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Register a new userMinimal dimensional representations of reduced enveloping algebras for $mathfrak{gl}_n$
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Let $mathfrak g = mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N in mathbb Z_{ge 1}$. Let $chi in mathfrak g^*$ and denote by $U_chi(mathfrak g)$ the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional $U_chi(mathfrak g)$-module has dimension divisible by $p^{d_chi}$, where $d_chi$ is half the dimension of the coadjoint orbit of $chi$. Our main theorem gives a classification of $U_chi(mathfrak g)$-modules of dimension $p^{d_chi}$. As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for $U_0(mathfrak h)$ for a certain Levi subalgebra $mathfrak h$ of $mathfrak g$; we view this as a modular analogue of M{oe}glins theorem on completely primitive ideals in $U(mathfrak{gl}_N(mathbb C))$. To obtain these results, we reduce to the case $chi$ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted $W$-algebra.
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The affine evaluation map is a surjective homomorphism from the quantum toroidal ${mathfrak {gl}}_n$ algebra ${mathcal E}_n(q_1,q_2,q_3)$ to the quantum affine algebra $U_qwidehat{mathfrak {gl}}_n$ at level $kappa$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=kappa^2$. We discuss ${mathcal E}_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${mathcal E}_n(q_1,q_2,q_3)$, which describes a deformation of the coset theory $widehat{mathfrak {gl}}_n/widehat{mathfrak {gl}}_{n-1}$.
We define a $mathfrak{gl}_N$-stratification of the Grassmannian of $N$ planes $mathrm{Gr}(N,d)$. The $mathfrak{gl}_N$-stratification consists of strata $Omega_{mathbf{Lambda}}$ labeled by unordered sets $mathbf{Lambda}=(lambda^{(1)},dots,lambda^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $(otimes_{i=1}^n V_{lambda^{(i)}})^{mathfrak{sl}_N} e 0$. Here $V_{lambda^{(i)}}$ is the irreducible $mathfrak{gl}_N$-module with highest weight $lambda^{(i)}$. We show that the closure of a stratum $Omega_{mathbf{Lambda}}$ is the union of the strata $Omega_{mathbfXi}$, $mathbf{Xi}=(xi^{(1)},dots,xi^{(m)})$, such that there is a partition ${I_1,dots,I_m}$ of ${1,2,dots,n}$ with $ {rm {Hom}}_{mathfrak{gl}_N} (V_{xi^{(i)}}, otimes_{jin I_i}V_{lambda^{(j)}}big) eq 0$ for $i=1,dots,m$. The $mathfrak{gl}_N$-stratification of the Grassmannian agrees with the Wronski map. We introduce and study the new object: the self-dual Grassmannian $mathrm{sGr}(N,d)subset mathrm{Gr}(N,d)$. Our main result is a similar $mathfrak{g}_N$-stratification of the self-dual Grassmannian governed by representation theory of the Lie algebra $mathfrak {g}_{2r+1}:=mathfrak{sp}_{2r}$ if $N=2r+1$ and of the Lie algebra $mathfrak g_{2r}:=mathfrak{so}_{2r+1}$ if $N=2r$.
Let $mathfrak g$ 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 representations of $U(mathfrak g,e)$. For $mathfrak g$ of low rank, we report on computer calculations that have been used to determine the structure of $mathcal E(mathfrak g,e)$, and the action of the component group $Gamma_e$ of the centralizer of $e$ on $mathcal E(mathfrak g,e)$. As a consequence, we provide two examples where the nilpotent orbit of $e$ is induced, but there is a 1-dimensional $Gamma_e$-stable $U(mathfrak g,e)$-module which is not induced via Losevs parabolic induction functor. In turn this gives examples where there is a non-induced multiplicity free primitive ideal of $U(mathfrak g)$.
On a Fock space constructed from $mn$ free bosons and lattice ${Bbb {Z}}^{mn}$, we give a level $n$ action of the quantum toroidal algebra $mathscr {E}_m$ associated to $mathfrak{gl}_m$, together with a level $m$ action of the quantum toroidal algebra ${mathscr E}_n$ associated to ${mathfrak {gl}}_n$. We prove that the $mathscr {E}_m$ transfer matrices commute with the $mathscr {E}_n$ transfer matrices after an appropriate identification of parameters.
The essential feature of a root-graded Lie algebra L is the existence of a split semisimple subalgebra g with respect to which L is an integrable module with weights in a possibly non-reduced root system S of the same rank as the root system R of g. Examples include map algebras (maps from an affine scheme to g, S = R), matrix algebras like sl_n(A) for a unital associative algebra A (S = R = A_{n-1}), finite-dimensional isotropic central-simple Lie algebras (S properly contains R in general), and some equivariant map algebras. In this paper we study the category of representations of a root-graded Lie algebra L which are integrable as representations of g and whose weights are bounded by some dominant weight of g. We link this category to the module category of an associative algebra, whose structure we determine for map algebras and sl_n(A). Our results unify previous work of Chari and her collaborators on map algebras and of Seligman on isotropic Lie algebras.
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