Cuspidal character sheaves on graded exceptional Lie algebras: stable gradings
We determine cuspidal character sheaves explicitly for all (GIT) stably graded exceptional Lie algebras.
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arXiv subjects
Ting Xue
Publications and source records attributed to Ting Xue.
Cuspidal character sheaves on graded Lie algebras II
In this paper we give a complete classification of cyclically graded semisimple Lie algebras that afford cuspidal character sheaves and determine the support of the cuspidal character sheaves. This constitutes a major step towards the explicit classification of cuspidal character sheaves for graded Lie algebras.
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Level-Rank Dualities for Finite Reductive Groups
This is an extended abstract of our work "Level-Rank Dualities from $\Phi$-Cuspidal Pairs..." We present evidence for a family of surprising coincidences within the representation theory of a finite reductive group $G$: more precisely, dualities between blocks of cyclotomic Hecke algebras attached by Brou\'e-Malle to $\Phi$-cuspidal pairs of $G$, where the Hecke parameters are specialized not to the order of the underlying finite field, but to roots of unity. For the groups $G = \mathrm{GL}_n(\mathbf{F}_q)$, these coincidences can be expressed very concretely in terms of the combinatorics of partitions, and the whole story recovers an avatar of the level-rank duality studied by Frenkel, Uglov, Chuang-Miyachi, and others.
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Cuspidal character sheaves on graded Lie algebras
We show in this paper that in the context of graded Lie algebras, all cuspidal character sheaves arise from a nearby-cycle construction followed by a Fourier--Sato transform in a very specific manner. Combined with results of the last two named authors, this completes the classification of cuspidal character sheaves for Vinberg's type I graded classical Lie algebras.
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Level-Rank Dualities from $\Phi$-Cuspidal Pairs and Affine Springer Fibers
We propose a generalization of the level-rank dualities arising from Uglov's work on higher-level Fock spaces. The statements use Hecke algebras defined by Brou\'{e}-Malle, which conjecturally describe the endomorphisms of Lusztig induction modules, and a generalization of Harish-Chandra theory due to Brou\'{e}-Malle-Michel. For any generic finite reductive group $\mathbb{G}$ and integers $e, m > 0$, we conjecture that: (1) the intersection of a $\Phi_e$-Harish-Chandra series and a $\Phi_m$-Harish-Chandra series is parametrized by a union of blocks of the Hecke algebra of the $\Phi_e$-cuspidal pair at an $m$th root of unity, and similarly for the Hecke algebra of the $\Phi_m$-cuspidal pair at an $e$th root of unity; (2) these parametrizations match the blocks on the two sides; (3) when two blocks match, the bijection between them lifts to a derived equivalence between associated blocks of rational DAHAs. Surprisingly, these structures also appear in bimodules formed from the cohomology of affine Springer fibers studied by Oblomkov-Yun. When $\mathbb{G} = \mathbb{GL}_n$ and $e, m$ are coprime, we show that (1)-(3) hold, and that (3) recovers the level-rank dualities conjectured by Chuang-Miyachi and later proved through the work of several other people. Finally, we verify for many cases where $\mathbb{G}$ is exceptional that Brou\'{e}-Malle's parameters are numerically compatible with our conjectures.
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The Ariki--Koike algebras and Rogers--Ramanujan type partitions
In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike algebras $\mathcal{H}_{\mathbb{C},q;Q_1,\ldots, Q_m}\big(G(m, 1, n)\big)$ (when the parameters are roots of unity and $q\neq 1$) are labeled by the so-called Kleshchev multipartitions. This together with Ariki's categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl--Kac character formula. In this paper, we revisit this generating function for the $q=-1$ case. This $q=-1$ case is particularly interesting, for the corresponding Kleshchev multipartitions have a very close connection to generalized Rogers--Ramanujan type partitions when $Q_1=\cdots=Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Based on this connection, we provide an analytic proof of the result of Ariki and Mathas for $q=Q_1=\cdots Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Our second objective is to investigate simple modules of the Ariki--Koike algebra in a fixed block. It is known that these simple modules in a fixed block are labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic is also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we provide two bivariate generating function identities when $m=2$.
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Control theoretically explainable application of autoencoder methods to fault detection in nonlinear dynamic systems
This paper is dedicated to control theoretically explainable application of autoencoders to optimal fault detection in nonlinear dynamic systems. Autoencoder-based learning is a standard machine learning method and widely applied for fault (anomaly) detection and classification. In the context of representation learning, the so-called latent (hidden) variable plays an important role towards an optimal fault detection. In ideal case, the latent variable should be a minimal sufficient statistic. The existing autoencoder-based fault detection schemes are mainly application-oriented, and few efforts have been devoted to optimal autoencoder-based fault detection and explainable applications. The main objective of our work is to establish a framework for learning autoencoder-based optimal fault detection in nonlinear dynamic systems. To this aim, a process model form for dynamic systems is firstly introduced with the aid of control theory, which also leads to a clear system interpretation of the latent variable. The major efforts are made on the development of a control theoretic solution to the optimal fault detection problem, in which an analog concept to minimal sufficient statistic, the so-called lossless information compression, is introduced and proven for dynamic systems and fault detection specifications. In particular, the existence conditions for such a latent variable are derived, based on which a loss function and further a learning algorithm are developed. This learning algorithm enables optimally training of autoencoders to achieve an optimal fault detection in nonlinear dynamic systems. A case study on three-tank system is given at the end of this paper to illustrate the capability of the proposed autoencoder-based fault detection and to explain the essential role of the latent variable in the proposed fault detection system.
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Character sheaves for classical graded Lie algebras
In this note we study character sheaves for graded Lie algebras arising from inner automorphisms of special linear groups and Vinberg's type II classical graded Lie algebras.
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Invariant systems and character sheaves for graded Lie algebras
In this paper we introduce a new ingredient, invariant systems of differential equations, to our study of character sheaves on graded Lie algebras. The character sheaves we construct in this paper, together with the ones constructed in [VX1, X2], constitute all cuspidal character sheaves for Vinberg's type I classical graded Lie algebras.
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Character sheaves for symmetric pairs: spin groups
We determine character sheaves for symmetric pairs associated to spin groups. In particular, we determine the cupsidal character sheaves and show that they can be obtained via the nearby cycle construction of [GVX] and its generalisation in [VX2].
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Character sheaves for symmetric pairs: special linear groups
We give an explicit description of character sheaves for the symmetric pairs associated to inner involutions of the special linear groups. We make use of the general strategy given in [VX1] and central character consideration. We also determine the cuspidal character sheaves.
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A note on Hessenberg varieties
We give a short proof based on Lusztig's generalized Springer correspondence of some results of [BrCh,BaCr,P].
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Springer correspondence, hyperelliptic curves, and cohomology of Fano varieties
In \cite{CVX3}, we have established a Springer theory for the symmetric pair $(\operatorname{SL}(N),\operatorname{SO}(N))$. In this setting we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations arise from cohomology of families of certain (Hessenberg) varieties. In this paper we determine the Springer correspondence explicitly for IC sheaves supported on order 2 nilpotent orbits. In this process we encounter universal families of hyperelliptic curves. As an application we calculate the cohomolgy of Fano varieties of $k$-planes in the smooth intersection of two quadrics in an even dimensional projective space.
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Nearby Cycle Sheaves for Stable Polar Representations
Let G|V, G connected, reductive over C, be a stable polar representation in the sense of [DK], satisfying some mild additional hypotheses. Given a G-equivariant rank one local system L on the general fiber of the quotient map f : V --> V/G, we compute the Fourier transform of the corresponding nearby cycle sheaf P on the zero-fiber of f. This provides a partial generalization of the results of [Gr1] and [GVX1]. Our main intended application is to the theory of character sheaves for graded Lie algebras over C.
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Character Sheaves for Graded Lie Algebras: Stable Gradings
In this paper we construct full support character sheaves for stably graded Lie algebras. Conjecturally these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection groups at roots of unity enter the description. We do so by analysing the Fourier transform of the nearby cycle sheaves constructed in [GVX2].
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The null-cone and cohomology of vector bundles on flag manifolds
We study the null-cone of a semi-simple algebraic group acting on a number of copies of its Lie algebra via the diagonal adjoint action. We show that the null cone has rational singularities in the case of SL3. We observe by example that the null-cone is not normal in general and that the normalization of the null-cone does not have rational singularities in general. This is achieved by computing cohomology of certain vector bundles on flag varieties.
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Hessenberg varieties, intersections of quadrics, and the Springer correspondence
In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [CVX2].
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Springer correspondence for exceptional Lie algebras and their duals in small characteristic
We describe the Springer correspondence explicitly for exceptional Lie algebras of type $G_2$ and $F_4$ and their duals in bad characteristics, i.e. in characteristics 2 and 3.
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