# Representation Theory of the American Mathematical Society (Represent Theor )

Publisher: American Mathematical Society, American Mathematical Society

## Description

This electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content.

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- WebsiteRepresentation Theory website
- Other titlesRepresentation theory
- ISSN1088-4165
- OCLC34602921
- Material typeDocument, Internet resource
- Document typeInternet Resource, Computer File, Journal / Magazine / Newspaper

## Publisher details

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- Author can archive a pre-print version

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- Author can archive a post-print version

- Conditions
- Must include set publisher statement - (First published in [Publication] in [volume and number, or year], published by the American Mathematical Society)
- Publisher's version/PDF may be used
- On open access repositories
- Non-commercial
- Eligible UK authors may deposit in OpenDepot

- Classification green

## Publications in this journal

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**ABSTRACT:**For a local non-Archimedean field $K$ we construct ${\rm GL}_{d+1}(K)$-equivariant coherent sheaves ${\mathcal V}_{{\mathcal O}_K}$ on the formal ${\mathcal O}_K$-scheme ${\mathfrak X}$ underlying the symmetric space $X$ over $K$ of dimension $d$. These ${\mathcal V}_{{\mathcal O}_K}$ are ${\mathcal O}_K$-lattices in (the sheaf version of) the holomorphic discrete series representations (in $K$-vector spaces) of ${\rm GL}_{d+1}(K)$ as defined by P. Schneider \cite{schn}. We prove that the cohomology $H^t({\mathfrak X},{\mathcal V}_{{\mathcal O}_K})$ vanishes for $t>0$, for ${\mathcal V}_{{\mathcal O}_K}$ in a certain subclass. The proof is related to the other main topic of this paper: over a finite field $k$, the study of the cohomology of vector bundles on the natural normal crossings compactification $Y$ of the Deligne-Lusztig variety $Y^0$ for ${\rm GL}_{d+1}/k$ (so $Y^0$ is the open subscheme of ${\mathbb P}_k^d$ obtained by deleting all its $k$-rational hyperplanes).Representation Theory of the American Mathematical Society 08/2014; -
##### Article: Geometric local theta correspondence for dual reductive pairs of type II at the Iwahori level

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**ABSTRACT:**In this paper we are interested in the geometric local theta correspondence at the Iwahori level for dual reductive pairs $(G,H)$ of type II over a non-Archimedean field of characteristic $p\neq 2$ in the framework of the geometric Langlands program. We consider the geometric version of the $I_{H}\times I_{G}$-invariants of the Weil representation $\mathcal{S}^{I_{H}\times I_{G}}$ as a bimodule under the of action Iwahori-Hecke algebras $\mathcal{H}_{I_{G}}$ and $\mathcal{H}_{I_{H}}$ and we give some partial geometric description of the corresponding category under the action of Hecke functors. We also define geometric Jacquet functors for any connected reductive group $G$ at the Iwahori level and we show that they commute with the Hecke action of the $\mathcal{H}_{I_{L}}$-subelgebra of $\mathcal{H}_{I_{G}}$ for a Levi subgroup $L$.Representation Theory of the American Mathematical Society 10/2013; 17(21). - [Show abstract] [Hide abstract]

**ABSTRACT:**Generalizations and eigenvalue problems of Capelli-type identities have received a lot of attention. In [Colloq. Math. 118, No. 1, 349–364 (2010; Zbl 1194.22015)], R. Howe and S.-T. Lee considered a certain O n -invariant Capelli-type differential operator, a product of the determinants of matrices of variables and corresponding partial derivatives, in the context of Grassmannians of k planes in ℂ n . In the article, they raised an eigenvalue problem for the differential operator. When k=1, the problem is reduced to the classical theory of harmonic polynomials, and they solved the problem when k=2. The article under review answers the question for general k. More precisely, the author solves the eigenvalue problem for a certain family of O n -invariant Cappelli-type differential operators, to which the differential operator that Howe and Lee considered belongs. This is done by methods, which are completely different from those Howe and Lee used. The author achieved it by imbedding the problem into a more general setting of the symmetric space SO n (ℝ)/( SO k (ℝ)× SO l (ℝ)). In the appendix, by combining the results of the article with those of M. Alexander and M. Nazarov [Math. Ann. 313, No. 2, 315–357 (1999; Zbl 0989.17006)] and M. Itoh [J. Lie Theory 10, No. 2, 463–489 (2000; Zbl 0981.17005)], the author also gives new Capelli-type identities for invariant differential operators for orthogonal Lie algebras.Representation Theory of the American Mathematical Society 06/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**Following Kashiwara's algebraic approach, we construct crystal bases and canonical bases for quantum supergroups with no isotropic odd roots and for their integrable modules.Representation Theory of the American Mathematical Society 04/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**We define and study a correspondence between the set of distinguished G^0-conjugacy classes in a fixed connected component of a reductive group G (with G^0 almost simple) and the set of (twisted) elliptic conjugacy classes in the Weyl group. We also prove a homogeneity property related to this correspondence.Representation Theory of the American Mathematical Society 04/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**(The present paper complements the author’s work [Compos. Math. 51, 333–399 (1984; Zbl 0624.22011)].) Let 𝔽 be a local field, non-Archimedean and of characteristic not 2. Let (V,Q) be a nondegenerate quadratic space over 𝔽 of dimension n. Let M r be the direct sum of r copies of V. We prove that, for r<n, there is no nonzero distribution on M r which under the action of the orthogonal group transforms according to the character determinant.Representation Theory of the American Mathematical Society 03/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**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.Representation Theory of the American Mathematical Society 02/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**We introduce categories of homogeneous strict polynomial functors, $\Pol^\I_{d,\k}$ and $\Pol^\II_{d,\k}$, defined on vector superspaces over a field $\k$ of characteristic not equal 2. These categories are related to polynomial representations of the supergroups $GL(m|n)$ and Q(n), respectively. In particular, we prove an equivalence between $\Pol^\I_{d,\k}$, $\Pol^\II_{d,\k}$ and the category of finite dimensional supermodules over the Schur superalgebra $\Sc(m|n,d)$, $\Qc(n,d)$ respectively provided $m,n \ge d$. We also discuss some aspects of Sergeev duality from the viewpoint of the category $\Pol^\II_{d,\k}$.Representation Theory of the American Mathematical Society 01/2013; 17(20). -
##### Article: The 2-blocks of defect 4

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**ABSTRACT:**We show that the major counting conjectures of modular representation theory are satisfied for 2-blocks of defect at most 4 except one possible case. In particular, we determine the invariants of such blocks.Representation Theory of the American Mathematical Society 01/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**The author studies the Bernstein center for GL 2 over a non-Archimedean local field using the work of A. Moy and M. Tadić [Represent. Theory 6, 313–329 (2002); erratum ibid. 9, 455–456 (2005; Zbl 1019.22007)]. Author’s summary: Let F be a nonarchimedean local field of residue characteristic p, and let r be an odd natural number less than p. Using the work of Moy and Tadić (loc. cit.), we find an element z of the Bernstein center of G= GL 2 (F) that acts on any representation π of G by the scalar z(π)= tr Frob ; Sym r ∘φ π I F , the trace of any geometric Frobenius element Frob of the absolute Weil group W F of F, acting on the inertia-fixed points of the representation Sym r ∘φ π of W F , where φ π :W F →G ^ is the restriction to W F of the Langlands parameter of π. This element z is specified by giving the functions obtained by convolving it with the characteristic functions of a large class of compact open subgroups of G, that includes all the groups of both the congruence and the Iwahori filtrations of G having depth at least one.Representation Theory of the American Mathematical Society 01/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**We give a counterexample to the most optimistic analogue (due to Kleshchev and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras associated to simply-laced Dynkin diagrams. The first counterexample occurs in type A_5 for p = 2 and involves the same singularity used by Kashiwara and Saito to show the reducibility of the characteristic variety of an intersection cohomology D-module on a quiver variety. Using recent results of Polo one can give counterexamples in type A in all characteristics.Representation Theory of the American Mathematical Society 12/2012; -
##### Article: Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the universal case

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**ABSTRACT:**Let $(W,S)$ be a Coxeter system and let $w \mapsto w^*$ be an involution of $W$ which preserves the set of simple generators $S$. Lusztig has recently shown that the set of twisted involutions (i.e., elements $w \in W$ with $w^{-1} = w^*$) naturally generates a module of the Hecke algebra of $(W,S)$ with two distinguished bases. The transition matrix between these bases defines a family of polynomials $P^\sigma_{y,w}$ which one can view as "twisted" analogs of the much-studied Kazhdan-Lusztig polynomials of $(W,S)$. The polynomials $P^\sigma_{y,w}$ can have negative coefficients, but display several conjectural positivity properties of interest. This paper reviews Lusztig's construction and then proves three such positivity properties for Coxeter systems which are universal (i.e., having no braids relations), generalizing previous work of Dyer. Our methods are entirely combinatorial and elementary, in contrast to the geometric arguments employed by Lusztig and Vogan to prove similar positivity conjectures for crystallographic Coxeter systems.Representation Theory of the American Mathematical Society 11/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**Infinitesimal Cherednik algebras, first introduced in [EGG], are continuous analogues of rational Cherednik algebras, and in the case of gl_n, are deformations of universal enveloping algebras of the Lie algebras sl_{n+1}. Despite these connections, infinitesimal Cherednik algebras are not widely-studied, and basic questions of intrinsic algebraic and representation theoretical nature remain open. In the first half of this paper, we construct the complete center of H_\zeta(gl_n) for the case of n=2 and give one particular generator of the center, the Casimir operator, for general n. We find the action of this Casimir operator on the highest weight modules to prove the formula for the Shapovalov determinant, providing a criterion for the irreducibility of Verma modules. We classify all irreducible finite dimensional representations and compute their characters. In the second half, we investigate Poisson-analogues of the infinitesimal Cherednik algebras and use them to gain insight on the center of H_\zeta(gl_n). Finally, we investigate H_\zeta(sp_{2n}) and extend various results from the theory of H_\zeta(gl_n), such as a generalization of Kostant's theorem.Representation Theory of the American Mathematical Society 10/2012; - Representation Theory of the American Mathematical Society 10/2012; 16(14).
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**ABSTRACT:**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 provide a simple formula for the limit of characters. 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.Representation Theory of the American Mathematical Society 09/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**Given a nilpotent orbit O of a real, reductive algebraic group, a necessary condition is given for the existence of a tempered representation pi such that O occurs in the wave front cycle of pi. The coefficients of the wave front cycle of a tempered representation are expressed in terms of volumes of precompact submanifolds of an affine space.Representation Theory of the American Mathematical Society 09/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**Let G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K and let V be a unipotent representation of G(K) (for example, an Iwahori-spherical representation). We calculate the character of V at compact very regular elements of G(K) under some restrictions on the characteristic.Representation Theory of the American Mathematical Society 08/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**We prove the conjecture of Frenkel, Kac and Wakimoto on the existence of two-sided BGG resolutions of G-integrable admissible representations of affine Kac-Moody algebras at fractional levels. As an application we establish the semi-infintie analogue of the generalized Borel-Weil theorem for mimimal parabolic subalgebras which enables an inductive study of admissible representations.Representation Theory of the American Mathematical Society 07/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**According to an old result of Sch\"utzenberger, the involutions in a given two-sided cell of the symmetric group $\SG_n$ are all conjugate. In this paper, we study possible generalisations of this property to other types of Coxeter groups. We show that Sch\"utzenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz' conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types; combined with previous work, this leaves type $E_8$ as the only remaining open case.Representation Theory of the American Mathematical Society 05/2012;

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