifying a conjecture of Vogan in 1984.

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arXiv2015

Quantization of Special Symplectic Nilpotent Orbits and Normality of their Closures

We study the regular function ring $R(\mathcal{O})$ for all symplectic nilpotent orbits $\mathcal{O}$ with even column sizes. We begin by recalling the quantization model for all such orbits by Barbasch using unipotent representations. With this model, we express the multiplicities of fundamental representations appearing in $R(\mathcal{O})$ by a parabolically induced module. Finally, we will use this formula to give a criterion on the normality of the Zariski closure $\bar{\mathcal{O}}$ of $\mathcal{O}$.

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arXiv2015

Local Systems on Classical Nilpotent Orbits and Maximal Length Elements

It is known that there is a bijection between dominant weights of a complex reductive Lie group $G$ and the set $\mathcal{N}_{\mathcal{O},r}$ whose elements are of the form $(\mathcal{O},ρ)$, where $\mathcal{O}$ is a nilpotent orbit and $ρ$ is an irreducible, algebraic representation of the stabilizer group $G^e$ of an element $e$ in the nilpotent orbit $\mathcal{O}$. We would like to study the above bijection when $G$ is classical and $ρ$ corresponds to a local system of $\mathcal{O}$. In particular, we will prove Conjecture 3.1 in \cite{AS} and Conjecture 7.4' in \cite{Ac2} in the classical setting.

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arXiv2015

Regular Functions of Symplectic Spherical Nilpotent Orbits and their Quantizations

We study the ring of regular functions of classical spherical orbits $R(\mathcal{O})$ for $G = Sp(2n,\mathbb{C})$. In particular, treating $G$ as a real Lie group with maximal compact subgroup $K$, we focus on a quantization model of $\mathcal{O}$ when $\mathcal{O}$ is the nilpotent orbit $(2^{2p}1^{2q})$. With this model, we verify a conjecture by McGovern and another conjecture by Achar and Sommers for such orbits. Assuming the results in [Barbasch 2008], we will also verify the Achar-Sommers conjecture for a larger class of nilpotent orbits.

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