arXiv subjects

Shilin Yu

Publications and source records attributed to Shilin Yu.

15 recordsLinked to original sources

arXiv2026

Special unipotent representations and the coadjoint orbit method

For any linear real reductive group or metapletic group, this article gives a geometric classification of special unipotent representations in the weak Arthur/Adams-Barbasch-Vogan (ABV) packets attached to quasi-distinguished nilpotent orbits in the Langlands or metaplectic dual Lie algebra in terms of their associated cycles. We provide a uniform construction of the Harish-Chandra modules of these representations via deformation quantization of admissible vector bundles over certain Lagrangian subvarieties of the affinizations of the universal covers of special nilpotent orbits in question, which aligns with the coadjoint orbit method philosophy of Kirillov, Kostant, and Vogan. As consequences, all such Harish-Chandra modules have irreducible associated cycles, and are unitarizable by results from the theory of mixed Hodge modules. Conjecturally, the unitarity of all ABV packets can be reduced to the case when the dual orbits are distinguished. Our approach highlights the application of Lusztig's conjecture on the geometry of special pieces, which has been proven for classical Lie algebras by Kraft and Procesi, and for all cases by Juteau, Levy, Sommers and the author.

math.RT↗

arXiv2026

Lusztig's special pieces conjecture

Let $\mathcal O$ be a special nilpotent orbit in the Lie algebra $\mathfrak g$ of a simple algebraic group $G$. We give two proofs of the result that every special piece ${\mathcal P}(\mathcal O)$ in $\mathfrak g$ is the quotient of a smooth $G$-variety $X$ by the action of a certain finite group $H$. We first deduce the result from a similar result for transverse slices, established in earlier work of the first three authors and Fu. Then we give a more explicit construction of $X$, as a subvariety of the closure of a $G$-orbit in the direct sum of $\mathfrak g$ and some fundamental weight representations of $G$. Both methods apply to classical $\mathfrak g$, where we give new proofs of this result, which was first proved by Kraft and Procesi. The result in the exceptional groups was conjectured by Lusztig. Our first proof shows that there can be several $G$-varieties $X$ that satisfy the conjecture, related to a natural embedding of $H$ in the fundamental group of $\mathcal O$. In an appendix, we relate this natural embedding to Lusztig's definition of $H$ that arises from the family in the Weyl group of $G$ attached to $\mathcal O$ and from the Springer correspondence.

math.RT↗

arXiv2025

Weak unipotence and Langlands duality

Weak unipotence of primitive ideals is a crucial property in the study of unitary representations of reductive groups. We establish a sufficient condition, referred to as mild unipotence, which guarantees weak unipotence and is more accessible in practice. We establish mild unipotence for both the $q$-unipotent ideals defined by McGovern and unipotent ideals attached to nilpotent orbit covers defined by Losev-Mason-Brown-Matvieievskyi (arXiv:2108.03453 [math.RT]). Our proof is conceptual and uses the bijection between special orbits in type $D$ and metaplectic special orbits in type $C$ found by Barbasch-Ma-Sun-Zhu (arXiv:2010.16089 [math.RT]) in an essential way.

math.RT↗

arXiv2024

PrecLLM: A Privacy-Preserving Framework for Efficient Clinical Annotation Extraction from Unstructured EHRs using Small-Scale LLMs

Large Language Models (LLMs) have demonstrated remarkable proficiency in automated text annotation within natural language processing. However, their deployment in clinical settings is severely constrained by strict privacy regulations and the prohibitive computational cost of processing voluminous unstructured Electronic Health Records (EHRs). In this study, we developed a resource-efficient preprocessing technique that can be adopted in existing LLM procedures. This approach is particularly useful for smaller LLMs, which are often more accuracy-challenged, and forms a compact LLM framework optimized for local deployment in computational environments with stringent privacy requirements and restricted access to high-performance GPUs (PrecLLM). The preprocessing step includes both regular expressions (regex) and Retrieval-Augmented Generation (RAG) to extract and highlight key information from unstructured clinical notes. Pre-filtering long and unstructured texts enhanced the performance of smaller LLMs on EHR-related tasks. Evaluation was performed on two distinct cohorts: a locally curated private EHR dataset from the EPIC system for a Head and Neck Cancer (HNC) cohort, and the publicly available EHR dataset (MIMIC-IV). Using MIMIC-IV, we further compared PrecLLM against fine-tuned LLMs. Results demonstrated that PrecLLM substantially enhanced the performance of the original smaller LLMs in terms of sensitivity, specificity, and F1 scores, making it well-suited for privacy-sensitive and resource-constrained applications. This study offers optimized LLM performance for local, secure, and efficient healthcare applications, and provides practical guidance for clinical LLM deployment while addressing challenges related to privacy, computational feasibility, and clinical applicability.

cs.AI↗

arXiv2023

Unipotent Representations of Complex Groups and Extended Sommers Duality

Let $G$ be a complex reductive algebraic group. In arXiv:2108.03453, we have defined a finite set of irreducible admissible representations of $G$ called `unipotent representations', generalizing the special unipotent representations of Arthur and Barbasch-Vogan. These representations are defined in terms of filtered quantizations of symplectic singularities and are expected to form the building blocks of the unitary dual of $G$. In this paper, we provide a description of these representations in terms of the Langlands dual group $G^{\vee}$. To this end, we construct a duality map $D$ from the set of pairs $(\mathbb{O}^{\vee},\bar{C})$ consisting of a nilpotent orbit $\mathbb{O}^{\vee} \subset \mathfrak{g}^{\vee}$ and a conjugacy class $\bar{C}$ in Lusztig's canonical quotient $\bar{A}(\mathbb{O}^{\vee})$ to the set of finite covers of nilpotent orbits in $\mathfrak{g}^*$.

math.RT↗

arXiv2023

On Harish-Chandra modules over quantizations of nilpotent orbits

Let $G$ be a semisimple algebraic group over the complex numbers and $K$ be a connected reductive group mapping to $G$ so that the Lie algebra of $K$ gets identified with a symmetric subalgebra of $\mathfrak{g}$. So we can talk about Harish-Chandra $(\mathfrak{g},K)$-modules, where $\mathfrak{g}$ is the Lie algebra of $G$. The goal of this paper is to give a geometric classification of irreducible Harish-Chandra modules with full support over the filtered quantizations of the algebras of the form $\mathbb{C}[\mathbb{O}]$, where $\mathbb{O}$ is a nilpotent orbit in $\mathfrak{g}$ with codimension of the boundary at least $4$. Namely, we embed the set of isomorphism classes of irreducible Harish-Chandra modules into the set of isomorphism classes of irreducible $K$-equivariant suitably twisted local systems on $\mathbb{O}\cap \mathfrak{k}^\perp$. We show that under certain conditions, for example when $K\subset G$ or when $\mathfrak{g}\cong \mathfrak{so}_n,\mathfrak{sp}_{2n}$, this embedding is in fact a bijection. On the other hand, for $\mathfrak{g}=\mathfrak{sl}_n$ and $K=\operatorname{Spin}_n$, the embedding is not bijective and we give a description of the image. Finally, we perform a partial classification for exceptional Lie algebras.

math.RT↗

arXiv2018

Equivariant deformation quantization and coadjoint orbit method

The purpose of this paper is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive groups. We first prove some general results on the existence of equivariant deformation quantization of vector bundles on closed Lagrangian subvarieties, which lie in smooth symplectic varieties with Hamiltonian group actions. Then we apply them to orbit method and construct nontrivial irreducible Harish-Chandra modules for certain coadjoint orbits. Our examples include new geometric construction of representations associated to certain orbits of real exceptional Lie groups.

math.RT↗

arXiv2018

A geometric formula for multiplicities of $K$-types of tempered representations

Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be compact. Under a condition on $K$, which holds in particular if $K$ is maximal compact, we give a geometric expression for the multiplicities of the $K$-types of any tempered representation (in fact, any standard representation) $\pi$ of $G$. This expression is in the spirit of Kirillov's orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of $\pi|_K$ obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of $\mathrm{SU}(p,1)$, $\mathrm{SO}_0(p,1)$ and $\mathrm{SO}_0(2,2)$ restrict multiplicity-freely to maximal compact subgroups.

math.DG↗

arXiv2017

Mackey analogy as deformation of $\mathcal{D}$-modules

Given a real reductive group Lie group $G_\mathbb{R}$, the Mackey analogy is a bijection between the set of irreducible tempered representations of $G_\mathbb{R}$ and the set of irreducible unitary representations of its Cartan motion group. We show that this bijection arises naturally from families of twisted $\mathcal{D}$-modules over the flag variety of $G_\mathbb{R}$.

math.RT↗

arXiv2017

A geometric realisation of tempered representations restricted to maximal compact subgroups

Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be maximal compact. For a tempered representation $\pi$ of $G$, we realise the restriction $\pi|_K$ as the $K$-equivariant index of a Dirac operator on a homogeneous space of the form $G/H$, for a Cartan subgroup $H<G$. (The result in fact applies to every standard representation.) Such a space can be identified with a coadjoint orbit of $G$, so that we obtain an explicit version of Kirillov's orbit method for $\pi|_K$. In a companion paper, we use this realisation of $\pi|_K$ to give a geometric expression for the multiplicities of the $K$-types of $\pi$, in the spirit of the quantisation commutes with reduction principle. This generalises work by Paradan for the discrete series to arbitrary tempered representations.

math.RT↗

arXiv2016

Mackey analogy via $\mathcal{D}$-modules in the example of $SL(2,\mathbb{R})$

A conjecture by Mackey and Higson claims that there is close relationship between irreducible representations of a real reductive group and those of its Cartan motion group. The case of irreducible tempered unitary representations has been verified recently by Afgoustidis. We study the admissible representations of $SL(2,\mathbb{R})$ by considering families of $\D$-modules over its flag varieties. We make a conjecture which gives a geometric understanding of the Makcey-Higson bijection in the general case.

math.RT↗

arXiv2015

Todd class via homotopy perturbation theory

We compute the quantized cycle class of a closed embedding of complex manifolds defined by Grivaux using homotopy perturbation theory. In the case of a diagonal embedding, our approach provides a novel perspective of the usual Todd class of a complex manifold.

math.AG↗

arXiv2015

Dolbeault dga and $L_\infty$-algebroid of the formal neighborhood

We continue the study the Dolbeault dga of the formal neighborhood of an arbitary closed embedding of complex manifolds previously defined by the author in \cite{DolbeaultDGA}. The special case of the diagonal embedding has been studied in \cite{Diagonal}. We describe the Dolbeault dga explicitly in terms of the formal differential geometry of the embedding. Moreover, we show that the Dolbeault dga is the completed Chevalley-Eilenberg dga an $L_\infty$-algebroid structure on the shifted normal bundle of the submanifold. This generlizes the result of Kapranov on the diagonal embedding and Atiyah class.

math.AG↗

arXiv2012

The Dolbeault dga of the formal neighborhood of the diagonal

A well-known theorem of Kapranov states that the Atiyah class of the tangent bundle $TX$ of a complex manifold $X$ makes the shifted tangent bundle $TX[-1]$ into a Lie algebra object in the derived category $D(X)$. Moreover, he showed that there is an $L_\infty$-algebra structure on the Dolbeault resolution of $TX[-1]$ and wrote down the structure maps explicitly in the case when $X$ is K\"ahler. The corresponding Chevalley-Eilenberg complex is isomorphic to the Dolbeault resolution of the jet bundle $\mathcal{J}^\infty_X$ via the construction of the holomorphic exponential map of the K\"ahler manifold. In this paper, we show that the Dolbeault resolution of the jet bundle is naturally isomorphic to the Dolbeault dga associated to the formal neighborhood of the diagonal of $X \times X$ which we introduced in a previous paper. We also give an alternative proof of Kapranov's theorem by obtaining an explicit formula for the pullback of functions via the holomorphic exponential map, which allows us to study the general case of an arbitrary embedding later.

math.AG↗

arXiv2012

Dolbeault dga of a formal neighborhood

Inspired by a work of Kapranov, we define the notion of Dolbeault complex of the formal neighborhood of a closed embedding of complex manifolds. This construction allows us to study coherent sheaves over the formal neighborhood via complex analytic approach, as in the case of usual complex manifolds and their Dolbeault complexes. Moreover, our the Dolbeault complex as a differential graded algebra can be associated with a dg-category according to Block. We show this dg-category is a dg-enhancement of the bounded derived category over the formal neighborhood under the assumption that the submanifold is compact. This generalizes a similar result of Block in the case of usual complex manifolds.

math.AG↗