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Register a new userMultiplets of representations, twisted Dirac operators and Vogans conjecture in affine setting
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Authors
Victor G. Kac
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We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogans conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.
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We prove the twisted Whittaker category on the affine flag variety and the category of representations of the mixed quantum group are equivalent.
Let G be a simple simply-connected group over an algebraically closed field k, X be a smooth connected projective curve over k. In this paper we develop the theory of geometric Eisenstein series on the moduli stack Bun_G of G-torsors on X in the setting of the quantum geometric Langlands program (for etale l-adic sheaves) in analogy with [3]. We calculate the intersection cohomology sheaf on the version of Drinfeld compactification in our twisted setting. In the case G=SL_2 we derive some results about the Fourier coefficients of our Eisenstein series. In the case of G=SL_2 and X=P^1 we also construct the corresponding theta-sheaves and prove their Hecke property.
We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $hat{sell}_{2|1}$ (resp. $hat{psell}_{2|2}$) can be modified, using Zwegers real analytic corrections, to form a modular (resp. $S$-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N=2 (resp. N=4) superconformal algebras with central charge $3(1-frac{2m+2}{M})$, where $m in mathbb{Z}_{geq 0}$, $Min mathbb{Z}_{geq 2}$, $gcd(2m+2,M)=1$ if $m>0$ (resp. $6(frac{m}{M}-1)$, where $m in mathbb{Z}_{geq 1}, Min mathbb{Z}_{geq 2}$, $gcd(2m,M)=1$ if $m>1$), whose modified characters and supercharacters form a modular invariant family.
We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra $widehat{sl}_{2|1}$ can be modified, using Zwegers real analytic corrections, to form an $SL_2(mathbf{Z})$-invariant family of functions. Using a variation of Zwegers correction, we obtain a similar result for $widehat{osp}_{3|2}$. Applying the quantum Hamiltonian reduction, this leads to new families of positive energy modules over the $N=2$ (resp. $N=3$) superconformal algebras with central charge $c=3 (1-frac{2m+2}{M})$, where $m in mathbf{Z}_{geq 0}, M in mathbf{Z}_{geq 2}$, gcd$(2m+2,M)=1$ if $m>0$ (resp. $c=-3frac{2m+1}{M}$, where $m in mathbf{Z}_{geq 0}, M in mathbf{Z}_{geq 2}$ gcd$(4m +2, M) =1)$, whose modified supercharacters form an $SL_2(mathbf{Z})$-invariant family of functions.
We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ mathfrak{g}. $ For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers type modifier is used. We show that the span of the resulting modified normalized supercharacters is $ SL_2(mathbb{Z}) $-invariant, with the transformation matrix equal, in the case the Killing form on $mathfrak{g}$ is non-degenerate, to that for the subalgebra $ mathfrak{g}^! $ of $ mathfrak{g}, $ orthogonal to a maximal isotropic set of roots of $ mathfrak{g}. $
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