G. Lusztig
In a 1983 paper the author has established a (decategorified) Satake equivalence for affine Hecke algebras. In this paper we give new proofs for some results of that paper, one based on the theory of J-rings and one based on the known character formula for rational representations of a reductive group in positive, large, characteristic. We also give an extension of that character formula to disconnected groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
For certain characters of the compact torus of a reductive $p$-adic group, which we call strongly parabolic characters, we prove Satake-type isomorphisms. Our results generalize those of Satake, Howe, Bushnell and Kutzko, and Roche.
We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an irreducible algebraic representation of $G$.
For an arbitrary split adjoint group we identify the unramified Whittaker space with the space of skew-invariant functions on the lattice of coweights and deduce from it the Casselman-Shalika formula.
Let G denote a connected reductive group over a nonarchimedean local field F. Let K denote a special maximal parahoric subgroup of G(F). We establish a Satake isomorphism for the Hecke algebra H of K-bi-invariant compactly supported functions on G(F). The key ingredient is a Cartan decomposition describing the double coset space K\G(F)/K. We also describe how our results relate to the treatment of Cartier, where K is replaced by a special maximal compact open subgroup K' of G(F) and where a Satake isomorphism is established for the Hecke algebra of K'-bi-invariant compactly supported functions…
Let F be a local field with finite residue field of characteristic p and k an algebraic closure of the residue field. Let G be the group of F-points of a F-split connected reductive group. In the apartment corresponding to a chosen maximal split torus of T, we fix a hyperspecial vertex and denote by K the corresponding maximal compact subgroup of G. Given an irreducible smooth k-representation $ρ$ of K, we construct an isomorphism from the affine semigroup k-algebra of the dominant cocharacters of T onto the Hecke algebra $H(G, ρ)$. In the case when the derived subgroup of G is simply connecte…
Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported, K-biequivariant functions f: G(F) \to End V. These Hecke algebras were first considered by Barthel-Livne for GL_2. They play a role in the recent mod p and p-adic Langlands correspondences for GL_2(Q_p), in generalisations of Serre's conjecture on the modularity of mod p …
We study the center of the pro-p Iwahori-Hecke ring H of a connected split p-adic reductive group G. For k an algebraically closed field with characteristic p, we prove that the center of the k-algebra H_k:= H\otimes_Z k contains an affine semigroup algebra which is naturally isomorphic to the Hecke algebra attached to any irreducible smooth k-representation of a given hyperspecial maximal compact subgroup of G. This isomorphism is obtained using the inverse Satake isomorphism constructed in arXiv:1207.5557. We apply this to classify the simple supersingular H_k-modules, study the supersingula…
This article constructs the Satake parameter for any irreducible smooth $J$-spherical representation of a $p$-adic group, where $J$ is any parahoric subgroup. This parametrizes such representations when $J$ is a special maximal parahoric subgroup. The main novelty is for groups which are not quasi-split, and the construction should play a role in formulating a geometric Satake isomorphism for such groups over local function fields.
Let $G$ be any connected reductive $p$-adic group. Let $K\subset G$ be any special parahoric subgroup and $V,V'$ be any two irreducible smooth $\overline {\mathbb F}_p[K]$-modules. The main goal of this article is to compute the image of the Hecke bi-module $\operatorname{End}_{\overline {\mathbb F}_p[K]}(\operatorname{c-Ind}_K^G V, \operatorname{c-Ind}_K^G V')$ by the generalized Satake transform and to give an explicit formula for its inverse, using the pro-$p$ Iwahori Hecke algebra of $G$. This immediately implies the "change of weight theorem" in the proof of the classification of mod $p$ …
Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism
该研究利用Satake同构与Hall-Littlewood多项式，刻画非阿基米德域上分裂约化群乘积的奇异数与角元，推广已有结果并证明其最小分离性，推导相关概率极限定理，建立代数分解与概率极限的统一框架。
We prove that the geometric Satake correspondence admits quantum corrections for minuscule Grassmannians of Dynkin types $A$ and $D$. We find, as a corollary, that the quantum connection of a spinor variety $OG(n,2n)$ can be obtained as the half-spinorial representation of that of the quadric $Q_{2n-2}$. We view the (quantum) cohomology of these Grassmannians as endowed simultaneously with two structures, one of a module over the algebra of symmetric functions, and the other, of a module over the Langlands dual Lie algebra, and investigate the interaction between the two. In particular, we stu…
The integral motivic Satake equivalence for ramified groups
本文构建了非阿基米德局部域上准分裂reductive群的几何萨塔克等价式，扩展了Gaussent-Littelmann的工作，并推广了Zhu的积分萨塔克同构到分裂群，定义了通用球面海格尔代数并构造了通用萨塔克和伯恩斯坦同构。
We prove the geometrical Satake isomorphism for a reductive group defined over F=k((t)), and split over a tamely ramified extension. As an application, we give a description of the nearby cycles on certain Shimura varieties via the Rapoport-Zink-Pappas local models.
Spectral mod p Satake isomorphism for GL_n
本文为 GL_n 的非 Steinberg Serre 权对应的 Emerton--Gee 栈不可约分支构造了到 Hecke 代数的自然单射，并在一般情形下证明其为同构，从而得到带抛物结构的分层。
Some applications of the geometric Satake equivalence to modular representation theory
本研究将几何Satake等价应用于模表示论，给出正特征域上约化代数群不可分解倾斜模的特征标描述，为与G. Williamson的合作成果。
Geometrization of the Satake transform for mod $p$ Hecke algebras
本文通过几何方法推导了模p海肯代数的萨塔克变换公式，并探讨了其在局部朗兰兹对应中的应用。
This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dime…
Block decomposition via the geometric Satake equivalence
通过萨塔克范畴和Smith-Treumann理论，重新证明了正特征域上代数群表示的块结构，并给出了块分解的量子群在根号l次单位时的新的证明。
We prove the Casselman-Shalika formula for unramified groups over a non-archimedean local field by studying the action of the spherical Hecke algebra on the space of compact spherical Whittaker functions via the twisted Satake transform. This method provides a conceptual explanation of the appearance of characters of a dual group in the Casselman-Shalika formula.
Let $k$ be an algebraically closed field of characteristic $p$. Denote by $W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified finite extension of $W(k)[1/p]$ and $\mathcal{O}$ the its ring of integers. For a connected reductive group scheme $G$ over $\mathcal{O}$, we study the category $P_{L^+G}(Gr_G,Λ)$ of $L^+G$-equivariant perverse sheaves in $Λ$-coefficient on the affine Grassmannian $Gr_G$ where $Λ=\mathbb{Z}_{\ell}$ and $\mathbb{F}_{\ell}$ and prove it is equivalent as a tensor category to the category of finitely generated $Λ$-representations of the Langlands dual …
The asymptotic Plancherel formula and Lusztig's asymptotic algebra for $\tilde{\mathsf{A}}_n$
本文通过构造平衡的单元模系统，证明渐近Plancherel定理，并发展每个双侧Kazhdan-Lusztig单元的相对Satake同构，给出Lusztig渐近代数的新显式构造。
Derived geometric Satake equivalence on the Beilinson-Drinfeld Grassmannian with one leg in mixed characteristic
本文研究了在混合特征下，导出几何萨塔克等价性在格拉斯曼流形上的应用，证明了局部海克栈上微分层导出范畴与L群等变完美复叠的等价性。
Let $\mathrm{G}$ be a split reductive group, $K$ be a non-Archimedean local field, and $O$ be its ring of integers. Satake isomorphism identifies the algebra of compactly supported invariants $\mathbb{C}_c[\mathrm{G}(K)/\mathrm{G}(O))]^{\mathrm{G}(O)}$ with a complexification of the algebra of characters of finite-dimensional representations $\mathcal{O}(\mathrm{G}^L(\mathbb{C}))^{\mathrm{G}^L(\mathbb{C})}$ of the Langlands dual group. In this note we report on the results of the study of analogues of such an isomorphism for finite groups. In our setup we replaced Gelfand pair $\mathrm{G}(O)\s…
Quantum Satake in Type A: The General and Generic Case
本文在泛化条件下证明了A_{n-1}型（n≥4）的量子Satake等价，将代数重构及其q-变形从A₁、A₂推广到一般情形，并推广了相关文献结果。
For an almost split Kac-Moody group G over a local non-archimedean field, the last two authors constructed a spherical Hecke algebra H (over the complex numbers C, say) and its Satake isomorphism with the commutative algebra of Weyl invariant elements in some formal series algebra C[[Y]].In this article, we prove a Macdonald's formula, i.e. an explicit formula for the image of a basis element of H. The proof involves geometric arguments in the masure associated to G and algebraic tools, including the Cherednik's representation of the Bernstein-Lusztig-Hecke algebra (introduced in a previous ar…
The geometric Satake isomorphism is an equivalence between the categories of spherical perverse sheaves on affine Grassmanian and the category of representations of the Langlands dual group. We provide a similar description for derived categories of l-adic sheaves on an affine flag variety which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra; of the anti-spherical module over this algebra; and of the space of Iwahori-invariant Whitakker functions.
The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an…
As a generalization of the modular isomorphism problem we study the behavior of defect groups under Morita equivalence of blocks of finite groups over algebraically closed fields of positive characteristic. We prove that the Morita equivalence class of a block B of defect at most 3 determines the defect groups of B up to isomorphism. In characteristic 0 we prove similar results for metacyclic defect groups and 2-blocks of defect 4. In the second part of the paper we investigate the situation for p-solvable groups G. Among other results we show that the group algebra of G itself determines if G…
We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups and only depends on the classification by means of Artin-Tits' simple order theorem.
Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p$, and assume that $p$ is a very good prime for $G$. Let $P$ be a parabolic subgroup whose unipotent radical $U_P$ has nilpotence class less than $p$. We show that there exists a particularly nice Springer isomorphism for $G$ which restricts to a certain canonical isomorphism $\text{Lie}(U_P) \xrightarrow{\sim} U_P$ defined by J.-P. Serre. This answers a question raised both by G. McNinch in \cite{M2}, and by J. Carlson \textit{et. al} in \cite{CLN}. For the groups $SL_n, SO_n$, and $Sp_{2n}$, viewed in …
Let $p$ be a prime and let $G$ be a finite $p$-group. We show that the isomorphism type of the maximal abelian direct factor of $G$, as well as the isomorphism type of the group algebra over $\mathbb F_p$ of the non-abelian remaining direct factor, are determined by $\mathbb F_p G$, generalizing the main result in arXiv:2110.10025 over the prime field. In order to do this, we address the problem of finding characteristic subgroups of $G$ such that their relative augmentation ideals depend only on the $k$-algebra structure of $kG$, where $k$ is any field of characteristic $p$, and relate it to …
On the modular isomorphism problem for groups of nilpotency class 2 with cyclic center
该研究解决了具有循环中心的幂零类2群的模同构问题，证明p元域下这类p-群的群代数同构可推出群同构，奇数阶群在任意特征p域下也成立，偶数阶群需附加群或域的假设。
We prove a comparison theorem between locally analytic group cohomology and Lie algebra cohomology for locally analytic representations of a Lie group over a nonarchimedean field of characteristic 0. The proof is similar to that of van-Est's isomorphism and uses only a minimum of functional analysis.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。