اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSome categories of modules for toroidal Lie algebras
326
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper, we use basic formal variable techniques to study certain categories of modules for the toroidal Lie algebra $tau$. More specifically, we define and study two categories $mathcal{E}_{tau}$ and $mathcal{C}_{tau}$ of $tau$-modules using generating functions, where $mathcal{E}_{tau}$ is proved to contain the evaluation modules while $mathcal{C}_{tau}$ contains certain restricted $tau$-modules, the evaluation modules, and their tensor product modules. Furthermore, we classify the irreducible integrable modules in categories $mathcal{E}_{tau}$ and $mathcal{C}_{tau}$.
قيم البحث
اقرأ أيضاً
We identify level one global Weyl modules for toroidal Lie algebras with certain twists of modules constructed by Moody-Eswara Rao-Yokonuma via vertex operators for type ADE and by Iohara-Saito-Wakimoto and Eswara Rao for general type. The twist is g
iven by an action of $mathrm{SL}_{2}(mathbb{Z})$ on the toroidal Lie algebra. As a byproduct, we obtain a formula for the character of the level one local Weyl module over the toroidal Lie algebra and that for the graded character of the level one graded local Weyl module over an affine analog of the current Lie algebra.
We prove that the tensor product of a simple and a finite dimensional $mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $mathfrak{q}(n)$-supermodules to that of simple $mathfrak{sl}_n$-modules. Rough st
ructure of simple $mathfrak{q}(n)$-supermodules, considered as $mathfrak{sl}_n$-modules, is described in terms of the combinatorics of category $mathcal{O}$.
The rank $n$ symplectic oscillator Lie algebra $mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spa
ces over $mathfrak{g}_n$. When $dot z eq 0$, it is shown that there is an equivalence between the full subcategory $mathcal{O}_{mathfrak{g}_n}[dot z]$ of the BGG category $mathcal{O}_{mathfrak{g}_n}$ for $mathfrak{g}_n$ and the BGG category $mathcal{O}_{mathfrak{sp}_{2n}}$ for $mathfrak{sp}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over $mathfrak{g}_n$ with finite-dimensional weight spaces.
We provide a micro-local necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let $bf G$ be a complex algebraic reductive group, and $bf Hsubset G$ be a spherical algebraic su
bgroup. Let $mathfrak{g},mathfrak{h}$ denote the Lie algebras of $bf G$ and $bf H$, and let $mathfrak{h}^{bot}$ denote the annihilator of $mathfrak{h}$ in $mathfrak{g}^*$. A $mathfrak{g}$-module is called $mathfrak{h}$-distinguished if it admits a non-zero $mathfrak{h}$-invariant functional. We show that the maximal $bf G$-orbit in the annihilator variety of any irreducible $mathfrak{h}$-distinguished $mathfrak{g}$-module intersects $mathfrak{h}^{bot}$. This generalizes a result of Vogan. We apply this to Casselman-Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that as suggested by Prasad, if $H$ is a symmetric subgroup of a real reductive group $G$, the existence of a tempered $H$-distinguished representation of $G$ implies the existence of a generic $H$-distinguished representation of $G$. Many models studied in the theory of automorphic forms involve an additive character on the unipotent radical of $bf H$, and we devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of W-algebras. As an application we derive necessary conditions for the existence of Rankin-Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan-Gross-Prasad conjectures for non-generic representations. We also prove more general results that ease the sphericity assumption on the subgroup, and apply them to local theta correspondence in type II and to degenerate Whittaker models.
The deformed current Lie algebra was introduced by the author to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
