Remarks on q-difference opers arising from quantum toroidal algebras

Abstract We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="fraktur">gl</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> algebra on relaxed Verma modules, and q -hypergeometric opers with apparent singularities. We introduce alongside the notion of apparent singularities for linear q -difference operators and discuss some of their properties. We also touch on a generalization to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="fraktur">gl</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> .