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

## Journal 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.

## Current impact factor: 0.48

## Impact Factor Rankings

2015 Impact Factor | Available summer 2015 |
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2013 / 2014 Impact Factor | 0.475 |

## Additional details

5-year impact | 0.00 |
---|---|

Cited half-life | 0.00 |

Immediacy index | 0.00 |

Eigenfactor | 0.00 |

Article influence | 0.00 |

Website | Representation Theory website |

Other titles | Representation theory |

ISSN | 1088-4165 |

OCLC | 34602921 |

Material type | Document, Internet resource |

Document type | Internet Resource, Computer File, Journal / Magazine / Newspaper |

## Publisher details

- Pre-print
- Author can archive a pre-print version

- Post-print
- 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

- Representation Theory of the American Mathematical Society 03/2015; 19(3):9-23. DOI:10.1090/S1088-4165-2015-00464-9
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**ABSTRACT:**Let a reductive algebraic group over an algebraically closed field of good characteristic be given. Attached to a nilpotent element of its Lie algebra, we consider a family of algebraic varieties, which incorporates classical objects such as Springer fiber, Spaltenstein varieties, and Hessenberg varieties. When the nilpotent element is of standard Levi type, we show that the varieties of this family admit affine pavings that can be obtained by intersecting with the Schubert cells corresponding to a suitable Borel subgroup.Representation Theory of the American Mathematical Society 10/2014; 18:341-360. - [Show abstract] [Hide abstract]

**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; DOI:10.1090/S1088-4165-05-00259-1 - [Show abstract] [Hide abstract]

**ABSTRACT:**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.Representation Theory of the American Mathematical Society 07/2014; 19(2). DOI:10.1090/S1088-4165-2015-00463-7 -
##### Article: On the theta correspondence for $(\mathrm {GSp}(4), \mathrm {GSO}(4,2))$ and Shalika periods

Representation Theory of the American Mathematical Society 04/2014; 18(3):28-87. DOI:10.1090/S1088-4165-2014-00451-5 - [Show abstract] [Hide abstract]

**ABSTRACT:**Assume $\mathbf{G}$ is a connected reductive algebraic group defined over an algebraic closure $\mathbb{K} = \overline{\mathbb{F}}_p$ of the finite field of prime order $p>0$. Furthermore, assume that $F : \mathbf{G} \to \mathbf{G}$ is a Frobenius endomorphism of $\mathbf{G}$. In this article we give a formula for the value of any $F$-stable character sheaf of $\mathbf{G}$ at a unipotent element. This formula is expressed in terms of class functions of $\mathbf{G}^F$ which are supported on a single unipotent class of $\mathbf{G}$. In general these functions are not determined, however we give an expression for these functions under the assumption that $Z(\mathbf{G})$ is connected, $\mathbf{G}/Z(\mathbf{G})$ is simple and $p$ is a good prime for $\mathbf{G}$. In this case our formula is completely explicit.Representation Theory of the American Mathematical Society 03/2014; 18(10). DOI:10.1090/S1088-4165-2014-00457-6 -
##### 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). DOI:10.1090/S1088-4165-2013-00448-X - [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; DOI:10.1090/S1088-4165-2013-00434-X - [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; DOI:10.1090/S1088-4165-2014-00453-9 - [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; DOI:10.1090/S1088-4165-2014-00455-2 - [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; DOI:10.1090/S1088-4165-2013-00428-4 - [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; DOI:10.1090/S1088-4165-2014-00449-7 - [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). DOI:10.1090/S1088-4165-2013-00445-4 - [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; DOI:10.1090/S1088-4165-2013-00427-2 - [Show abstract] [Hide abstract]

**ABSTRACT:**We study a family of abelian categories O_{c, t} depending on complex parameters c, t which are interpolations of the O-category for the rational Cherednik algebra H_c(t) of type A, where t is a positive integer. We define the notion of a Verma object in such a category (a natural analogue of the notion of Verma module). We give some necessary conditions and some sufficient conditions for the existence of a non-trivial morphism between two such Verma objects. We also compute the character of the irreducible quotient of a Verma object for sufficiently generic values of parameters c, t, and prove that a Verma object of infinite length exists in O_{c, t} only if c is rational and c < 0. We also show that for every rational c < 0 there exists a rational t < 0 such that there exists a Verma object of infinite length in O_{c, t}. The latter result is an example of a degeneration phenomenon which can occur in rational values of t, as was conjectured by P. Etingof.Representation Theory of the American Mathematical Society 01/2013; 18. DOI:10.1090/S1088-4165-2014-00459-X

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