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Register a new userVarieties of modules over the quantum plane
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The quantum plane is the non-commutative polynomial algebra in variables $x$ and $y$ with $xy=qyx$. In this paper, we study the module variety of $n$-dimensional modules over the quantum plane, and provide an explicit description of its irreducible components and their dimensions. We also describe the irreducible components and their dimensions of the GIT quotient of the module variety with respect to the conjugation action of ${rm GL}_n$.
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$imath$quantum groups are generalizations of quantum groups which appear as coideal subalgebras of quantum groups in the theory of quantum symmetric pairs. In this paper, we define the notion of classical weight modules over an $imath$quantum group, and study their properties along the lines of the representation theory of weight modules over a quantum group. In several cases, we classify the finite-dimensional irreducible classical weight modules by a highest weight theory.
In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.
In the present paper, using the technique of localization, we determine the center of the quantum Schr{o}dinger algebra $S_q$ and classify simple modules with finite-dimensional weight spaces over $S_q$, when $q$ is not a root of unity. It turns out that there are four classes of such modules: dense $U_q(mathfrak{sl}_2)$-modules, highest weight modules, lowest weight modules, and twisted modules of highest weight modules.
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 subgroup. 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.
Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $mathcal C(mathfrak p) = {(X,Y) in mathfrak p times mathfrak p mid [X,Y] = 0}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $mathcal C(mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $mathfrak p$. As a consequence, for the case $P = B$ a Borel subgroup of $G$, we give a classification of when $mathcal C(mathfrak b)$ is irreducible; this builds on a partial classification given by Keeton. Further, in cases where $mathcal C(mathfrak p)$ is irreducible, we consider whether $mathcal C(mathfrak p)$ is a normal variety. In particular, this leads to a classification of when $mathcal C(mathfrak b)$ is normal.
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