Search

Let G be a classical group defined over a local field F of characteristic zero. For any irreducible admissible representation \pi of G(F), which is of Casselman-Wallach type if F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F. In particular, if \pi has a generic local L-parameter, we introduce the spectral first occurrence index {\mathfrak {f}}_{\mathfrak {s}}(\pi) and the arithmetic first occurrence index {\mathfrak {f}}_{{\mathfrak {a}}}(\pi) of \pi and prove in this paper that {\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{{\mathfrak {a}}}(\pi). Based on the theory of consecutive descents of enhanced L-parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L-packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality. [ABSTRACT FROM AUTHOR]

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur–Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B, and module categories of endomorphism algebras of generator-relative cogenerators over B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n so that the endomorphism algebra of a generator-cogenerator provides an n-faithful cover, in the sense of Rouquier, of B. The degree n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple modules of B. We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category \mathcal {O} establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one. [ABSTRACT FROM AUTHOR]

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer's cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a "cellular algebra Clifford theory" for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(\ell,p,n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(\ell,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck's result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(\ell,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of "adjustment matrices" in positive characteristic. [ABSTRACT FROM AUTHOR]

Let \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of \omega-semistable properly supported sheaves on Y with fixed slope \mu and \zeta-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on \zeta depending on the polarization \omega and \mu. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of \omega-semistable properly supported sheaves on Y with fixed slope \mu: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit realization of all irreducible modules in Chari's category \widetilde {\mathcal O} for the affine Kac-Moody algebra A_{1}^{(1)} by using the idea of free fields. We work on a much more general setting which also gives us explicit realizations of all simple weight modules for certain current algebra of \mathfrak {sl}_2(\mathbb {C}) with finite weight multiplicities, including the polynomial current algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t], the loop algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1}] and the three-point Lie algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1},(t-1)^{-1}] arisen in the work by Kazhdan-Lusztig. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let g be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D on the coordinate ring 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 well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g is of type An or D4, the map D takes similar distinguished values on the set of all flag minors of C[N], raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D on flag minors belonging to the same standard seed, and we show that in any ADE type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh's monoidal categorification of the cluster structure of C[N] via representations of quiver Hecke algebras. [ABSTRACT FROM AUTHOR]

In a previous paper the authors have attached to each Dynkin quiver an associative algebra. The definition is categorical and the algebra is used to construct desingularizations of arbitrary quiver Grassmannians. In the present paper we prove that this algebra is isomorphic to an algebra constructed by Hernandez-Leclerc defined combinatorially and used to describe certain graded Nakajima quiver varieties. This approach is used to get an explicit realization of the orbit closures of representations of Dynkin quivers as affine quotients. [ABSTRACT FROM AUTHOR]

This paper provides a comparison between the K-structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type D. Precisely, let G0 = Spin(2n,C) be the Spin complex group as a real group, and let K ≅ G0 be the complexification of the maximal compact subgroup of G0. We compute K-spectra of the regular functions on some small nilpotent orbits O transforming according to characters Ψ of CK(O) trivial on the connected component of the identity CK(≅)0. We then match them with the K-types of the genuine (i.e., representations which do not factor to SO(2n,C)) unipotent representations attached to O. [ABSTRACT FROM AUTHOR]

Let G be a classical p-adic group and let π be a smooth irreducible representation of G. In this paper, we consider the problem of calculating the dual (in the sense of Aubert and Schneider-Stuhler) .... More precisely, if π is specified by its Langlands data, the problem is to determine the Langlands data for .... This problem reduces (based on supercuspidal support) to two main cases: half-integral reducibility and integral reducibility; the latter is addressed here. [ABSTRACT FROM AUTHOR]

The \imathHall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 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 \imathquantum loop algebra, which is a generalization of the \imathquantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The \imathHall algebra of the \imathquiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two \imathHall algebras, explaining the isomorphism of the \imathquantum group of split affine type A under the two presentations. [ABSTRACT FROM AUTHOR]

Let G be an almost simple, simply connected algebraic group over an algebraically closed field of characteristic p > 0. In this paper we restate our conjecture from 1979 on the characters of irreducible modular representations of G so that it is now directly applicable to any dominant highest weight. [ABSTRACT FROM AUTHOR]

Let G be an affine algebraic group over an algebraically closed field whose identity component G0 is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G0 is unipotent. In this paper we define a map from the set of "twisted conjugacy classes" in W to the set of unipotent G0-conjugacy classes in D generalizing an earlier construction which applied when G is connected. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig's nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver. [ABSTRACT FROM AUTHOR]

We denote by A the ring of Laurent polynomials in the indeterminate v and by K its field of fractions. In this paper, we are interested in representation theory of the "generic" q-Schur algebra Sq(n, r) over A. We will associate to every symmetrising trace form τ on KSq(n, r) a subalgebra Jτ of KSq(n, r) which is isomorphic to the "asymptotic" algebra J (n, r)A defined by J. Du. As a consequence, we give a new hypothesis which implies James' conjecture. [ABSTRACT FROM AUTHOR]

The equivariant \mathcal W-algebra of a simple Lie algebra \mathfrak {g} is a BRST reduction of the algebra of chiral differential operators on the Lie group of \mathfrak {g}. We construct a family of vertex algebras A[\mathfrak {g}, \kappa, n] as subalgebras of the equivariant \mathcal W-algebra of \mathfrak {g} tensored with the integrable affine vertex algebra L_n(\check {\mathfrak {g}}) of the Langlands dual Lie algebra \check {\mathfrak {g}} at level n\in \mathbb {Z}_{>0}. They are conformal extensions of the tensor product of an affine vertex algebra and the principal \mathcal {W}-algebra whose levels satisfy a specific relation. When \mathfrak {g} is of type ADE, we identify A[\mathfrak {g}, \kappa, 1] with the affine vertex algebra V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g}), giving a new and efficient proof of the coset realization of the principal \mathcal W-algebras of type ADE. [ABSTRACT FROM AUTHOR]

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 a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases. [ABSTRACT FROM AUTHOR]

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 weight \lambda. This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V(\lambda)\otimes V(\mu). Then, we prove the corresponding geometric results including the higher cohomology vanishing on the \mathcal {G}-Schubert varieties in the product partial flag variety \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of \mathcal {G}-sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map. [ABSTRACT FROM AUTHOR]

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 and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra U_qL\mathfrak {sl}_{2}. [ABSTRACT FROM AUTHOR]

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 \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi)\right)^K of the K-invariant elements of \mathscr U(\mathfrak g)/\operatorname {ker}(\pi). It turns out that this algebra is commutative if and only if the restriction \pi |_K of \pi to K has finite multiplicities (cf. Baklouti and Fujiwara [J. Math. Pures Appl. (9) 83 (2004), pp. 137-161]). In this article we suppose this eventuality and we provide a proof of the polynomial conjecture asserting that D_{\pi }(G)^K is isomorphic to the algebra \mathbb C[\Omega (\pi)]^K of K-invariant polynomial functions on \Omega (\pi). The conjecture was partially solved in our previous works (Baklouti, Fujiwara, and Ludwig [Bull. Sci. Math. 129 (2005), pp. 187-209]; J. Lie Theory 29 (2019), pp. 311-341). [ABSTRACT FROM AUTHOR]

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 algebras of the partition algebra are cellular. We also prove similar results for the half partition algebras. [ABSTRACT FROM AUTHOR]

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 over the quantum loop algebra U_{q}(L\mathfrak {g}) via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with Q and Hernandez-Leclerc's category \mathcal {C}_{Q}, assuming the simpleness of some poles of normalized R-matrices for type \mathrm {E}. [ABSTRACT FROM AUTHOR]

Let $mathscr {S}(varLambda )$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $mathscr {H}_{n,r}$, introduced by Dipper, James and Mathas. In this paper, we consider $v$-decomposition numbers of $mathscr {S}(varLambda )$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji and Wada, a product formula for $v$-decomposition numbers of $mathscr {S}(varLambda )$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike algebras $mathscr {H}_{n_i,r_i}$ of smaller rank. Moreover, we prove a similar formula for $v$-decomposition numbers of $mathscr {H}_{n,r}$ by using a Schur functor. [ABSTRACT FROM AUTHOR]

Let $G$ be a connected, simply-connected, and almost simple algebraic group, and let $sigma $ be a Dynkin automorphism on $G$. Then $(G,G^sigma )$ is a symmetric pair. In this paper, we get a bijection between the set of $sigma $-invariant MV cycles (polytopes) for $G$ and the set of MV cycles (polytopes) for $G^sigma $, which is the fixed point subgroup of $G$; moreover, this bijection can be restricted to the set of MV cycles (polytopes) in irreducible representations. As an application, we obtain a new proof of the twining character formula. [ABSTRACT FROM AUTHOR]

Let G be a connected reductive group defined over a locally compact non-archimedean field F, let P be a parabolic subgroup with Levi M and compatible with a pro-p Iwahori subgroup of G ≔ G(F). Let R be a commutative unital ring. We introduce the parabolic pro-p Iwahori–Hecke R-algebra HR(P) of P ≔ P(F) and construct two R-algebra morphisms ΘPM : HR(P) → HR(M) and &#926PG : HR(P) → HR(G) into the pro-p Iwahori–Hecke R-algebra of M ≔ M(F) and G, respectively. We prove that the resulting functor Mod − HR(M) → Mod − HR(G) from the category of right HRR(M)-modules to the category of right HR(G)-modules (obtained by pulling back via ΘPM and extension of scalars along ΞPG) coincides with the parabolic induction due to Ollivier–Vignéras. The maps ΘPM and ΞPG factor through a common subalgebra HR(M,G) of HR(G) which is very similar to HR(M). Studying these algebras HR(M,G) for varying (M,G) we prove a transitivity property for tensor products. As an application we give a new proof of the transitivity of parabolic induction. [ABSTRACT FROM AUTHOR]

We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D(Mod −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 generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D(Mod −A). This leads us to classification results over some classes of commutative rings and over finite dimensional hereditary algebras. [ABSTRACT FROM AUTHOR]

We present a quantization of an isomorphism of Mirković and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in gln to a slice between spherical Schubert varieties in the affine Grassmannian of PGLn (with weights encoded by the Jordan types of the nilpotent orbits). A quantization of the former variety is provided by a parabolic W-algebra and of the latter by a truncated shifted Yangian. Building on earlier work of Brundan and Kleshchev, we define an explicit isomorphism between these non-commutative algebras and show that its classical limit is a variation of the original isomorphism of Mirković and Vybornov. As a corollary, we deduce that the W-algebra is free as a left (or right) module over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and Ovsienko. [ABSTRACT FROM AUTHOR]

For any simple Lie algebra g and an element μ ∈ g∗, the corresponding commutative subalgebra Aμ of U(g) is defined as a homomorphic image of the Feigin-Frenkel centre associated with g. It is known that when μ is regular this subalgebra solves Vinberg's quantisation problem, as the graded image of Aμ coincides with the Mishchenko-Fomenko subalgebra Aμ of S(g). By a conjecture of Feigin, Frenkel, and Toledano Laredo, this property extends to an arbitrary element μ. We give sufficient conditions on μ which imply the property. In particular, this proves the conjecture in type C and gives a new proof in type A. We show that the algebra Aμ is free in both cases and produce its generators in an explicit form. Moreover, we prove that in all classical types generators of Aμ can be obtained via the canonical symmetrisation map from certain generators of Aμ. The symmetrisation map is also used to produce free generators of nilpotent limits of the algebras Aμ and to give a positive solution of Vinberg's problem for these limit subalgebras. [ABSTRACT FROM AUTHOR]

Let l be a prime number. We show that the Morita Frobenius number of an l-block of a quasi-simple finite group is at most 4 and that the strong Frobenius number is at most 4|D|²!, where D denotes a defect group of the block. We deduce that a basic algebra of any block of the group algebra of a quasi-simple finite group over an algebraically closed field of characteristic l is defined over a field with la elements for some a ≤ 4. We derive consequences for Donovan's conjecture. In particular, we show that Donovan's conjecture holds for l-blocks of special linear groups. [ABSTRACT FROM AUTHOR]

The pro-p-Iwahori Hecke algebra has an involution ι defined in terms of the Iwahori-Matsumoto basis. Then for a module π of pro-p-Iwahori Hecke, πι = π o ι is also a module. We calculate πι for simple modules π. We also calculate the dual of π. These calculations will be used for calculating the extensions between simple modules. [ABSTRACT FROM AUTHOR]

We study fixed-point loci of Nakajima varieties under symplectomorphisms and their antisymplectic cousins, which are compositions of a diagram isomorphism, a reflection functor, and a transpose defined by certain bilinear forms. These subvarieties provide a natural home for geometric representation theory of symmetric pairs. In particular, the cohomology of a Steinberg-type variety of the symplectic fixed-point subvarieties is conjecturally related to the universal enveloping algebra of the subalgebra in a symmetric pair. The latter symplectic subvarieties are further used to geometrically construct an action of a twisted Yangian on a torus equivariant cohomology of Nakajima varieties. In the type A case, these subvarieties provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices of classical groups and associated symmetric spaces, which leads to a rectangular symmetry and a refinement of Kraft-Procesi row/column removal reductions. [ABSTRACT FROM AUTHOR]

An "automatic continuity" question has naturally occurred since Roger Howe established the local theta correspondence over ℝ: Does the algebraic version of local theta correspondence over ℝ agree with the smooth version? We show that the answer is yes, at least when the concerning dual pair has no quaternionic type I irreducible factor. [ABSTRACT FROM AUTHOR]

Simple representations of KLR algebras can be used to realize the infinity crystal for the corresponding symmetrizable Kac-Moody algebra. It was recently shown that, in finite and affine types, certain sub-categories of "cuspidal" representations realize crystals for sub-KacMoody algebras. Here we put that observation on a firmer categorical footing by exhibiting a corresponding functor between the category of representations of the KLR algebra for the sub-Kac-Moody algebra and the category of cuspidal representations of the original KLR algebra. [ABSTRACT FROM AUTHOR]

By results of Rognerud, a source algebra equivalence between two p-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above mentioned derived equivalence. We show further that a splendid stable equivalence of Morita type between two blocks induces an equivalence between the categories of cohomological Mackey functors which vanish at the trivial group. We observe that the module categories of a block, the category of cohomological Mackey functors, and the category of cohomological Mackey functors which vanish at the trivial group arise in an idempotent recollement. Finally, we extend a result of Tambara on the finitistic dimension of cohomological Mackey functors to blocks. [ABSTRACT FROM AUTHOR]

Following Kashiwara's algebraic approach, we construct crystal bases and canonical bases for quantum supergroups of anisotropic type and for their integrable modules. [ABSTRACT FROM AUTHOR]

For a minimal affinization over a quantum loop algebra of type BC, we provide a character formula in terms of Demazure operators and multiplicities in terms of crystal bases. We also prove the formula for the limit of characters conjectured by Mukhin and Young. These are achieved by verifying that its graded limit (a variant of a classical limit) is isomorphic to some multiple generalization of a Demazure module, and by determining the defining relations of the graded limit. [ABSTRACT FROM AUTHOR]

Torus fixed points of quiver moduli spaces are given by stable representations of the universal (abelian) covering quiver. As far as the Kronecker quiver is concerned they can be described by stable representations of certain bipartite quivers coming along with a stable colouring. By use of the glueing method it is possible to construct a huge class of such quivers implying a lower bound for the Euler characteristic. For certain roots it is even possible to construct all torus fixed points. [ABSTRACT FROM AUTHOR]

We further develop and simplify the general theory of distinguished tame supercuspidal representations of reductive p-adic groups due to Hakim and Murnaghan, as well as the analogous theory for finite reductive groups due to Lusztig. We apply our results to study the representations of GLn(F), with n odd and F a nonarchimedean local field, that are distinguished with respect to an orthogonal group in n variables. In particular, we determine precisely when a supercuspidal representation is distinguished with respect to an orthogonal group and, if so, that the space of distinguishing linear forms has dimension one. [ABSTRACT FROM AUTHOR]

We prove that certain parabolic Kazhdan-Lusztig polynomials calculate the graded decomposition matrices of &#x03C5-Schur algebras given by the Jantzen filtration of Weyl modules, confirming a conjecture of Leclerc and Thibon. [ABSTRACT FROM AUTHOR]

The article focuses on the center of infinitesimal Cherednik algebras of gln. It informs that these algebras are subalgebras of continuous Cheredmik algebras introduced by mathematicians P. Etingof, W.L. Gan and V. Ginzburg, as natural continuous analogs of the widely studied rational Cherednik algebras. It states that there is an isomorphic relationship between the center of an infinitesimal Cherednik algebtra of gln and polynomial algebra of several variables.

The article presents information related to rational Schur algebras. Information is given that rational Schur algebras are a part of the family of type A classical Schur algebras which are used to determine the polynomial representation theory of general linear groups over an infinite field. Several formulas applied to prove various theorems related to Schur algebra are also presented. As stated, rational Schur algebras are quasihereditary over any field and have finite global dimension.

The article presents information on the regularity of prime(p)-adic distribution algebras over arbitrary base fields. The proof related to the distribution algebra of a compact p-adic Lie group over an arbitrary base field is also discussed. Several formulas used to solve the theorems related to p-adic distribution algebras are also presented. Fréchet-Stein theory of the distribution algebra is also discussed and it is stated that it depends heavily on the presence of uniform subgroups.