We discuss the 5d AGT correspondence of supergroup gauge theories with A-type
supergroups. We introduce two intertwiners called positive and negative
intertwiners to compute the instanton partition function. The positive
intertwiner is the ordinary Awata-Feigin-Shiraishi intertwiner while the
negative intertwiner is an intertwiner obtained by using central charges with
negative levels. We show that composition of them gives the basic Nekrasov
factors appearing in supergroup partition functions. We explicitly derive the
instanton partition functions of supergroup gauge theories with A and D-type
quiver structures. Using the intertwiners, we briefly study the Gaiotto state,
$qq$-characters and the relation with quiver W-algebra. Furthermore, we show
that the negative intertwiner corresponds to the anti-refined topological
vertex recently defined by Kimura and Sugimoto. We also discuss how superquiver
theories should appear in our formalism if they exist. The existence of the AGT
correspondence of the theories we study in this paper implies that there is a
broader 2d/4d (5d/$q$-algebra) correspondence, or more generally the BPS/CFT
correspondence, where new non-unitary theories play important roles.
