arXiv subjects
Publications and source records attributed to Qixian Zhao.
This is an expository paper based on the authors' joint works. The goal is to explain the statements and the ideas behind two conjectures on associated varieties and simple modules of simple affine vertex algebras $L_k(\mathfrak{g})$ for a simple and simply-laced Lie algebra $\mathfrak{g}$ and a integer level $k$ above the critical level.
We present a conjecture for associated varieties of simple affine vertex algebras $L_k(\mathfrak{g})$ attached to a simple Lie algebra $\mathfrak{g}$ of simply-laced type and any rational level $k$ greater than the critical level. The key new ingredient compared to the integral case is the covering duality map introduced by Gao-Liu-Lo-Shahidi. We provide evidence for the conjecture.
We establish an explicit correspondence of certain Arthur packets between real unitary groups and $p$-adic symplectic or orthogonal groups. This allows one to compute Arthur packets of real unitary groups by translating results from the $p$-adic side. A main ingredient in our proof is an explicit relation between Zuckerman's translation functor on the real side and the Jacquet functor on the $p$-adic side. To achieve this, we construct a correspondence of stacks of Langlands parameters with fixed infinitesimal characters between the relevant real and $p$-adic groups. Our approach also allows one to relate the Kazhdan-Lusztig polynomials and the microlocal geometry between real and $p$-adic sides.
In this paper, we introduce and study two cyclotomic level maps defined respectively on the set of nilpotent orbits $\underline{\mathcal{N}}$ in a complex semi-simple Lie algebra $\mathfrak{g}$ and the set of conjugacy classes $\underline{W}$ in its Weyl group, with values in positive integers. We show that these maps are compatible under Lusztig's map $\underline{W} \to \underline{\mathcal{N}}$, which is also the minimal reduction type map as shown by Yun. We also discuss their relationship with two-sided cells in affine Weyl groups. We use these maps to formulate a conjecture on the associated varieties of simple affine vertex algebras attached to $\mathfrak{g}$ at non-admissible integer levels, and provide some evidence for this conjecture.
In this paper, we use geometric methods to study the relations between admissible representations of $\mathbf{GL}_n(\mathbb{C})$ and unramified representations of $\mathbf{GL}_m(\mathbb{Q}_p)$. We show that the geometric relationship between Langlands parameter spaces of $\mathbf{GL}_n(\mathbb{C})$ and $\mathbf{GL}_m(\mathbb{Q}_p)$ constructed by the first named author is compatible with the functor recently defined algebraically by Chan-Wong. We then show that the said relationship intertwines translation functors on representations of $\mathbf{GL}_n(\mathbb{C})$ and partial Bernstein-Zelevinskii derivatives on representations of $\mathbf{GL}_m(\mathbb{Q}_p)$, providing purely geometric counterparts to some results of Chan-Wong. In the sequels, the techniques of this work will be extended to real and $p$-adic classical groups and used to study their Arthur packets.
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We give a description of characters of irreducible Whittaker modules for $\mathfrak{g}$ with any infinitesimal character, along with a Kazhdan-Lusztig algorithm for computing them. This generalizes Milicic-Soergel's and Romanov's results for integral infinitesimal characters. As a special case, we recover the non-integral Kazhdan-Lusztig conjecture for Verma modules.