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Kostant's pair of Lie type and conformal embeddingsFeb 08 2018We deal with some aspects of the theory of conformal embeddings of affine vertex algebras, providing a new proof of the Symmetric Space Theorem and a criterion for conformal embeddings of equal rank subalgebras. We finally study some examples of embeddings ... More
Representations of Compact Lie Groups of Low CohomogeneityFeb 08 2018We survey different tools to classify representations of compact Lie groups according to their cohomogeneity and apply these methods to the case of irreducible representations of cohomogeneity 6, 7 and 8.
3D Current Algebra and Twisted K TheoryFeb 08 2018Equivariant twisted K theory classes on compact Lie groups $G$ can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra $LG$ using a supersymmetric ... More
The structure and homological properties of generalized standard Auslander-Reiten componentsFeb 08 2018We describe the structure and homological properties of arbitrary generalized standard Auslander-Reiten components of artin algebras. In particular, we prove that for all but finitely many indecomposable modules in such components the Euler characteristic ... More
Groups with frames of translatesFeb 07 2018Let $G$ be a locally compact group with left regular representation $\lambda_{G}.$ We say that $G$ admits a frame of translates if there exist a countable set $\Gamma\subset G$ and $\varphi\in L^{2}(G)$ such that $(\lambda_{G}(x) \varphi)_{x \in\Gamma}$ ... More
Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representationsFeb 07 2018A fundamental difficulty in the study of automorphic representations, representations of $p$-adic groups and the Langlands program is to handle the non-generic case. In this work we develop a complete local and global theory of tensor product $L$-functions ... More
Ordinary $GL_2(F)$-representations in characteristic two via affine Deligne-Lusztig constructionsFeb 07 2018The group $\GL_2$ over a local field with (residue) characteristic $2$ possesses much more smooth supercuspidal $\ell$-adic representations, than over a local field of residue characteristic $> 2$. One way to construct these representations is via the ... More
$\mathfrak F$-categories and $\mathfrak F$-functors in Representation Theory IIFeb 06 2018This is a partial derivative of \cite{MR94g:17044}. We give a list of examples/problems that some will find amusing.
Projective modules over classical Lie algebras of infinite rank in the parabolic categoryFeb 06 2018We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As ... More
Analogues of centralizer subalgebras for fiat 2-categories and their 2-representationsFeb 06 2018The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $\mathcal{J}$ of a fiat $2$-category $\cC$ and the set of equivalence classes of faithful simple transitive ... More
Resolutions of DG-modules and their applications to commutative DG-algebrasFeb 06 2018A DG-version of projective resolution and injective resolution of DG-modules over DG-algebra are already known. In the first half of this paper, we introduce another DG-version for DG-modules over a connective DG-algebra and show that they behave nicely ... More
Classification of $A_{q}(λ)$ modules by their Dirac cohomology for type $D$ and $\mathfrak{sp}(2n,\mathbb{R})$Feb 06 2018Let $G$ be a connected real reductive group with maximal compact subgroup $K$ of the same rank as $G$. In the recent paper of Huang, Pand\v{z}i\'{c} and Vogan, it was shown that the admissible $\Theta$-stable parabolic subalgebras $\mathfrak{q}$ of $\mathfrak{g}$ ... More
Theta correspondence and the Prasad conjecture for SL(2)Feb 06 2018We use relations between the base change representations and theta lifts, to give a new proof to the local period problems of SL(2) over a nonarchimedean quadratic field extension E/F. Then we will verify the Prasad conjecture for SL(2). With a similar ... More
Cuspidal cohomology of stacks of shtukasFeb 05 2018Let $G$ be a connected split reductive group over a finite field ${\mathbb F}_q$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalisation of the space of automorphic forms over a function field. In this paper, we construct a constant term ... More
Brauer characters and normal Sylow $p$-subgroupsFeb 03 2018In this paper, we study some variations of the well-known It\^{o}-Michler theorem for $p$-Brauer characters using various inequalities involving the $p$-Brauer character degrees of finite groups. Several new criteria for the existence of a normal Sylow ... More
Finitistic dimension and Endomorphism algebras of Gorenstein projective modulesFeb 02 2018Let $A$ be an Artin algebra, $M$ be a Gorenstein projective $A$-module and $B =$ End$_A M$, then $M$ is a $A$-$B$-bimodule. We use the restricted flat dimension of $M_B$ to give a characterization of the homological dimensions of $A$ and $B$, and obtain ... More
$(\mathfrak{g},K)$-module of $\mathrm{O}(p,q)$ associated with the finite-dimensional representation of $\mathfrak{sl}_2$Jan 31 2018The main aim of this paper is to construct irreducible $(\mathfrak{g},K)$-modules of $\mathrm{O}(p,q)$ corresponding to the finite-dimensional representation of $\mathfrak{sl}_2$ of dimension $m+1$ under the Howe duality, to find the $K$-type formula, ... More
Weil representations of unitary groups over ramified extensions of finite local rings with odd nilpotency lengthJan 31 2018We find the irreducible decomposition of the Weil representation of the unitary group $\mathrm{U}_{2m}(A)$, where $A$ is a ramified extension of a finite, principal local ring $R$ and the nilpotency degree of the maximal ideal of $A$ is odd. We show in ... More
Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$Jan 30 2018Feb 05 2018We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on $\mathrm{GL}_2$ over a number field which is not totally real, improving the one obtained by Marshall. The main ... More
Standard modules, Jones-Wenzl projectors, and the valenced Temperley-Lieb algebraJan 30 2018This article concerns a generalization of the Temperley-Lieb algebra, motivated by applications to conformal field theory. We call this algebra the valenced Temperley-Lieb algebra. We prove salient facts concerning this algebra and its representation ... More
An Algorithm to Decompose Permutation Representations of Finite Groups: Polynomial Algebra ApproachJan 29 2018We describe an algorithm for splitting a permutation representation of a finite group into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projection ... More
Operational calculus for Fourier transform on the group $GL(2,R)$Jan 29 2018Consider the Fourier transform on the group $GL(2,R)$ of real $2\times 2$-matrices. We show that Fourier-images of polynomial differential operators on $GL(2,R)$ are differential-difference operators with coefficients meromorphic in parameters of representations. ... More
Gelfand-Tsetlin Theory for Rational Galois AlgebrasJan 28 2018In the present paper we study Gelfand-Tsetlin modules defined in terms of BGG differential operators. The structure of these modules is described with the aid of the Postnikov-Stanley polynomials introduced in [PS09]. These polynomials are used to identify ... More
Reducible subgroups of exceptional algebraic groupsJan 28 2018Let $G$ be a simple algebraic group over an algebraically closed field. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if, whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi factor of $P$. ... More
The totally nonnegative part of G/P is a ballJan 26 2018We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball.
Color Lie rings and PBW deformations of skew group algebrasJan 26 2018We investigate color Lie rings over finite group algebras and their universal enveloping algebras. We exhibit these universal enveloping algebras as PBW deformations of skew group algebras: Every color Lie ring over a finite group algebra with a particular ... More
After Plancherel formulaJan 26 2018We discuss two topics related to Fourier transforms on Lie groups and on homogeneous spaces: the operational calculus and the Gelfand--Gindikin problem (program) about separation of non-uniform spectra. Our purpose is to indicate some non-solved problems ... More
On Uniform Admissibility of Unitary and Smooth RepresentationsJan 26 2018Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth representations of $G$ are ... More
Representations of meromorphic open-string vertex algebrasJan 26 2018We study representations of the meromorphic open-string vertex algebra (MOSVAs hereafter) defined in [H3], a noncommutative generalization of vertex (operator) algebra. We start by recalling the definition of a MOSVA $V$ and left $V$-modules in [H3]. ... More
Restricted One-dimensional Central Extensions of the Restricted Filiform Lie Algebras ${\frak m}_0^λ(p)$Jan 24 2018We show, for a field ${\mathbb F}$ of prime characteristic $p>0$, that the truncated filiform Lie algebra ${\frak m}_0(p)$ admits a family ${\frak m}_0^\lambda(p)$ of restricted Lie algebra structures parameterized by elements $\lambda\in {\mathbb F}^p$. ... More
Overgroups of exterior powers of an elementary group. I. Levels and normalizersJan 24 2018In the present paper, we prove the first part in the standard description of groups $H$ lying between $m$-th exterior power of elementary group $E(n,R)$ and the general linear group $GL_{\binom{n}{m}}(R)$. We study structure of the exterior power of elementary ... More
Algebras with finitely many conjugacy classes of left ideals versus algebras of finite representation typeJan 24 2018Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple ... More
Blob algebra approach to modular representation theoryJan 22 2018Two decades ago P. Martin and D. Woodcock made a surprising (and prophetic) link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix ... More
Generically free representations III: exceptionally bad characteristicJan 22 2018In parts I and II, we determined which irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero, with some ... More
Unicity and non-unicity of types for a class of essentially tame supercuspidal representationsJan 20 2018For toral tame supercuspidal representations of a simply connected and semisimple $p$-adic group $G$, constructed as per Adler-Yu, we determine which components of their restriction to a maximal compact subgroup are types. We give conditions under which ... More
Cohomology of $p$-adic Stein spacesJan 20 2018We compute the $p$-adic \'etale and the pro-\'etale cohomologies of the Drinfeld half-space of any dimension. The main input is a new comparison theorem for the $p$-adic pro-\'etale cohomology of $p$-adic Stein spaces.
Super BundlesJan 19 2018In this paper we give a brief account of the main aspects of the theory of associated and principal super bundles. As an application, we review the Borel-Weil-Bott Theorem in the super setting, and some results on projective embeddings of homogeneous ... More
Jones Index Theorem revisitedJan 17 2018A new shorter proof of the Jones Index Theorem is given. Our approach is based on the notion of a cluster C*-algebra.
Modular group algebras whose group of unitary units is locally nilpotentJan 16 2018We characterize those modular group algebras FG whose group of unitary units is locally nilpotent under the classical involution of FG.
Group algebra whose unit group is locally nilpotentJan 16 2018We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units V(FG) of the group algebra FG is locally nilpotent; (ii) the group algebra FG has a finite number of nilpotent elements and V(FG) is an Engel group.
Triviality properties of principal bundles on singular curves-IIJan 15 2018For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing ... More
Regular orbits of sporadic simple groupsJan 14 2018Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. We ... More
On semisimplification of tensor categoriesJan 13 2018We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic $p$ in terms of representations ... More
A characterisation of $τ$-tilting finite algebrasJan 12 2018We prove that a finite dimensional algebra is $\tau$-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a $\tau$-tilting finite algebra $A$ there is a bijection between isomorphism classes of basic support ... More
The affine VW supercategoryJan 12 2018We 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 ... More
Reducible characteristic cycles of Harish-Chandra modules for $\mathrm{U}(p,q)$ and the Kashiwara-Saito singularityJan 10 2018We give examples of reducible characteristic cycles for irreducible Harish-Chandra modules for $\mathrm{U}(p,q)$ by analyzing a four-dimensional singular subvariety of $\mathbb{C}^8$. We relate this singularity to the Kashiwara-Saito singularity arising ... More
Cuspidal prehomogeneous vector spaces for reductive Lie groups and related algebraic and geometric structuresJan 09 2018In this paper, we study the relations between left-invariant flat connections on Lie groups, left-symmetric algebras, symplective Lie algebras, Frobenius Lie algebras and cuspidal prehomogeneous vector spaces. More specifically, we establish a one-to-one ... More
Modified trace is a symmetrised integralDec 31 2017A modified trace for a finite k-linear pivotal category is a family of linear forms on endomorphism spaces of projective objects which has cyclicity and so-called partial trace properties. We show that a non-degenerate modified trace defines a compatible ... More
Symmetry breaking for representations of rank one orthogonal groups IIDec 30 2017For a pair $(G,G')=(O(n+1,1), O(n,1))$ of reductive groups, we investigate intertwining operators (symmetry breaking operators) between principal series representations $I_\delta(V,\lambda)$ of $G$, and $J_\varepsilon(W,\nu)$ of the subgroup $G'$. The ... More
Characterizing moonshine functions by vertex-operator-algebraic conditionsDec 29 2017Given a holomorphic $C_2$-cofinite vertex operator algebra $V$ with graded dimension $j-744$, Borcherds's proof of the Monstrous Moonshine Conjecture implies any finite order automorphism of $V$ has graded trace given by a completely replicable function, ... More
Reduction of a pair of skew-symmetric matrices to its canonical form under congruenceDec 23 2017Let $(A,B)$ be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum \[ (\underline{\underline A},\underline{\underline B})\oplus (A_1,B_1)\oplus\dots\oplus(A_t,B_t) \] that is congruent ... More
La théorie de Hodge des bimodules de Soergel (d'après Soergel et Elias-Williamson)Nov 07 2017Soergel bimodules are certain bimodules over polynomial algebras, associated with Coxeter groups, and introduced by Soergel in the 1990's while studying the category O of complex semisimple Lie algebras. Even though their definition is algebraic and rather ... More
On Levi-Malcev theorem for Leibniz algebrasOct 30 2017The present paper is devoted to provide conditions for the Levi--Malcev theorem to hold or not to hold (i.e. for two Levi subalgebras to be or not conjugate by an inner automorphism) in the context of finite-dimensional Leibniz algebras over a field of ... More
A Counterexample to the First Zassenhaus ConjectureOct 24 2017Nov 20 2017Hans J. Zassenhaus conjectured that for any unit $u$ of finite order in the integral group ring of a finite group $G$ there exists a unit $a$ in the rational group algebra of $G$ such that $a^{-1}\cdot u \cdot a=\pm g$ for some $g\in G$. We disprove this ... More
Acyclic cluster algebras, reflection groups, and curves on a punctured discSep 29 2017Dec 05 2017We establish a bijective correspondence between certain non-self-intersecting curves in an $n$-punctured disc and positive ${\mathbf c}$-vectors of acyclic cluster algebras whose quivers have multiple arrows between every pair of vertices. As a corollary, ... More
Representations of Lie algebras of vector fields on affine varietiesSep 26 2017For an irreducible affine variety $X$ over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on $X$ - gauge modules and Rudakov modules, which admit a compatible action of the ... More
On the variety of 1-dimensional representations of finite $W$-algebras in low rankAug 29 2017Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e \in \mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(\mathfrak g,e)$ associated to $e$ and the problem of determining the variety $\mathcal E(\mathfrak g,e)$ of 1-dimensional ... More
When is the heart of a t-structure a Grothendieck category?Aug 24 2017Let $\mathcal D$ be a triangulated category endowed with a $t$-structure $\mathfrak t=(\mathcal U,\Sigma \mathcal V)$ and denote by $\mathcal H:=\mathcal U\cap \Sigma\mathcal V$ its heart. In this paper we study the following well-known problem: Under ... More
Symplectic spinors and Hodge theoryAug 07 2017Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the ... More
Modified Ringel-Hall Algebras and Derived Hall AlgebrasJul 26 2017In this paper we define the modified Ringel-Hall algebra $\mathcal{M}\mathcal{H}(\mathcal{A})$ of a hereditary abelian category $\mathcal{A}$ from the category $C^b(\mathcal{A})$ of bounded $\mathbb{Z}$-graded complexes, and prove that in certain twisted ... More
On the Humphreys conjecture on support varieties of tilting modulesJul 24 2017Jul 27 2017Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent ... More
Hochschild Cohomology and the Modular GroupJul 13 2017It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group ... More
Yang-Baxter representations of the infinite symmetric groupJul 01 2017Every unitary involutive solution of the quantum Yang-Baxter equation ("R-matrix") defines an extremal character and a representation of the infinite symmetric group $S_\infty$. We give a complete classification of all such Yang-Baxter characters and ... More
Koszul duality for Kac-Moody groups and characters of tilting modulesJun 01 2017We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p-Kazhdan-Lusztig polynomials, for p>h the Coxeter number. Using results of Andersen, one may deduce a character formula ... More
Separable equivalence, complexity and representation typeMay 31 2017Oct 31 2017We generalise the notion of separable equivalence, originally presented by Linckelmann (2011), to an equivalence relation on additive categories. We use this generalisation to show that from an initial equivalence between two algebras we may build equivalences ... More
Contractibility of the stability manifold for silting-discrete algebrasMay 30 2017We show that any bounded t-structure in the bounded derived category of a silting-discrete algebra is algebraic, i.e. has a length heart with finitely many simple objects. As a corollary, we obtain that the space of Bridgeland stability conditions for ... More
Modular finite $W$-algebrasMay 17 2017Nov 03 2017Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$-algebra $U(\mathfrak g,e)$ associated ... More
Notes on the geometric Satake equivalenceMar 21 2017Dec 13 2017These notes are devoted to a detailed exposition of the proof of the Geometric Satake Equivalence for general coefficients, following Mirkovic-Vilonen.
Free-monodromic mixed tilting sheaves on flag varietiesMar 16 2017In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category ... More
Metaplectic Covers of Kac-Moody Groups and Whittaker FunctionsMar 15 2017Starting from some linear algebraic data (a Weyl-group invariant bilinear form) and some arithmetic data (a bilinear Steinberg symbol), we construct a cover of a Kac-Moody group generalizing the work of Matsumoto. Specializing our construction over non-archimedean ... More
Singular Hochschild cohomology and algebraic string operationsMar 11 2017Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns ... More
The center of the small quantum group II: singular blocksMar 07 2017Dec 01 2017We generalize to the case of singular blocks the result in \cite{BeLa} that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in \cite{LQ1}, ... More
Questions on mod p representations of reductive p-adic groupsMar 06 2017This is a list of questions raised by our joint work arXiv:1412.0737 and its sequels.
Geometric monodromy -- semisimplicity and maximalityFeb 22 2017Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded ... More
Symmetry of the Definition of Degeneration in Triangulated CategoriesDec 17 2016Dec 22 2016Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs ... More
Hopf algebra structures and tensor products for group algebrasDec 14 2016The modular group algebra of an elementary abelian p-group is isomorphic to the restricted enveloping algebra of commutative restricted Lie algebra. The different ways of regarding this algebra result in different Hopf algebra structures that determine ... More
Dg algebras with enough idempotents, their dg modules and their derived categoriesDec 14 2016May 02 2017We develop the theory dg algebras with enough idempotents and their dg modules and show their equivalence with that of small dg categories and their dg modules. We introduce the concept of dg adjunction and show that the classical covariant tensor-Hom ... More
Glider representations of group algebra filtrations of nilpotent groupsDec 08 2016We continue the study of glider representations of finite groups $G$ with given structure chain of subgroups $e \subset G_1 \subset \ldots \subset G_d = G$. We give a characterization of irreducible gliders of essential length $e \leq d$ which in the ... More
Representations of unipotent reduction for SO(2n+1), II: endoscopyDec 08 2016For the groups SO(2n+1,F), where F is a p-adic field, we consider the tempered irr{\'e}ducible representations of unipotent reduction. Lusztig has contructed and parametrized these representations. We prove that they satisfy the expected endoscopic identities ... More
The Auslander-Reiten duality via morphisms determined by objectsDec 08 2016Given an exact category $\mathcal{C}$, we denote by $\mathcal{C}_l$ the smallest additive subcategory containing injectives and indecomposable objects which appear as the first term of an almost split conflation. We prove that a deflation is right determined ... More
Faithful Actions from Hyperplane ArrangementsDec 08 2016An axiomatic framework is developed, under which the tilting modules of an algebra produce a faithful group action on its derived category. As a consequence, if X is a quasi-projective 3-fold admitting a flopping contraction, then the fundamental group ... More
Representation Embeddings of Cartesian TheoriesDec 08 2016A representation embedding between cartesian theories can be defined to be a functor between respective categories of models that preserves indecomposable projective models and that preserves and reflects certain epimorphisms. This recalls standard definitions ... More
Interlacing adjacent levels of $β$-Jacobi corners processesDec 07 2016We study the asymptotic of the global fluctuations for the difference between two adjacent levels in the $\beta$-Jacobi corners process (multilevel and general $\beta$ extension of the classical Jacobi ensemble of random matrices). The limit is identified ... More
Bounded linear endomorphisms of rigid analytic functionsDec 06 2016Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the algebra $\mathcal{E}$ ... More
Another approach to Juhl's conformally covariant differential operators from $S^n$ to $S^{n-1}$Dec 06 2016A new construction of Juhl's conformally covariant differential operators from $S^n$ to $S^{n-1}$ (or from $\mathbb R^n$ to $\mathbb R^{n-1}$) is proposed. They are obtained as the composition of a new family of differential operators on $S^n$, covariant ... More
Sanya lectures on harmonic analysis for real spherical spacesDec 06 2016We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces $Z=G/H$ attached to a real reductive Lie group $G$. A special emphasis is made to the case where $Z$ is real spherical.
On completions of Hecke algebrasDec 06 2016Let G be a reductive p-adic group and let H(G)^s be a Bernstein block of the Hecke algebra of G. We consider two important topological completions of H(G)^s: a direct summand S(G)^s of the Harish-Chandra--Schwartz algebra of G and a two-sided ideal C*_r ... More
Supercharacters of queer Lie superalgebrasDec 06 2016Let $\mathfrak g=\mathfrak g_{\bar 0}\oplus\mathfrak g_{\bar 1}$ be the queer Lie superalgebra and let $L$ be a finite-dimensional non-trivial irreducible $\mathfrak g$-module. Restricting the $\mathfrak g$-action on $L$ to $\mathfrak g_{\bar 0}$, we ... More
Noncommutative enhancements of contractionsDec 06 2016Given a contraction of a variety X to a base Y, we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions ... More
Derived Recollements and Generalised AR FormulasDec 06 2016The Defect Recollement, Evaluation Recollement, Restriction Recollement, and Auslander-Gruson-Jensen-Recollement are shown to be instances of a general construction using derived functors and methods from stable module theory. The right derived functors ... More
Derived Recollements and Generalised AR FormulasDec 06 2016Dec 07 2016The Defect Recollement, Evaluation Recollement, Restriction Recollement, and Auslander-Gruson-Jensen-Recollement are shown to be instances of a general construction using derived functors and methods from stable module theory. The right derived functors ... More
Preprojective algebras of tree-type quiversDec 05 2016Let $Q$ be a tree-type quiver, $\mathbf{k}Q$ be its path algebra, and $\lambda$ be a nonzero element in a base field $\mathbf{k}$. We construct irreducible morphisms in the Auslander-Reiten quiver of the bounded derived category of $\mathbf{k}Q$ that ... More
Matrix multiplication algorithms from group orbitsDec 05 2016We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of orbits under ... More
Stability of the Chari-Loktev bases for local Weyl modules of $\mathfrak{sl}_{r+1}[t]$Dec 05 2016We prove stability of the Chari-Loktev bases with respect to the inclusions of local Weyl modules of the current algebra $\mathfrak{sl}_{r+1}[t]$. This is conjectured in \cite{RRV2} and proved the $r=1$ case in \cite{RRV1}. Local Weyl modules being known ... More
On Extensions of supersingular representations of ${\rm SL}_2(\mathbb{Q}_p)$Dec 05 2016In this note for $p>5$ we calculate the dimensions of ${\rm Ext}^1_{{\rm SL}_2(\mathbb{Q}_p)}(\tau, \sigma)$ for any two irreducible supersingular representations $\tau$ and $\sigma$ of ${\rm SL}_2(\mathbb{Q}_p)$.
Root multiplicities for Borcherds algebras and graph coloringDec 05 2016We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed formula for ... More
Parabolic inductions for pro-$p$-Iwahori Hecke algebrasDec 05 2016We give some properties of parabolic inductions and their adjoint functors for pro-$p$-Iwahori Hecke algebras.
Regular functions on spherical nilpotent orbits in complex symmetric pairs: classical Hermitian casesDec 05 2016Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. We show that all such orbit closures are normal and describe ... More
Fibered Biset FunctorsDec 04 2016The theory of biset functors, introduced by Serge Bouc, gives a unified treatment of operations in representation theory that are induced by permutation bimodules. In this paper, by considering fibered bisets, we introduce and describe the basic theory ... More
The Ismagilov conjecture over a finite field ${\mathbb F}_p$Dec 04 2016We construct the so-called quasiregular representations of the group of infinite upper triangular matrices with coefficients in a finite field and give the criteria of theirs irreducibility in terms of the initial measure. These representations are particular ... More