We start by giving a survey to the theory of ({ ext {Borel}}^{*}(kappa )) sets in the generalized Baire space ({ ext {Baire}}(kappa )=kappa ^{kappa }). In particular we look at the relation of this complexity class to other complexity classes which we denote by ({ ext {Borel}}(kappa )), ({Delta _1^1}(kappa )) and ({Sigma _1^1}(kappa )) and the connections between ({ ext {Borel}}^*(kappa )) sets and the infinitely deep language id="IEq8">(M_{kappa ^+kappa }). In the end of the paper we will prove the consistency of ({ ext {Borel}}^{*}(kappa ) e Sigma ^{1}_{1}(kappa )).