Exhaustion conjecture for simple bounded cuspidal affine modules

Let g\mathfrak{g} be the basic classical Lie superalgebra, let L(g)L(\mathfrak{g}) be its loop superalgebra, and let g^\widehat{\mathfrak{g}} be the corresponding affine Lie superalgebra. For a simple bounded cuspidal L(g)L(\mathfrak{g})-module VV, write Li(V)L^i(V) for the simple modules in the decomposition associated with the affine construction. Exhaustion conjecture. If MM is a simple bounded cuspidal g^\widehat{\mathfrak{g}}-module, then there is a simple bounded cuspidal L(g)L(\mathfrak{g})-module VV for which

MLi(V)M\cong L^i(V)

for some iNi\in\mathbb{N}. The conjecture asserts that the family obtained from loop-algebra modules accounts for all simple bounded cuspidal affine modules; the source does not provide evidence of a resolution.

Sources & referencesView supporting material

Primary source

Lucas Calixto, Vyacheslav Futorny and Henrique Rocha, “Harish-Chandra modules for map and affine Lie superalgebras”, arXiv:2104.07517 (2025).

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