# The inclusion of Siegel sets of the general linear groups

In some paper, it is written that none of Siegel set of $$GL_{n}$$ is not contained in any Siegel set of $$GL_{n+1}$$. I don't understand this because I think it should hold.

To explain my question more precisely, let me first introduce some notations.

Let $$G'=GL_n \subset GL_{n+1}$$ be general linear groups defined over $$\mathbb{Q}$$. Let $$P_0=U_0M_0$$ and $$P_0'=U_0'M_0'$$ be the standard parabolic subgroups of $$G$$ and $$G'$$, respectively. Let $$T_0' \subset T_0$$ be the maximal split torus of $$G'$$ and $$G$$ and $$A_0$$ and $$A_0'$$ be the identity component of $$T_0(\mathbb{R})$$ and $$T_0'(\mathbb{R})$$.

Let $$K \subset K'$$ be the standard maximal compact subgroup of $$G$$ and $$G'$$. Then by the Iwasawa decomposition, $$G=P_0K$$ and $$G'=P_0'K'$$.

Let $$\mathfrak{a}_0^*$$ be the $$\mathbb{R}$$-vector space spanned by $$\mathbb{Q}$$-rational characters of $$M_0$$ and $$\mathfrak{a}_0$$ its dual space. Let $$\Delta_0$$ be the set of simple roots of $$\mathfrak{a}_0$$.

Let $$H:G \mathbb(A_\mathbb{Q}) \to \mathfrak{a}_0$$ be the Harish-Chandra function.

For sufficiently negative $$t_0 \in \mathfrak{a}_0$$, let $$A_0(t_0)=\{a \in A_0 \mid \langle \alpha,H(a)-t_0\rangle > 0 \text{ for } \alpha \in \Delta_0\}$$.

Then a Siegel set of $$G'$$ is a set $$w'A_0'(t_0')K'$$, where $$w'$$ is a compact subset in $$U_0'\mathbb(A_\mathbb{Q})(M_0')^1\mathbb(A_\mathbb{Q})$$ and $$t_0' \in \mathfrak{a}_0'$$. Then for a Siegel set $$w'A_0'(t_0')K' \subset G$$, can we take a compact subset $$w \subset U_0\mathbb(A_\mathbb{Q})(M_0)^1\mathbb(A_\mathbb{Q})$$ and $$t \in \mathfrak{a}_0$$ such that $$w'A_0'(t_0')K' \in wA_0(t_0)K$$?

I think this is always possible because $$P_0' \subset P_0$$ and $$K' \subset K$$. But some paper says that it is not possible. Is there something I am confusing or missing?