Let $G=GL_n(\mathbb{C})$.
Scenerio 1: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $ g.(x,y)=(gx,y). $
Scenerio 2: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $ g.(x,y)=(xg^{-1},gyg^{-1}). $
Is there a geometric difference between the two scenerios?