We show that the algebras describing blocks of the category of cuspidal weight (respectively generalized weight) $\mathfrak{sl}_n$-modules are one-parameter (respectively multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley-Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and Schur algebras and in the study of the Milnor fiber of Kleinian singularities.
We give a description of all the cuspidal representations of $\mathrm{GL}_4(\mathfrak{o}_2)$, where $\mathfrak{o}_2$ is a finite ring coming from the ring of integers in a local field, modulo the square of its maximal ideal $\mathfrak{p}$. This shows in particular the existence of representations which are cuspidal, yet are not strongly cuspidal, that is, do not have orbit with irreducible characteristic polynomial mod $\mathfrak{p}$. It has been shown by Aubert, Onn, and Prasad that this phenomenon cannot occur for $\mathrm{GL}_n$, when $n$ is prime.
We show that every block of the category of cuspidal generalized weight modules with finite dimensional generalized weight spaces over the Lie algebra sp(2n)(C) is equivalent to the category of finite dimensional C[[t_1,t_2,...,t_n]]-modules.
In this paper we study the partial Brauer $\mathbb{C}$-algebras $\mathfrak{R}_n(δ,δ')$, where $n \in \mathbb{N}$ and $δ,δ'\in\mathbb{C}$. We show that these algebras are generically semisimple, construct the Specht modules and determine the Specht module restriction rules for the restriction $\mathfrak{R}_{n-1} \hookrightarrow \mathfrak{R}_n$. We also determine the corresponding decomposition matrix, and the Cartan decomposition matrix.
We introduce the marked Brauer algebra and the marked Brauer category. These generalize the analogous constructions for the ordinary Brauer algebra to the setting of a homogeneous bilinear form on a $\mathbb{Z}_2$-graded vector space. We classify the simple modules of the marked Brauer algebra over any field of characteristic not two. Under suitable assumptions we show that the marked Brauer algebra is in Schur-Weyl duality with the Lie superalgebra, $\mathfrak{g}$, of linear maps which leave the bilinear form invariant. We also provide a classification of the indecomposable summands of the te…
Let $A$ be the locally unital algebra associated to a cyclotomic oriented Brauer category over an arbitrary algebraically closed field $\Bbbk$ of characteristic $p\ge 0$. The category of locally finite dimensional representations of $A $ is used to give the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra $\mathfrak g$, where $\mathfrak g$ is a direct sum of copies of $\mathfrak {sl}_\infty$ (resp., $ \hat{\mathfrak {sl}}_p$ ) if $p=0$ (resp., $p>0$)…
Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of order $q$. When $q$ is small, we make further assumptions on $G$. Then we determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive $p$-adic groups.
Let $m, n\in{\mathbb N}$. In this paper we study the right permutation action of the symmetric group ${\mathfrak S}_{2n}$ on the set of all the Brauer $n$-diagrams. A new basis for the free ${\mathbb Z}$-module ${\mathfrak B}_n$ spanned by these Brauer $n$-diagrams is constructed, which yields Specht filtrations for ${\mathfrak B}_n$. For any $2m$-dimensional vector space $V$ over a field of arbitrary characteristic, we give an explicit and characteristic free description of the annihilator of the $n$-tensor space $V^{\otimes n}$ in the Brauer algebra ${\mathfrak B}_n(-2m)$. In particular, we …
We formulate Nazarov-Wenzl type algebras ${\widehat{P}_d^-}$ for the representation theory of the Periplectic Lie superalgebras $\mathfrak{p}(n)$. We establish a Arakawa-Suzuki type theorem to provide a connection between $\mathfrak{p}(n)$-representations and $\widehat{P}_d^-$-representations. We also consider various tensor product representations for $\widehat{P}_d^-$. The periplectic Brauer algebra $A_d$ defined by Moon is an quotient of $\widehat{P}_d^-$. In particular, actions induced by Jucys-Murphy elements can be obtained under the tensor product representation of $\widehat{P}_d^-$. Al…
A proof of Harish-Chandra's integrability theorem for cuspidal representations of $\mathrm{GL}_n(\mathbb F_\ell((t)))$
Weizmann Institute of Science（魏茨曼科学研究所）；Northwestern University（西北大学）；The Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Ben Gurion University of the Negev（内盖夫本-古里安大学）
Let $\mathcal{L}^{S}\left(s,π,χ,\operatorname{\mathfrak{st}}\right)$ be a standard twisted partial $\mathcal{L}$-function of degree $7$ of the cuspidal automorphic representation $π$ of the exceptional group of type $G_2$. In this paper we construct a family of Rankin-Selberg integrals representing this $\mathcal{L}$-function. As an application, we prove that the representations attaining certain prescribed poles are exactly the representations attained by $θ$-lift from a group of finite type.
We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras ${\mathsf {BC}}_{r,s}(q)$. The superalgebra ${\mathsf {BC}}_{r,s}(q)$ is a quantum deformation of the walled Brauer-Clifford superalgebra ${\mathsf {BC}}_{r,s}$ and a super version of the quantum walled Brauer algebra. We prove that ${\mathsf {BC}}_{r,s}(q)$ is the centralizer superalgebra of the action of ${\mathfrak U}_{q}({\mathfrak q}(n))$ on the mixed tensor space $\mathbf{V}_{q}^{r,s}=\mathbf{V}_{q}^{\otimes r} \otimes (\mathbf{V}_q^*)^{\otimes s}$ when $n \ge r+s$, where ${\mathbf V}_{q}=\mathbb…
Let $F$ be a non-Archimedean local field and let $\mathcal{O}_{F}$ be its ring of integers. The orbit of an irreducible representation $ρ$ of $\mathrm{GL}_n(\mathcal{O}_F)$ is a conjugacy class in $\mathfrak{gl}_n(\mathcal{O}_F)$ attached to $ρ$ by means of Clifford's theory. We give a description of orbits of cuspidal types on $\mathrm{GL}_{p}( \mathcal{O}_{F})$, with $p$ prime. We determine which of them are regular and we provide an example which shows that the orbit of a representation does not always determine whether it is a cuspidal type or not.
A representation of $\mathfrak{gl}(V)=V \otimes V^*$ is a linear map $μ\colon \mathfrak{gl}(V) \otimes M \to M$ satisfying a certain identity. By currying, giving a linear map $μ$ is equivalent to giving a linear map $a \colon V \otimes M \to V \otimes M$, and one can translate the condition for $μ$ to be a representation to a condition on $a$. This alternate formulation does not use the dual of $V$, and makes sense for any object $V$ in a tensor category $\mathcal{C}$. We call such objects representations of the curried general linear algebra on $V$. The currying process can be carried out fo…
Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let $Λ^+$ be an indexing set of all simple modules in $A\lmod$. We show that if there is an integer $r\in\N$ such that for any $λ\inΛ^+$, there is an embedding $Δ(λ)\hookrightarrow T^{\oplus r}$ as well as an epimorphism $T^{\oplus r}\twoheadrightarrow\overline{\nabla}(λ)$ as $A$-modules, then $T$ is a faithful $A$-module and $A$ has the double centraliser property with respect to $T$. As applications, we prove that if $A$ is quasi-hereditary with a simple preserving duality and $T$ a given faithful …
We define an extension of the affine Brauer algebra, the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group and it naturally acts on $END_K(X \otimes V^{\otimes k})$ for Orthogonal and Symplectic groups. Thus we obtain a compact analogue of Schur-Weyl duality. We study functors $F_{μ,k}$ from the category of admissible $O(p,q)$ or $Sp_{2n}(\mathbb{R})$ modules to representations of the type B/C affine Brauer algebra $\mathfrak{B}_k^θ$. Thus providing a Akawaka-Suzuki-esque link between $O(p,q)$ (or $Sp_{2n}(\mathbb{R})$) and $\mathfrak{B}_k^θ$. Furthermore thes…
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $\mathcal{C}^0_{\mathfrak{g}}$ be Hernandez-Leclerc's category. For a duality datum $\mathcal{D}$ in $\mathcal{C}^0_{\mathfrak{g}}$, we denote by $\mathcal{F}_{\mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $\mathcal{D}$ to provide the functor $\mathcal{F}_{\mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in $\mathcal{C}^0_{\mathfrak{g}}$, and show that all simple modules in $\mathcal{C}…
We classify Jet modules for the Lie (super)algebras $\mathfrak{L}=W\ltimes(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}])$, where $W$ is the Witt algebra and $\mathfrak{g}$ is a Lie superalgebra with an even diagonlizable derivation. Then we give a concept method to classify all simple cuspidal modules for $\mathfrak{L}$ and the map superalgebras, which are of the form $\mathfrak{L}\otimes R$, where $R$ is a Noetherian unital supercommutative associative superalgebra.
Given a complex orthosymplectic superspace $V$, the orthosymplectic Lie superalgebra $\mathfrak {osp}(V)$ and general linear algebra ${\mathfrak {gl}}_N$ both act naturally on the coordinate super-ring $\mathcal{S}(N)$ of the dual space of $V\otimes{\mathbb C}^N$, and their actions commute. Hence the subalgebra $\mathcal{S}(N)^{\mathfrak {osp}(V)}$ of $\mathfrak {osp}(V)$-invariants in $\mathcal{S}(N)$ has a ${\mathfrak {gl}}_N$-module structure. We introduce the space of super Pfaffians as a simple ${\mathfrak {gl}}_N$-submodule of $\mathcal{S}(N)^{\mathfrak {osp}(V)}$, give an explicit formu…
We prove Schur--Weyl duality between the Brauer algebra $\mathfrak{B}_n(m)$ and the orthogonal group $O_{m}(K)$ over an arbitrary infinite field $K$ of odd characteristic. If $m$ is even, we show that each connected component of the orthogonal monoid is a normal variety; this implies that the orthogonal Schur algebra associated to the identity component is a generalized Schur algebra. As an application of the main result, an explicit and characteristic-free description of the annihilator of $n$-tensor space $V^{\otimes n}$ in the Brauer algebra $mathfrak{B}_n(m)$ is also given.
Let $F$ be a $p$-adic field and let $G$ be a connected reductive group defined over $F$. We assume $p$ is big. Denote $\mathfrak{g}$ the Lie algebra of $G$. To each vertex $s$ of the reduced Bruhat-Tits' building of $G$, we associate as usual a reductive Lie algebra $\mathfrak{g}_{s}$ defined over the residual field ${\mathbb F}_{q}$. We normalize suitably a Fourier-transform $f\mapsto \hat{f}$ on $C_{c}^{\infty}(\mathfrak{g}(F))$. We study the subspace of functions $f\in C_{c}^{\infty}(\mathfrak{g}(F))$ such that the orbital integrals of $f$ and of $\hat{f}$ are $0$ for each element of $ \mat…
In the modular representation theory of finite unitary groups when the characteristic $\ell$ of the ground field is a unitary prime, the $\hat{\mathfrak{sl}}_e$-crystal on level $2$ Fock spaces graphically describes the Harish-Chandra branching of unipotent representations restricted to the tower of unitary groups. However, how to determine the cuspidal support of an arbitrary unipotent representation has remained an open question. We show that for $\ell$ sufficiently large, the $\mathfrak{sl}_\infty$-crystal on the same level $2$ Fock spaces provides the remaining piece of the puzzle for the …
We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group $\mathfrak{u}_q(\mathfrak{g})$, where $q$ is a root of unity.
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $\mathcal{C}_{\mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $\mathcal{C}_{\mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Λ$ and $Λ^\infty$ introduced by the authors. We next define the reflections $\mathcal{S}_k$ and $\mathcal{S}^{-1}_k$ acting on str…
A new class of locally unital and locally finite dimensional algebras $A$ over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of an upper finite triangular decomposition. Any locally unital algebra which admits an upper finite Cartan decomposition is Morita equivalent to some special locally unital algebra $A$ which admits an upper finite weakly triangular decomposition. It is established that the category $A$-lfdmod of locally finite dimensional left $A$-modules is an upper finite fully stratified ca…
Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. In this paper, we study $G_0$-equivariant parity sheaves on $\mathfrak{g}_n$, under some assumptions on the field $\Bbbk$ and the group $G$. The assumption on $G$ holds for $GL_n$ and for any $G$, it recovers results of Lusztig in characteristic $0$. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspid…
A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations
Sorbonne Université（索邦大学）；Université Paris Cité（巴黎西岱大学）；CNRS, IMJ-PRG（法国国家科学研究中心，IMJ-PRG）；School of Mathematics, Southampton University（南安普顿大学数学学院）；School of Mathematics, Manchester University（曼彻斯特大学数学学院）
An irreducible weight module of an affine Kac-Moody algebra $\mathfrak{g}$ is called dense if its support is equal to a coset in $\mathfrak{h}^{*}/Q$. Following a conjecture of V. Futorny about affine Kac-Moody algebras $\mathfrak{g}$, an irreducible weight $\mathfrak{g}$-module is dense if and only if it is cuspidal (i.e. not a quotient of an induced module). The conjecture is confirmed for $\mathfrak{g}=A_{2}^{\left(1\right)}$, $A_{3}^{\left(1\right)}$ and$A_{4}^{\left(1\right)}$ and a classification of the supports of the irreducible weight $\mathfrak{g}$-modules obtained. For all $A_{n}^{\…
For a $p$-adic local field $F$ of characteristic 0, with residue field $\mathfrak{f}$, we prove that the Rankin-Selberg gamma factor of a pair of level zero representations of linear general groups over $F$ is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over $\mathfrak{f}$. Our results can be used to prove a variant of Jacquet's conjecture.
We identify the dimension of the centralizer of the symmetric group $\mathfrak{S}_d$ in the partition algebra $\mathcal{A}_d(δ)$ and in the Brauer algebra $\mathcal{B}_d(δ)$ with the number of multidigraphs with $d$ arrows and the number of disjoint union of directed cycles with $d$ arrows, respectively. Using Schur-Weyl duality as a fundamental theory, we conclude that each centralizer is related with the $G$-invariant space $P^d(M_n(\mathbf{k}))^G$ of degree $d$ homogeneous polynomials on $n \times n$ matrices, where $G$ is the orthogonal group and the group of permutation matrices, respecti…
Let $G$ be a simple simply connected complex algebraic group and let $\mathfrak{g}_*$ be a $\mathbf{Z}/m$-grading on its Lie algebra $\mathfrak{g}$. In a recent series of articles, G. Lusztig and Z. Yun, studied the classification of simple $G_0$-equivariant perverse sheaves on the nilpotent cone of $\mathfrak{g}_i$ for $i\in \mathbf{Z}/m$, where $G_0$ is the exponentiation of the degree zero piece $\mathfrak{g}_0$. They proved a decomposition of the equivariant derived category of $\ell$-adic sheaves on the nilpotent cone of $\mathfrak{g}_i$ into blocks, each generated by a certain cuspidal l…
We define the affine VW supercategory $\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$, which arises from studying the action of the periplectic Lie superalgebra $\mathfrak{p}(n)$ on the tensor product $M\otimes V^{\otimes a}$ of an arbitrary representation $M$ with several copies of the vector representation $V$ of $\mathfrak{p}(n)$. It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in $\mathfrak{p}(n)\otimes \mathfrak{p}(n)$. When $M$ is the…
Let $π$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field $F$. Under the assumption that $π$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(\frac{1}{2},π\timesχ)$, with a certain automorphic character $χ$ of ${\rm U}_1$, for the case of ${\mathfrak n}=2,3,4$, and for the general ${\mathfrak n}\geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results…
We use duality theorems to obtain presentations of some categories of modules. To derive these presentations we generalize a result of Cautis-Kamnitzer-Morrison [arXiv:1210.6437v4]: Let $\mathfrak{g}$ be a reductive Lie algebra, and $A$ an algebra, both over $\mathbb{C}$. Consider a $(\mathfrak{g} , A)$-bimodule $P$ in which (a) $P$ has a multiplicity free decomposition into irreducible $(\mathfrak{g} , A)$-bimodules. (b) $P$ is "saturated" i.e. for any irreducible $\mathfrak{g}$-module $V$, if every weight of $V$ is a weight of $P$, then $V$ is a submodule of $P$. We show that statements (a) …
Let $\mathscr{H}_n$ denote the Iwahori-Hecke algebra corresponding to the symmetric group $\mathfrak{S}_n$. We set up a Green correspondence for bimodules of these Hecke algebras, and a Brauer correspondence between their blocks. We examine Specht modules for $\mathscr{H}_n$ and compute the vertices of certain Specht modules, before using this to give a classification of the vertices of blocks of $\mathscr{H}_n$ in any characteristic. Finally, we apply this classification to resolve the Dipper-Du conjecture about the structure of vertices of indecomposable $\mathscr{H}_n$-modules.
In this paper, a notion of cyclotomic (or level $k$) walled Brauer algebras $\mathscr B_{k, r, t}$ is introduced for arbitrary positive integer $k$. It is proven that $\mathscr B_{k, r, t}$ is free over a commutative ring with rank $k^{r+t}(r+t)!$ if and only if it is admissible. Using super Schur-Weyl duality between general linear Lie superalgebras $\mathfrak{gl}_{m|n}$ and $\mathscr B_{2, r, t}$, we give a classification of highest weight vectors of $\mathfrak{gl}_{m|n}$-modules $M_{pq}^{rt}$, the tensor products of Kac-modules with mixed tensor products of the natural module and its dual. …
