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We prove a simple formula that calculates the associated variety of a highest weight Harish-Chandra module directly from its highest weight. We also give a formula for the Gelfand--Kirillov dimension of highest weight Harish-Chandra module which is uniform across Cartan types and is valid for arbitrary infinitesimal character., Comment: 34 pages

Let $\mathfrak{g}$ be a simple complex Lie algebra with a Cartan subalgebra $\mathfrak{h}$. We fix a standard parabolic subalgebra $\mathfrak{p}\supset \mathfrak{h}$. The socular simple modules play an important role in the parabolic versions of category $\mathcal{O}^{\mathfrak{p}}$. From Irving's work, we know that these modules are just those modules with largest possible Gelfand-Kirillov dimension in $\mathcal{O}^{\mathfrak{p}}$. In this article, we will give an explicit characterization for these modules of $\mathfrak{sl}(n,\mathbb{C})$. Our characterization is given in the information of the corresponding highest weight and Young tableau., Comment: The main results are covered by other papers

Selective data protection is a promising technique to defend against the data leakage attack. In this paper, we revisit technical challenges that were neglected when applying this protection to real applications. These challenges include the secure input channel, granularity conflict, and sensitivity conflict. We summarize the causes of them and propose corresponding solutions. Then we design and implement a prototype system for selective data protection and evaluate the overhead using the RISC-V Spike simulator. The evaluation demonstrates the efficiency (less than 3% runtime overhead with optimizations) and the security guarantees provided by our system.

In this paper, we present a uniform formula of Lusztig's $ \mathbf{a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand-Kirillov dimensions of simple highest weight modules of classical Lie algebras, whose highest weight is not necessarily regular or integral. To deal with type $ D $, we prove an interesting property about domino tableaux by introducing an invariant, called the hollow tableau. As an application, the associated varieties of all the simple highest weight Harish-Chandra modules are explicitly determined, including the exceptional cases., Comment: 32 pages. Comments welcome

For a complex simple Lie algebra $\mathfrak{g}$, it is well-known that the associated variety of a highest weight module $L(\lambda)$ of $\mathfrak{g}$ is the union of some orbital varieties associated with the nilpotent orbit $\mathcal{O}_{\Ann{L(\lambda)}}$ whose closure is the associated variety of $\Ann{L(\lambda)}$. But in general, it is a very difficult problem to determine the associated variety of a highest weight module. The highest weight modules with minimal Gelfand-Kirillov dimension (or $\mathcal{O}_{\Ann{L(\lambda)}}=\mathcal{O}_{\rm min}$ is the unique minimal nilpotent orbit of $\mathfrak{g}$) play important roles in the study of representations of Lie groups and Lie algebras. Joseph found some necessary and sufficient condition for weak quantization (if there is a $U(\mathfrak{g})$-module whose associated variety is the given orbital variety) of these minimal orbital varieties, and he also classified those minimal highest weight modules whose annihilator ideals are completely prime. In this paper, we will give a classification of minimal highest weight modules with their annihilator ideals being not necessarily completely prime for all complex simple Lie algebras. We also describe the associated varieties of these modules. By comparing with Joseph's work, we have found all minimal highest weight modules which can weakly quantize a given minimal orbital variety (in the sense of Joseph)., Comment: 19pages

Let $G$ be a Hermitian type Lie group with complexified Lie algebra $\mathfrak{g}$ and maximal compact subgroup $K$. We use $L(\lambda)$ to denote a highest weight Harish-Chandra $G$-module with infinitesimal character $\lambda$. Let $w$ be an element in the Weyl group $W$. We use $L_w$ to denote a highest weight module with highest weight $-w\rho-\rho$. In this paper we will give a characterization for those $w$ such that $L_w$ is a highest weight Harish-Chandra module and the associated variety of $L(\lambda)$ will be characterized by the information of the Kazhdan-Lusztig right cell containing some special $w_{\lambda}$. We also count the number of those highest weight Harish-Chandra modules $L_w$ in a given Harish-Chandra cell., Comment: 33pages

We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3. Our proof enables us to give a description of the a-function and Kazhdan-Lusztig cells for these Coxeter groups., Comment: 40 pages, 2 figures

We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph, using deceasing induction on $ \mathbf{a} $-values and a kind of decomposition formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a description of the left, right, and two-sided cells. In the appendix, we prove P1-P15 for right-angled Coxeter groups by the same methods., Comment: 48 pages, in version 2, we add a new section \S9

Let $ (G,K) $ be an irreducible Hermitian symmetric pair of non-compact type with $G=SU(p,q)$, and let $ \lambda $ be an integral weight such that the simple highest weight module $ L(\lambda) $ is a Harish-Chandra $ (\mathfrak{g},K) $-module. We give a combinatoric algorithm for the Gelfand-Kirillov dimension of $ L(\lambda) $. This enables us to prove that the Gelfand-Kirillov dimension of $ L(\lambda) $ decreases as the integer $ \langle\lambda+\rho,\beta^\vee\rangle $ increases, where $\rho$ is the half sum of positive roots and $\beta$ is the maximal noncompact root. As a byproduct, we obtain a description on the associated variety of $ L(\lambda) $., Comment: We add a footnote on page 9 in this version

In this paper, we prove a decomposition formula for the Kazhdan-Lusztig basis of affine Hecke algebras of rank 2 with positive weight function. Then we discuss some applications of this kind of decomposition to Lusztig's conjectures P1-P15., Comment: 28 pages, 2 figures

Let $\mathbf{c}_0$ be the lowest generalized two-sided cell of an extended affine Weyl group W. We determine the structure of the based ring of $\mathbf{c}_0$. For this we show that certain conjectures of Lusztig on generalized cells (called P1-P15) hold for $\mathbf{c}_0$. As an application, we use the structure of the based ring to study certain simple modules of Hecke algebras of $ W $ with unequal parameters, namely those attached to $\mathbf{c}_0$. Also we give a set of prime ideals $\mathfrak{p}$ of the center $\mathcal{Z}$ of the generic affine Hecke algebra $\mathcal{H}$ such that the reduced affine Hecke algebra $k_\mathfrak{p}\mathcal{H}$ is simple over $k_\mathfrak{p}$, where $k_\mathfrak{p}=\rm{Frac}(\mathcal{Z}/\mathfrak{p})$ is the residue field of $\mathcal{Z}$ at $\mathfrak{p}$. In particular, we show that the algebra $\mathcal{H}\otimes_\mathcal{Z}\rm{Frac}(\mathcal{Z})$ is a split simple algebra over the field $ \rm{Frac}(\mathcal{Z})$., Comment: 23pages, 1 figure; second version; An error (in last section of last version), pointed by an anonymous referee , is corrected