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Recently a remarkable map between 4-dimensional superconformal field theories and vertex algebras has been constructed cite{BLLPRV15}. This has lead to new insights in the theory of characters of vertex algebras. In particular it was observed that in some cases these characters decompose in nice products cite{XYY16}, cite{Y16}. The purpose of this note is to explain the latter phenomena. Namely, we point out that it is immediate by our character formula cite{KW88}, cite{KW89} that in the case of a textit{boundary level} the characters of admissible representations of affine Kac-Moody algebras and the corresponding $W$-algebras decompose in products in terms of the Jacobi form $ vartheta_{11}(tau, z)$.
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A holomorphic continuation of Jacquet type integrals for parabolic subgroups with abelian nilradical is studied. Complete results are given for generic characters with compact stabilizer and arbitrary representations induced from admissible representations. A description of all of the pertinent examples is given. These results give a complete description of the Bessel models corresponding to compact stabilizer.
The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $mathfrak{sl}_3$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arxiv:1005.0185, arxiv:1910.13781]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral $mathsf{k}$ are always rational in category $mathscr{O}$, whilst they always admit nonsemisimple relaxed highest-weight modules unless $mathsf{k}+frac{3}{2} in mathbb{Z}_{ge0}$.
In this note, we show that the free generators of the Mishchenko-Fomenko subalgebra of a complex reductive Lie algebra, constructed by the argument shift method at a regular element, form a regular sequence. This result was proven by Serge Ovsienko in the type A at a regular and semisimple element. Our approach is very different, and is strongly based on geometric properties of the nilpotent bicone.
This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over C and over C[t_1, t_2,...,t_n]. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators studied by Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial, though the equivariant representation is not.
It is shown that, given a representation of a quiver over a finite field, one can check in polynomial time whether it is absolutely indecomposable.
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