A $$3\times 3$$ linear q-difference system with $$E_8^{(1)}$$-symmetry

Abstract

Abstract We present a linear q -difference equation of rank 3, which admits the affine Weyl group symmetry of type $$E_8^{(1)}$$ E 8 ( 1 ) . This equation has essentially 8 parameters and 2 accessory parameters. We characterize this equation by the spectral type of q -difference system which is introduced by Sakai–Yamaguchi. We further compare this equation with Moriyama–Yamada’s quantum curve which has $$W(E_8^{(1)})$$ W ( E 8 ( 1 ) ) -symmetry. We show that our equation can be regarded as an extension of Moriyama–Yamada’s curve. We introduce the point configuration of a higher-order scalar q -difference equation to characterize our equation and clarify the corresponding between our equation and the curve. The symmetry of our equation is provided by the q -middle convolution, defined by Sakai–Yamaguchi and reformulated by Arai–Takemura. In this paper, we provide a reconstruction of the q -middle convolution via a q -Okubo-type equation. This construction is a natural q -analog of the middle convolution of Fuchsian differential equation. Several properties for our q -middle convolution are also discussed.