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Register a new userWeight $q$-multiplicities for representations of the exceptional Lie algebra $mathfrak{g}_2$
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Given a simple Lie algebra $mathfrak{g}$, Kostants weight $q$-multiplicity formula is an alternating sum over the Weyl group whose terms involve the $q$-analog of Kostants partition function. For $xi$ (a weight of $mathfrak{g}$), the $q$-analog of Kostants partition function is a polynomial-valued function defined by $wp_q(xi)=sum c_i q^i$ where $c_i$ is the number of ways $xi$ can be written as a sum of $i$ positive roots of $mathfrak{g}$. In this way, the evaluation of Kostants weight $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a highest weight representation of $mathfrak{g}$. In this paper, we give closed formulas for computing weight $q$-multiplicities in a highest weight representation of the exceptional Lie algebra $mathfrak{g}_2$.
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Let ${mathcal W}_n$ be the Lie algebra of polynomial vector fields. We classify simple weight ${mathcal W}_n$-modules $M$ with finite weight multiplicities. We prove that every such nontrivial module $M$ is either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex on $mathbb C^n$.
Kostants weight $q$-multiplicity formula is an alternating sum over a finite group known as the Weyl group, whose terms involve the $q$-analog of Kostants partition function. The $q$-analog of the partition function is a polynomial-valued function defined by $wp_q(xi)=sum_{i=0}^k c_i q^i$, where $c_i$ is the number of ways the weight $xi$ can be written as a sum of exactly $i$ positive roots of a Lie algebra $mathfrak{g}$. The evaluation of the $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in an irreducible highest weight representation of $mathfrak{g}$. In this paper, we specialize to the Lie algebra $mathfrak{sp}_6(mathbb{C})$ and we provide a closed formula for the $q$-analog of Kostants partition function, which extends recent results of Shahi, Refaghat, and Marefat. We also describe the supporting sets of the multiplicity formula (known as the Weyl alternation sets of $mathfrak{sp}_6(mathbb{C})$), and use these results to provide a closed formula for the $q$-multiplicity for any pair of dominant integral weights of $mathfrak{sp}_6(mathbb{C})$. Throughout this work, we provide code to facilitate these computations.
We use analogues of Enrights and Arkhipovs functors to determine the quiver and relations for a category of $mathfrak{sl}_2 ltimes L(4)$-modules which are locally finite (and with finite multiplicities) over $mathfrak{sl}_2$. We also outline serious obstacles to extend our result to $mathfrak{sl}_2 ltimes L(k)$, for $k>4$.
Let $mathfrak l:= mathfrak q(n)timesmathfrak q(n)$, where $mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $mathfrak q(n)$, and hence is equipped with a canonical $mathfrak l$-module structure. We consider a distinguished basis ${D_lambda}$ of the algebra of $mathfrak l$-invariant super-polynomial differential operators on $V$, which is indexed by strict partitions of length at most $n$. We show that the spectrum of the operator $D_lambda$, when it acts on the algebra $mathscr P(V)$ of super-polynomials on $V$, is given by the factorial Schur $Q$-function of Okounkov and Ivanov. This constitutes a refinement and a new proof of a result of Nazarov, who computed the top-degree homogeneous part of the Harish-Chandra image of $D_lambda$. As a further application, we show that the radial projections of the spherical super-polynomials corresponding to the diagonal symmetric pair $(mathfrak l,mathfrak m)$, where $mathfrak m:=mathfrak q(n)$, of irreducible $mathfrak l$-submodules of $mathscr P(V)$ are the classical Schur $Q$-functions.
For the two Cartan type S subalgebras of the Witt algebra $W_n$, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra $h_n$. We also give all submodules of these modules.
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