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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 $N=2r$.
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 algebr
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 homog
We show that the Kazhdan-Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $widehat{mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure, and
We begin a study of the representation theory of quantum continuous $mathfrak{gl}_infty$, which we denote by $mathcal E$. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference oper
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-