Let $G={\rm GL}_n$ be the general linear group over an algebraically closed field $k$, let $\mathfrak g=\mathfrak gl_n$ be its Lie algebra and let $U$ be the subgroup of $G$ which consists of the upper uni-triangular matrices. Let $k[\mathfrak g]$ be the algebra of polynomial functions on $\mathfrak g$ and let $k[\mathfrak g]^G$ be the algebra of invariants under the conjugation action of $G$. In characteristic zero, we give for all dominant weights $χ\in\mathbb Z^n$ finite homogeneous spanning sets for the $k[\mathfrak g]^G$-modules $k[\mathfrak g]_χ^U$ of highest weight vectors. This result (with some mistakes) was already given without proof by J.~F.~Donin. Then we do the same for tuples of $n\times n$-matrices under the diagonal conjugation action. Furthermore we extend our earlier results in positive characteristic and give a general result which reduces the problem to giving spanning sets of the highest weight vectors for the action of ${\rm GL}_r\times{\rm GL}_s$ on tuples of $r\times s$ matrices. This requires the technique called "transmutation" by R.~Brylinsky which is based on an instance of Howe duality. In the cases that $χ_{{}_n}\ge -1$ or $χ_{{}_1}\le 1$ this leads to new spanning sets for the modules $k[\mathfrak g]_χ^U$.
We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $\mathfrak g$. In particular, our classification leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of $\mathfrak g$.
We demonstrate equivalence between two definitions of lower finite highest weight categories. We also show that, in the presence of a duality, a lower finite highest weight structure on a category is unique. Finally, we give a new proof for the fact that any abelian category is extension full in its ind-completion.
We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple.
Let $\mathfrak{g}$ be a finite or an affine type Lie algebra over $\mathbb{C}$ with root system $Δ$. We show a parabolic generalization of the partial sum property for $Δ$, which we term the parabolic partial sum property. It allows any root $β$ involving (any) fixed subset $S$ of simple roots, to be written as an ordered sum of roots, each involving exactly one simple root from $S$, with each partial sum also being a root. We show three applications of this property to weights of highest weight $\mathfrak{g}$-modules: (1)~We provide a minimal description for the weights of all non-integrable …
In this paper, we characterize quasi-integrable modules, of nonzero level, over twisted affine Lie superalgebras. We show that quasi-integrable modules are not necessarily highest weight modules. We prove that each quasi-integrable module is parabolically induced from a cuspidal module, over a finite dimensional Lie superalgebra having a Cartan subalgebra whose corresponding root system just contain real roots; in particular, the classification of quasi-integrable modules is reduced to the known classification of cuspidal modules over such Lie superalgebras.
In this paper, we begin the study of highest weight representations of the quantized enveloping superalgebra ${\mathfrak U}_q {\mathfrak p}_n$ of type $P$. We introduce a Drinfeld-Jimbo representation and establish a triangular-decomposition of ${\mathfrak U}_q {\mathfrak p}_n$. We explain how to relate modules over ${\mathfrak U}_q {\mathfrak p}_n$ to modules over ${\mathfrak p}_n$, the Lie superalgebra of type $P$, and we prove that the category of tensor modules over ${\mathfrak U}_q {\mathfrak p}_n$ is not semisimple.
We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.
In this article, we consider a dual pair $(G, G')$ in the symplectic group $Sp(W)$ with $G$ compact and let $(\tilde{G}, \tilde{G}')$ be the preimages of $G$ and $G'$ in the metaplectic group $\widetilde{Sp(W)}$. For every irreducible representation $Π$ of $\tilde{G}$ appearing in Howe correspondence, we compute explicitly the restriction of the character $Θ_{Π'}$ of the associated representation $Π'$ of $\tilde{G}'$ on the set of regular points on the compact Cartan subgroup $\tilde{H}'$ of $\tilde{G}'$.
