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?