Subgroups of finite index of the split maximal torus of $GL_n(\mathbb{Z}_p)$.

Question

Let $E_{i,j}$ be the $n \times n$ elementary matrices. Let $G=GL_n(\mathbb{Z}_p)$. Let $T_G$ be the split maximal torus of $GL_n(\mathbb{Z}_p)$. Let $\Theta$ be the subgroup of $T_G$ consisting of matrices with all the diagonal entries in $\mu_{p-1}$, the $(p-1)^{th}$ roots of unity in $\mathbb{Z}_p$. Let $K$ be the group $\mathbb{Z}_p^{\times}I_n$, where $I_n$ is the identity matrix. Let $T_S$ be the split maximal torus of $SL_n(\mathbb{Z}_p)$. Let $P:=\{g \in T_S|g\equiv 1 (\mod p\mathbb{Z}_p)$. Then my questions are

1) Is the group $T_G/K\Theta P$ finite?

2) Is the group $T_G/KP$ finite?

Thanks in advance for help.

Answer 1

Since $\Theta$ is finite, your two questions are equivalent.

The answer is yes.

Suppose that $A\in T_G$ with diagonal entries $a_1,\dots,a_n$, and so $\det(A)=a_1\dots a_n$. By multiplying by an element $B$ of $K$, we can multiply the determinant by any $n$th power in $\mathbb{Z}_p^\times$. It is well-known that $\mathbb{Z}_p^\times\cong\mu_{p-1}\times\mathbb{Z}_p$, and so $(\mathbb{Z}_p^\times)^n$ has finite index in $\mathbb{Z}_p^\times$ (the exact index depending on the highest power of $p$ dividing $n$ and on the highest common factor of $n$ and $p-1$). Let $R$ be a set of coset representatives. Then for a suitable choice of $B$, $\det(AB)\in R$.

Now by multiplying by a suitable element $C$ of $P$, we can ensure that the first $n-1$ diagonal entries of $ABC$ are in $\mu_{p-1}$. So the finite set of diagonal $n\times n$ matrices whose first $n-1$ diagonal entries are in $\mu_{p-1}$ and whose determinant is in $R$ contains a set of coset representatives of $KP$ in $T_G$.