Consider the algebraic torus $(\mathbb C^*)^n$. Let $G$ be a subgroup of $(\mathbb C^*)^n$ that is also a reductive group. Let $G^\circ$ be the connected component of $G$ containing the identity element. Then we know that $G^\circ$ is an irreducible affine algebraic group and hence an irreducible subvariety and sub group of $(\mathbb C^*)^n$. Thus $G^\circ$ is a torus.
Suppose $U$ is an affine toric variety with torus $(\mathbb C^*)^n$ then $(\mathbb C^*)^n$ acts on $U$ algebraically and is dense open subset of $U$. Since $G^\circ$ is a subgroup we have an action of $G^\circ$ on $U$. Let $A=G^\circ\cdot p$ be a $G^\circ$ - orbit in $U$. Then -
Is $\bar A$ an affine toric variety? Where $\bar A$ the closure of $A$ in $U$.
I need to get a torus as a dense open subset of $\bar A$. The obvious candidate should be $A$ but I not sure how it is open in $\bar A$ or a torus. I think we can identify $A$ with $G^\circ/G^\circ_p$ however I am not sure if this a torus. Any help would be appreciated.
Thank you.