Lucas Ruhstorfer
The Bonnafé-Rouquier equivalence can be seen as a modular analogue of Lusztig's Jordan decomposition for groups of Lie type. In this paper, we show that this equivalence can be lifted to include automorphisms of the finite group of Lie type. Moreover, we prove the existence of a local version of this equivalence which satisfies similar properties.
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Let $\mathcal{J}$ be the exceptional Jordan algebra and $V=\mathcal{J}\oplus \mathcal{J}$. We construct an equivariant map from $V$ to $\mathrm{Hom}_k(\mathcal{J}\otimes \mathcal{J},\mathcal{J})$ defined by homogeneous polynomials of degree $8$ such that if $x\in V$ is a generic point, then the image of $x$ is the structure constant of the isotope of $\mathcal{J}$ corresponding to $x$. We also give an alternative way to define the isotope corresponding to a generic point of $\mathcal{J}$ by an equivariant map from $\mathcal{J}$ to the space of trilinear forms.
Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer
本文证明有限约化群特征存在Galois等变的Jordan分解，并在连通中心化子情形下给出唯一性，类比Digne--Michel的连通中心结果。
As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type $A_{n-1}$, split or twisted. Key to the proofs is the study of certain characters of SL$_n(q)$ and SU$_n(q)$ related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types.
We prove an Induction Equivalence and a Kashiwara Equivalence for coadmissible equivariant D-modules on rigid analytic spaces. This allows us to completely classify such objects with support in a single orbit of a classical point with co-compact stabiliser. As an application, we use the locally analytic Beilinson-Bernstein equivalence to construct new examples of large families of topologically irreducible locally analytic representations of certain compact semisimple p-adic Lie groups.
We show that the family of Podles' spheres is complete under equivariant Morita equivalence (with respect to the action of quantum SU(2)), and determine the associated orbits. We also give explicit formulas for the actions which are equivariantly Morita equivalent with the quantum projective plane. In both cases, the computations are made by examining the localized spectral decomposition of a generalized Casimir element.
We establish Green equivalences for all Mackey 2-functors, without assuming Krull-Schmidt. By running through the examples of Mackey 2-functors, we recover all variants of the Green equivalence and Green correspondence known in representation theory and obtain new ones in several other contexts. Such applications include equivariant stable homotopy theory in topology and equivariant sheaves in geometry.
Equivariant Cleft Extensions and Singular Equivalences
研究阿贝尔范畴的裂扩张的等变提升及其对奇异范畴的影响，证明裂扩张的限制函子诱导奇异等价当且仅当其等变对应物也诱导奇异等价，并应用于斜群环。
Equivariant recollements and singular equivalences
本文研究了可微再分配与奇异等价，探讨了可除和三角范畴的再分配特性，并应用结果构建了导出范畴的可微再分配及斜群代数间的奇异等价。
The string group acts on the category of coherent sheaves over a weighted projective line by degree-shift actions. We study the equivariant equivalence relations induced by degree-shift actions between weighted projective lines. We prove that such an equivariant equivalence is characterized by an admissible homomorphism between the associated string groups. We classify all these equivariant equivalences for the weighted projective lines of domestic and tubular types.
For any positive integer $n$, $n$-derived-simple derived discrete algebras are classified up to derived equivalence. Furthermore, the Jordan-Hölder theorems for all kinds of derived categories of derived discrete algebras are obtained.
Commutativity via Spectral Equivalences of the Jordan Product in Banach Algebras
本文通过研究Banach代数中Jordan乘积的谱不变性，证明了谱等价条件可推出代数的交换性或元素的中心性。
$H$-Equivariant Morita equivalences of Loewy-graded comodule algebras
本文证明Loewy分次余模代数在零次部分满足H0-等变Morita等价时，可提升为H-等变Morita等价，并推广到余理想子代数情形，最后讨论Kac-Paljutkin Hopf代数特例。
As a step to establish the blockwise Alperin weight conjecture for all finite groups, we verify the inductive blockwise Alperin weight condition introduced by Navarro--Tiep and Späth for simple groups of Lie type $\mathsf A$, split or twisted. Key to the proofs is to reduce the verification of the inductive condition to the isolated (that means unipotent) blocks, using the Jordan decomposition for blocks of finite reductive groups given by Bonnafé, Dat and Rouquier.
New techniques for calculation of Jordan-Kronecker invariants for Lie algebras and Lie algebra representations
本文提出利用矩阵束严格等价分层及对偶表示关系两种技术，以简化李代数及其表示的 Jordan-Kronecker 不变量计算。
Equivariant deformation theory for nilpotent slices in symplectic Lie algebras
针对辛李代数中双块若尔当形幂零轨道的Slodowy切片非通用泊松形变问题，证明其幂零Slodowy簇与D型同构，C型切片为$\u2124_2$-等变通用泊松形变且有非交换对应。
Späth showed that the Alperin-McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin-McKay condition holds for all finite simple groups. In a previous article, we showed that the Bonnafé-Rouquier equivalence for blocks of finite groups of Lie type can be lifted to include automorphisms of groups of Lie type. We use our results to reduce the verification of the inductive condition for groups of Lie type to quasi-isolated blocks.
Transfer of abelian model structures to equivariant categories and homotopy squares
针对有限群G在满足|G|可逆的格罗滕迪克范畴𝒜上的作用，证明阿贝尔模型结构从𝒜到等变范畴𝒜^G的提升定理，构造关联导出函子的同伦方块并给出模范畴下的应用。
We provide a uniform construction of $L^2$-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite, and hence the previously used Jordan algebra methods have to be generalized to Jordan pairs. The crucial ingredients for the construction of the $L^2$-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier …
Orbits and equivariant local systems combinatorics in graded Lie algebras
本文用组合方法刻画四类无限族分次李代数（辛群与特殊正交群各两类）中的轨道、Jordan 块分解、维数、偏序及等变局部系统，为研究分次李代数产生的反常层提供工具。
Let $X$ be a smooth scheme with an action of an algebraic group $G$. We establish an equivalence of two categories related to the corresponding moment map $μ: T^*X \to Lie(G)^*$ - the derived category of G-equivariant coherent sheaves on the derived fiber $μ^{-1}(0)$ and the derived category of $G$-equivariant matrix factorizations on $T^*X \times Lie(G)$ with potential given by $μ$.
We give an easy proof of the Bernstein-Lunts equivalence of ordinary and equivariant derived categories of Harish-Chandra modules. This proof requires no boundedness assumptions. In the appendix we collect some needed, but not completely standard facts from homological algebra.
We study the moduli stack of degree $0$ semistable $G$-bundles on an irreducible curve $E$ of arithmetic genus $1$, where $G$ is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups $H$ of $G$ (the $E$-pseudo-Levi subgroups), where each stratum is computed in terms of $H$-bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan--Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where $E$ has a single cusp (res…
We formulate a version of Beck's monadicity theorem for abelian categories, which is applied to the equivariantization of abelian categories with respect to a finite group action. We prove that the equivariantization is compatible with the construction of quotient abelian categories by Serre subcategories. We prove that the equivariantization of the graded module category over a graded ring is equivalent to the graded module category over the same ring but with a different grading. We deduce from these results two equivalences between the category of (equivariant) coherent sheaves on a weighte…
We show that equivariant tilting modules over equivariant algebras induce equivalences of derived factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a Pfaffian variety is equivalent to the derived factorization category of a noncommutative gauged Landau-Ginzburg model $(Λ,χ, w)^{\mathbb{G}_m}$, where $Λ$ is a noncommutative resolution of the quotient singularity $W/\operatorname{GSp}(Q)$ arising from a certain representation $W$ of the symplectic similitude group $\operatorname{GSp}(Q)$ of a symplectic vector spa…
Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this paper is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent b…
Free and cofree equivariant spectra are important classes of equivariant spectra which represent equivariant cohomology theories on free equivariant spaces. Greenlees-Shipley and Pol and the author have given an algebraic model for rational (co)free equivariant spectra. In this paper, we extend this framework by proving that the Quillen functors of induction-restriction-coinduction between categories of (co)free rational equivariant spectra correspond to Quillen functors between the algebraic models in the case of connected compact Lie groups. This is achieved using new abstract techniques reg…
For any Kac-Moody group $G$ with Borel $B$, we give a monoidal equivalence between the derived category of $B$-equivariant mixed complexes on the flag variety $G/B$ and (a certain completion of) the derived category of $B^\vee$-monodromic mixed complexes on the enhanced flag variety $G^\vee/U^\vee$, here $G^\vee$ is the Langlands dual of $G$. We also prove variants of this equivalence, one of which is the equivalence between the derived category of $U$-equivariant mixed complexes on the partial flag variety $G/P$ and certain "Whittaker model" category of mixed complexes on $G^\vee/B^\vee$. In …
Representation theory of the principal equivariant $\mathcal{W}$-algebra and Langlands duality
研究受量子几何朗兰兹纲领推动，从约化群上的手征微分算子顶点代数出发，研究其结构与表示理论，构造\(\mathcal{W}_{G}^{\kappa}\)的单模，在特定群情形下建立基本局部等价，匹配朗兰兹对偶群表示理论组合学。
In math.RT/0201073 we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar ``Langlands dual'' description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik.
In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory $T[Σ,G]$ on $L(k,1) \times S^1$, the other is the $^LG$ "equivariant Verlinde formula", or equivalently partition function of $^LG_{\mathbb{C}}$ complex Chern-Simons theory on $Σ\times S^1$. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally $G$ and its Langlands dual $^LG$. When $G$ is not simply-connected, we provide a recipe of computing the index of $T[Σ,G]$ as summat…
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