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𝚤Hall algebras of weighted projective lines and quantum symmetric pairs Represent. Theory (IF 0.7) Pub Date : 2024-03-04 Ming Lu, Shiquan Ruan
The ı \imath Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 1 1 -periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the ı \imath quantum loop algebra, which is a generalization of the ı \imath quantum group arising from the quantum symmetric
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On input and Langlands parameters for epipelagic representations Represent. Theory (IF 0.7) Pub Date : 2024-02-12 Beth Romano
A paper of Reeder–Yu [J. Amer. Math. Soc. 27 (2014), pp. 437–477] gives a construction of epipelagic supercuspidal representations of p p -adic groups. The input for this construction is a pair ( λ , χ ) (\lambda , \chi ) where λ \lambda is a stable vector in a certain representation coming from a Moy–Prasad filtration, and χ \chi is a character of the additive group of the residue field. We say two
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Co-𝑡-structures on derived categories of coherent sheaves and the cohomology of tilting modules Represent. Theory (IF 0.7) Pub Date : 2024-02-02 Pramod Achar, William Hardesty
We construct a co- t t -structure on the derived category of coherent sheaves on the nilpotent cone N \mathcal {N} of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the “exotic parity objects” (considered by Achar, Hardesty, and Riche [Transform. Groups 24 (2019), pp. 597–657])
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Character formulas in category 𝒪_{𝓅} Represent. Theory (IF 0.7) Pub Date : 2024-01-05 Henning Andersen
Let O p \mathcal {O}_p denote the characteristic p > 0 p>0 version of the ordinary category O \mathcal {O} for a semisimple complex Lie algebra. In this paper we give some (formal) character formulas in O p \mathcal {O}_p . First we concentrate on the irreducible characters. Here we give explicit formulas for how to obtain all irreducible characters from the characters of the finitely many restricted
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L-packets over strong real forms Represent. Theory (IF 0.7) Pub Date : 2024-01-05 N. Arancibia Robert, P. Mezo
Langlands [On the classification of irreducible representations of real algebraic groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170] defined L L -packets for real reductive groups. In order to refine the local Langlands correspondence, Adams-Barbasch-Vogan [The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics
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Solid locally analytic representations of 𝑝-adic Lie groups Represent. Theory (IF 0.7) Pub Date : 2022-08-31 Joaquín Rodrigues Jacinto, Juan Rodríguez Camargo
Abstract:We develop the theory of locally analytic representations of compact $p$-adic Lie groups from the perspective of the theory of condensed mathematics of Clausen and Scholze. As an application, we generalise Lazard’s isomorphisms between continuous, locally analytic and Lie algebra cohomology to solid representations. We also prove a comparison result between the group cohomology of a solid
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Knizhnik–Zamolodchikov functor for degenerate double affine Hecke algebras: algebraic theory Represent. Theory (IF 0.7) Pub Date : 2022-08-30 Wille Liu
Abstract:In this article, we define an algebraic version of the Knizhnik–Zamolodchikov (KZ) functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo–Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for
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The integral geometric Satake equivalence in mixed characteristic Represent. Theory (IF 0.7) Pub Date : 2022-08-18 Jize Yu
Abstract: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 $\mathcal {O}$ its ring of integers. For a connected reductive group scheme $G$ over $\mathcal {O}$, we study the category $\mathrm {P}_{L^+G}(Gr_G,\Lambda )$ of $L^+G$-equivariant perverse sheaves in $\Lambda$-coefficient
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Reflexivity of Newton–Okounkov bodies of partial flag varieties Represent. Theory (IF 0.7) Pub Date : 2022-08-16 Christian Steinert
Abstract:Assume that the valuation semigroup $\Gamma (\lambda )$ of an arbitrary partial flag variety corresponding to the line bundle $\mathcal {L_\lambda }$ constructed via a full-rank valuation is finitely generated and saturated. We use Ehrhart theory to prove that the associated Newton–Okounkov body — which happens to be a rational, convex polytope — contains exactly one lattice point in its interior
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Root components for tensor product of affine Kac-Moody Lie algebra modules Represent. Theory (IF 0.7) Pub Date : 2022-07-26 Samuel Jeralds, Shrawan Kumar
Abstract:Let $\mathfrak {g}$ be an affine Kac-Moody Lie algebra and let $\lambda , \mu$ be two dominant integral weights for $\mathfrak {g}$. We prove that under some mild restriction, for any positive root $\beta$, $V(\lambda )\otimes V(\mu )$ contains $V(\lambda +\mu -\beta )$ as a component, where $V(\lambda )$ denotes the integrable highest weight (irreducible) $\mathfrak {g}$-module with highest
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Universal K-matrices for quantum Kac-Moody algebras Represent. Theory (IF 0.7) Pub Date : 2022-07-19 Andrea Appel, Bart Vlaar
Abstract:We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type
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Calculus of archimedean Rankin–Selberg integrals with recurrence relations Represent. Theory (IF 0.7) Pub Date : 2022-07-06 Taku Ishii, Tadashi Miyazaki
Abstract:Let $n$ and $n’$ be positive integers such that $n-n’\in \{0,1\}$. Let $F$ be either $\mathbb {R}$ or $\mathbb {C}$. Let $K_n$ and $K_{n’}$ be maximal compact subgroups of $\mathrm {GL}(n,F)$ and $\mathrm {GL}(n’,F)$, respectively. We give the explicit descriptions of archimedean Rankin–Selberg integrals at the minimal $K_n$- and $K_{n’}$-types for pairs of principal series representations
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Distinguished strata in a reductive group Represent. Theory (IF 0.7) Pub Date : 2022-06-30 G. Lusztig
Abstract:The set of strata of a reductive group can be viewed as an enlargement of the set of unipotent classes. In this paper the notion of distinguished unipotent class is extended to this larger set. The strata of a Weyl group are also introduced and studied.
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The contraction category of graphs Represent. Theory (IF 0.7) Pub Date : 2022-06-28 Nicholas Proudfoot, Eric Ramos
Abstract:We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes a graph to a graded piece of the homology of its unordered configuration space and the second takes a graph to an intersection homology group
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A proof of the polynomial conjecture for restrictions of nilpotent lie groups representations Represent. Theory (IF 0.7) Pub Date : 2022-06-02 Ali Baklouti, Hidenori Fujiwara, Jean Ludwig
Abstract:Let $G$ be a connected and simply connected nilpotent Lie group, $K$ an analytic subgroup of $G$ and $\pi$ an irreducible unitary representation of $G$ whose coadjoint orbit of $G$ is denoted by $\Omega (\pi )$. Let $\mathscr U(\mathfrak g)$ be the enveloping algebra of ${\mathfrak g}_{\mathbb C}$, $\mathfrak g$ designating the Lie algebra of $G$. We consider the algebra $D_{\pi }(G)^K \simeq
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Derived equivalences and equivariant Jordan decomposition Represent. Theory (IF 0.7) Pub Date : 2022-04-27 Lucas Ruhstorfer
Abstract: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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On the socles of certain parabolically induced representations of 𝑝-adic classical groups Represent. Theory (IF 0.7) Pub Date : 2022-04-25 Hiraku Atobe
Abstract:In this paper, we consider representations of $p$-adic classical groups parabolically induced from the products of shifted Speh representations and unitary representations of Arthur type of good parity. We describe how to compute the socles (the maximal semisimple subrepresentations) of these representations. As a consequence, we can determine whether these representations are reducible or
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Branching of metaplectic representation of 𝑆𝑝(2,ℝ) under its principal 𝕊𝕃(2,ℝ)-subgroup Represent. Theory (IF 0.7) Pub Date : 2022-04-25 GenKai Zhang
We study the branching problem of the metaplectic representation of S p ( 2 , R ) Sp(2, \mathbb R) under its principle subgroup S L ( 2 , R ) SL(2, \mathbb R) . We find the complete decomposition.
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On Donkin’s tilting module conjecture I: lowering the prime Represent. Theory (IF 0.7) Pub Date : 2022-04-04 Christopher Bendel,Daniel Nakano,Cornelius Pillen,Paul Sobaje
In this paper the authors provide a complete answer to Donkin’s Tilting Module Conjecture for all rank 2 2 semisimple algebraic groups and SL 4 ( k ) \operatorname {SL}_{4}(k) where k k is an algebraically closed field of characteristic p > 0 p>0 . In the process, new techniques are introduced involving the existence of ( p , r ) (p,r) -filtrations, Lusztig’s character formula, and the G r G_{r}
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An integral second fundamental theorem of invariant theory for partition algebras Represent. Theory (IF 0.7) Pub Date : 2022-04-01 Chris Bowman, Stephen Doty, Stuart Martin
Abstract:We prove that the kernel of the action of the group algebra of the Weyl group acting on tensor space (via restriction of the action from the general linear group) is a cell ideal with respect to the alternating Murphy basis. This provides an analogue of the second fundamental theory of invariant theory for the partition algebra over an arbitrary commutative ring and proves that the centraliser
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Representations of 2-transitive locally compact groups Represent. Theory (IF 0.7) Pub Date : 2022-03-25 Robert Bekes
Abstract:We show that noncompact representations of 2-transitive locally compact groups are irreducible.
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Base change and triple product 𝐿-series Represent. Theory (IF 0.7) Pub Date : 2022-03-25 Ming-Lun Hsieh,Shunsuke Yamana
Let π i \pi _i be an irreducible cuspidal automorphic representation of G L 2 \mathrm {GL}_2 with central character ω i \omega _i . When ω 1 ω 2 ω 3 \omega _1\omega _2\omega _3 is trivial, Atsushi Ichino proved a formula for the central value L ( 1 2 , π 1 × π 2 × π 3 ) L(\frac {1}{2}, \pi _1\times \pi _2\times \pi _3) of the triple product L L -series in terms of global trilinear forms. We will extend
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Parabolic induction and the Harish-Chandra 𝒟-module Represent. Theory (IF 0.7) Pub Date : 2022-03-24 Victor Ginzburg
Abstract:Let $G$ be a reductive group and $L$ a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between $\operatorname {Ad}$-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) ${\mathscr {D}}$-modules on $G$ and $L$, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors
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Homological invariants of the arrow removal operation Represent. Theory (IF 0.7) Pub Date : 2022-03-23 Karin Erdmann, Chrysostomos Psaroudakis, Øyvind Solberg
Abstract:In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra.
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Inverse Satake isomorphism and change of weight Represent. Theory (IF 0.7) Pub Date : 2022-03-21 N. Abe, F. Herzig, M. F. Vignéras
Abstract: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 bimodule $\operatorname {End}_{\overline {\mathbb {F}_p}[K]}(c-Ind_K^G V, c-Ind_K^G V’)$ by the generalized Satake transform and to give an explicit
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Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types Represent. Theory (IF 0.7) Pub Date : 2022-03-18 Ryo Fujita
Abstract:For a Dynkin quiver $Q$ (of type $\mathrm {ADE}$), we consider a central completion of the convolution algebra of the equivariant $K$-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc’s monoidal category $\mathcal {C}_{Q}$ of modules
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Stable maps, Q-operators and category 𝒪 Represent. Theory (IF 0.7) Pub Date : 2022-03-17 David Hernandez
Abstract:Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category $\mathcal {O}$ of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic
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Defect 2 spin blocks of symmetric groups and canonical basis coefficients Represent. Theory (IF 0.7) Pub Date : 2022-03-17 Matthew Fayers
This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect 2 2 , analogous to Richards’s formula for defect 2 2 blocks of symmetric groups.By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding
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Locally analytic 𝐸𝑥𝑡¹ for 𝐺𝐿₂(ℚ_{𝕡}) in de Rham non-trianguline case Represent. Theory (IF 0.7) Pub Date : 2022-03-08 Yiwen Ding
We prove Breuil’s conjecture on locally analytic E x t 1 \mathrm {Ext}^1 for G L 2 ( Q p ) \mathrm {GL}_2(\mathbb {Q}_p) in de Rham non-trianguline case.
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The dependence on parameters of the inverse functor to the 𝐾-finite functor Represent. Theory (IF 0.7) Pub Date : 2022-03-04 Nolan Wallach
Abstract:An interpretation of the Casselman-Wallach Theorem is that the $K$-finite functor is an isomorphism of categories from the category of finitely generated, admissible smooth Fréchet modules of moderate growth to the category of Harish-Chandra modules for a real reductive group, $G$ (here $K$ is a maximal compact subgroup of $G$). In this paper we study the dependence of the inverse functor
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Coxeter combinatorics for sum formulas in the representation theory of algebraic groups Represent. Theory (IF 0.7) Pub Date : 2022-03-02 Jonathan Gruber
Abstract:Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb {F}$ of characteristic $p \geq h$, the Coxeter number of $G$. We observe an easy ‘recursion formula’ for computing the Jantzen sum formula of a Weyl module with $p$-regular highest weight. We also discuss a ‘duality formula’ that relates the Jantzen sum formula to Andersen’s sum formula for tilting filtrations
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Microlocal characterization of Lusztig sheaves for affine quivers and 𝑔-loops quivers Represent. Theory (IF 0.7) Pub Date : 2022-02-11 Lucien Hennecart
Abstract:We prove that for extended Dynkin quivers, simple perverse sheaves in Lusztig category are characterized by the nilpotency of their singular support. This proves a conjecture of Lusztig in the case of affine quivers. For cyclic quivers, we prove a similar result for a larger nilpotent variety and a larger class of perverse sheaves. We formulate conjectures concerning similar results for quivers
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Coordinate rings and birational charts Represent. Theory (IF 0.7) Pub Date : 2022-01-05 Sergey Fomin, George Lusztig
Abstract:Let $G$ be a semisimple simply connected complex algebraic group. Let $U$ be the unipotent radical of a Borel subgroup in $G$. We describe the coordinate rings of $U$ (resp., $G/U$, $G$) in terms of two (resp., four, eight) birational charts introduced by Lusztig [Total positivity in reductive groups, Birkhäuser Boston, Boston, MA, 1994; Bull. Inst. Math. Sin. (N.S.) 14 (2019), pp. 403–459]
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Equivariant multiplicities of simply-laced type flag minors Represent. Theory (IF 0.7) Pub Date : 2021-12-16 Elie Casbi
Abstract:Let $\mathfrak {g}$ be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism $\overline {D}$ on the coordinate ring $\mathbb {C}[N]$ related to Brion’s equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as
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Typical representations via fixed point sets in Bruhat–Tits buildings Represent. Theory (IF 0.7) Pub Date : 2021-12-16 Peter Latham,Monica Nevins
For a tame supercuspidal representation π \pi of a connected reductive p p -adic group G G , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of G G , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of G G which is not inertially equivalent to π \pi . The consequence
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A proof of Casselman’s comparison theorem Represent. Theory (IF 0.7) Pub Date : 2021-12-01 Ning Li, Gang Liu, Jun Yu
Abstract:Let $G$ be a real linear reductive group and $K$ be a maximal compact subgroup. Let $P$ be a minimal parabolic subgroup of $G$ with complexified Lie algebra $\mathfrak {p}$, and $\mathfrak {n}$ be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation $V$ of $G$, the inclusion $V_{K}\subset V$ induces isomorphisms $H_{i}(\mathfrak
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Intertwining maps between 𝑝-adic principal series of 𝑝-adic groups Represent. Theory (IF 0.7) Pub Date : 2021-12-01 Dubravka Ban,Joseph Hundley
In this paper we study p p -adic principal series representation of a p p -adic group G G as a module over the maximal compact subgroup G 0 G_0 . We show that there are no non-trivial G 0 G_0 -intertwining maps between principal series representations attached to characters whose restrictions to the torus of G 0 G_0 are distinct, and there are no non-scalar endomorphisms of a fixed principal series
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Parametrization, structure and Bruhat order of certain spherical quotients Represent. Theory (IF 0.7) Pub Date : 2021-10-21 Pierre-Emmanuel Chaput, Lucas Fresse, Thomas Gobet
Abstract:Let $G$ be a reductive algebraic group and let $Z$ be the stabilizer of a nilpotent element $e$ of the Lie algebra of $G$. We consider the action of $Z$ on the flag variety of $G$, and we focus on the case where this action has a finite number of orbits (i.e., $Z$ is a spherical subgroup). This holds for instance if $e$ has height $2$. In this case we give a parametrization of the $Z$-orbits
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Quasi-split symmetric pairs of 𝑈(𝔰𝔩_{𝔫}) and Steinberg varieties of classical type Represent. Theory (IF 0.7) Pub Date : 2021-10-21 Yiqiang Li
We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type A n − 1 A_{n-1} . This is obtained inside the limit of a projective system of Borel-Moore homologies of the Steinberg varieties of n n -step isotropic flag varieties. Arising from the construction are a basis of homological origin for the idempotent form and
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Splitting fields of real irreducible representations of finite groups Represent. Theory (IF 0.7) Pub Date : 2021-10-14 Dmitrii Pasechnik
Abstract:We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb {R}$, is realisable over the field $E≔\mathbb {Q}(\zeta _n)\cap \mathbb {R}$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $\rho$ over $\mathbb {Q}(\zeta _n)$ to one over $E$.
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Local Langlands correspondence for unitary groups via theta lifts Represent. Theory (IF 0.7) Pub Date : 2021-10-13 Rui Chen, Jialiang Zou
Abstract:Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which is due to Mok) to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this paper provides an alternative approach to the
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Unipotent representations attached to the principal nilpotent orbit Represent. Theory (IF 0.7) Pub Date : 2021-10-07 Lucas Mason-Brown
Abstract:In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf {K}$-types, associated varieties, and Langlands parameters of all such representations.
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Parabolic induction via the parabolic pro-𝑝 Iwahori–Hecke algebra Represent. Theory (IF 0.7) Pub Date : 2021-10-05 Claudius Heyer
Abstract:Let $\mathbf {G}$ be a connected reductive group defined over a locally compact non-archimedean field $F$, let $\mathbf {P}$ be a parabolic subgroup with Levi $\mathbf {M}$ and compatible with a pro-$p$ Iwahori subgroup of $G ≔\mathbf {G}(F)$. Let $R$ be a commutative unital ring. We introduce the parabolic pro-$p$ Iwahori–Hecke $R$-algebra $\mathcal {H}_R(P)$ of $P ≔\mathbf {P}(F)$ and construct
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Mirković–Vilonen basis in type 𝐴₁ Represent. Theory (IF 0.7) Pub Date : 2021-09-29 Pierre Baumann, Arnaud Demarais
Abstract:Let $G$ be a connected reductive algebraic group over $\mathbb C$. Through the geometric Satake equivalence, the fundamental classes of the Mirković–Vilonen cycles define a basis in each tensor product $V(\lambda _1)\otimes \cdots \otimes V(\lambda _r)$ of irreducible representations of $G$. We compute this basis in the case $G=\mathrm {SL}_2(\mathbb C)$ and conclude that in this case it coincides
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Representation theoretic embedding of twisted Dirac operators Represent. Theory (IF 0.7) Pub Date : 2021-09-20 S. Mehdi, P. Pandžić
Abstract:Let $G$ be a non-compact connected semisimple real Lie group with finite center. Suppose $L$ is a non-compact connected closed subgroup of $G$ acting transitively on a symmetric space $G/H$ such that $L\cap H$ is compact. We study the action on $L/L\cap H$ of a Dirac operator $D_{G/H}(E)$ acting on sections of an $E$-twist of the spin bundle over $G/H$. As a byproduct, in the case of $(G,H
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The Dipper-Du conjecture revisited Represent. Theory (IF 0.7) Pub Date : 2021-09-03 Emily Norton
Abstract:We consider vertices, a notion originating in local representation theory of finite groups, for the category $\mathcal {O}$ of a rational Cherednik algebra and prove the analogue of the Dipper-Du Conjecture for Hecke algebras of symmetric groups in that setting. As a corollary we obtain a new proof of the Dipper-Du Conjecture over $\mathbb {C}$.
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Invariant measures on nilpotent orbits associated with holomorphic discrete series Represent. Theory (IF 0.7) Pub Date : 2021-08-18 Mladen Božičević
Abstract:Let $G_\mathbb R$ be a real form of a complex, semisimple Lie group $G$. Assume $G_\mathbb R$ has holomorphic discrete series. Let $\mathcal W$ be a nilpotent coadjoint $G_\mathbb R$-orbit contained in the wave front set of a holomorphic discrete series. We prove a limit formula, expressing the canonical measure on $\mathcal W$ as a limit of canonical measures on semisimple coadjoint orbits
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Parametrizing torsion pairs in derived categories Represent. Theory (IF 0.7) Pub Date : 2021-07-30 Lidia Angeleri Hügel, Michal Hrbek
Abstract:We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category $\mathrm {D}({\mathrm {Mod}}\text {-}A)$ of a ring $A$. To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in $A$, which is a natural extension of the construction of compactly
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Robinson–Schensted–Knuth correspondence in the representation theory of the general linear group over a non-archimedean local field Represent. Theory (IF 0.7) Pub Date : 2021-07-28 Maxim Gurevich, Erez Lapid
Abstract:We construct new “standard modules” for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson–Schensted–Knuth correspondence for Zelevinsky’s multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein (DRS) for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides
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Description of unitary representations of the group of infinite 𝑝-adic integer matrices Represent. Theory (IF 0.7) Pub Date : 2021-07-19 Yury Neretin
Abstract:We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of infinite matrices over a residue ring modulo $p^k$. Irreducible representations of the latter group are induced from finite-dimensional representations of
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Littlewood complexes for symmetric groups Represent. Theory (IF 0.7) Pub Date : 2021-07-13 Christopher Ryba
Abstract:We construct a complex $\mathcal {L}_\bullet ^\lambda$ resolving the irreducible representations $\mathcal {S}^{\lambda [n]}$ of the symmetric groups $S_n$ by representations restricted from $GL_n(k)$. This construction lifts to $\mathrm {Rep}(S_\infty )$, where it yields injective resolutions of simple objects. It categorifies stable Specht polynomials, and allows us to understand evaluations
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Irreducible tensor products of representations of covering groups of symmetric and alternating groups Represent. Theory (IF 0.7) Pub Date : 2021-06-25 Lucia Morotti
Abstract:In this paper we completely classify irreducible tensor products of covering groups of symmetric and alternating groups in characteristic $\not =2$.
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Gradings of Lie algebras, magical spin geometries and matrix factorizations Represent. Theory (IF 0.7) Pub Date : 2021-06-22 Roland Abuaf, Laurent Manivel
Abstract:We describe a remarkable rank $14$ matrix factorization of the octic $\mathrm {Spin}_{14}$-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular $\mathbb {Z}$-grading of $\mathfrak {e}_8$. Intriguingly
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On unitary representations of algebraic groups over local fields Represent. Theory (IF 0.7) Pub Date : 2021-06-10 Bachir Bekka, Siegfried Echterhoff
Abstract:Let $\mathbf {G}$ be an algebraic group over a local field $\mathbf {k}$ of characteristic zero. We show that the locally compact group $\mathbf {G}(\mathbf {k})$ consisting of the $\mathbf {k}$-rational points of $\mathbf {G}$ is of type I. Moreover, we complete Lipsman’s characterization of the groups $\mathbf {G}$ for which every irreducible unitary representation of $\mathbf {G}(\mathbf
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Eulerianity of Fourier coefficients of automorphic forms Represent. Theory (IF 0.7) Pub Date : 2021-06-07 Dmitry Gourevitch, Henrik Gustafsson, Axel Kleinschmidt, Daniel Persson, Siddhartha Sahi
Abstract:We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a ‘hidden’ invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations
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Quivers for 𝑆𝐿₂ tilting modules Represent. Theory (IF 0.7) Pub Date : 2021-06-03 Daniel Tubbenhauer, Paul Wedrich
Abstract:Using diagrammatic methods, we define a quiver with relations depending on a prime $\mathsf {p}$ and show that the associated path algebra describes the category of tilting modules for $\mathrm {SL}_{2}$ in characteristic $\mathsf {p}$. Along the way we obtain a presentation for morphisms between $\mathsf {p}$-Jones–Wenzl projectors.
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Equivalence of categories between coefficient systems and systems of idempotents Represent. Theory (IF 0.7) Pub Date : 2021-06-02 Thomas Lanard
Abstract:The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $\operatorname {Rep}_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to construct level 0 decompositions when $R=\overline {\mathbb {Z}}_{\ell }$, $\ell \neq p$, by Dat for $GL_{n}$ and the author for a more general
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On induction of class functions Represent. Theory (IF 0.7) Pub Date : 2021-05-07 G. Lusztig
Abstract:Let $G$ be a connected reductive group defined over a finite field $\mathbf {F}_q$ and let $L$ be a Levi subgroup (defined over $\mathbf {F}_q$) of a parabolic subgroup $P$ of $G$. We define a linear map from class functions on $L(\mathbf {F}_q)$ to class functions on $G(\mathbf {F}_q)$. This map is independent of the choice of $P$. We show that for large $q$ this map coincides with the known
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Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩}) Represent. Theory (IF 0.7) Pub Date : 2021-05-03 Zicheng Qian
Abstract:The primary purpose of this paper is to clarify the relation between previous results in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43-145], [Amer. J. Math. 141 (2019), pp. 661-703], and [Camb. J. Math. 8 (2020), p. 775-951] via the construction of some interesting locally analytic representations. Let be a sufficiently large finite extension of and be a -adic semi-stable representation such
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Finite dimensional semigroups of unitary endomorphisms of standard subspaces Represent. Theory (IF 0.7) Pub Date : 2021-04-27 Karl-H. Neeb
Abstract:Let be a standard subspace in the complex Hilbert space and be a finite dimensional Lie group of unitary and antiunitary operators on containing the modular group of and the corresponding modular conjugation . We study the semigroup and determine its Lie wedge , i.e., the generators of its one-parameter subsemigroups in the Lie algebra of . The semigroup is analyzed in terms of antiunitary