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Register a new userA family of representations of the affine Lie superalgebra $widehat{mathfrak{gl}_{m|n}}(mathbb{C})$
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In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra $mathfrak{gl}_{m|n}(mathbb{C})$. The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac-Moody Lie superalgebra $widehat{mathfrak{gl}_{m|n}}(mathbb{C})$ on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter $mu$, and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter $mu$ is nonzero.
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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.
The Capelli problem for the symmetric pairs $(mathfrak{gl}times mathfrak{gl},mathfrak{gl})$ $(mathfrak{gl},mathfrak{o})$, and $(mathfrak{gl},mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(mathfrak{g},mathfrak{g}):=(mathfrak{gl}(m|2n),mathfrak{osp}(m|2n))$ and $(mathfrak{gl}(m|n)timesmathfrak{gl}(m|n),mathfrak{gl}(m|n))$, acting on $W:=S^2(mathbb C^{m|2n})$ and $mathbb C^{m|n}otimes(mathbb C^{m|n})^*$. We also give an affirmative answer to the abstract Capelli problem for these cases.
We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $mathcal W_q(2(m|n))$ of $(m|n)$-type is introduced as the quantum differential operators (QDO for short) algebra $textrm{Diff}_q(Omega_q)$ defined over $Omega_q(m|n)$, which is a smash product of the quantum differential Hopf algebra $mathfrak D_q(m|n)$ (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra $Omega_q(m|n)$. An interested point of this approach here is that even though $mathcal W_q(2(m|n))$ itself is in general no longer a Hopf algebra, so are some interesting sub-quotients existed inside. This point of view gives us one of main expected results, that is, the quantum (restricted) Grassmann superalgebra $Omega_q$ is made into the $mathcal U_q(mathfrak g)$-module (super)algebra structure,$Omega_q=Omega_q(m|n)$ for $q$ generic, or $Omega_q(m|n, bold 1)$ for $q$ root of unity, and $mathfrak g=mathfrak{gl}(m|n)$ or $mathfrak {sl}(m|n)$, the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple $mathcal U_q(mathfrak g)$-modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra $Omega_q^!$ as $mathcal U_q(mathfrak g)$-module algebra.In the paper some examples of pointed Hopf algebras can arise from the QDOs, whose idea is an expansion of the spirit noted by Manin in cite{Ma}, & cite{Ma1}.
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.
Generalizations of the AGT correspondence between 4D $mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${mathbb {C}}^2$ with $Omega$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N = 2, 3, ldots$, on ${mathbb {C}}^2/{mathbb {Z}}_n$, $n = 2, 3, ldots$, with $Omega$-deformation and 2D conformal field theories with $mathcal{W}^{, para}_{N, n}$ ($n$-th parafermion $mathcal{W}_N$) symmetry and $widehat{mathfrak{sl}}(n)_N$ symmetry. In this work, we trivialize the factor with $mathcal{W}^{, para}_{N, n}$ symmetry in the 4D $SU(N)$ instanton partition functions on ${mathbb {C}}^2/{mathbb {Z}}_n$ (by using specific choices of parameters and imposing specific conditions on the $N$-tuples of Young diagrams that label the states), and extract the 2D $widehat{mathfrak{sl}}(n)_N$ WZW conformal blocks, $n = 2, 3, ldots$, $N = 1, 2, ldots, .$
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