Let $F$ be a $p$-adic local field and $X$ be a $n$-dimensional space over $F$. Let $0=X_0 \subset X_1 \subset \cdots X_{n-1} \subset X_n=X$ and $\{ x_1,\cdots,x_i \}$ be a basis of $X_i$.
Consider general linear group $GL_n(X)$.
For each $1\le i \le n$, let $P_i$ be the subgroup of $GL(X)$ that stabilize the flag $X_{n−i} \subset X_{n-i+1} \subset \cdots \subset X_{n-1}$ and fix $e_j$ modulo $X_{j−1}$ for $n−i+1 \le j \le n$. Note that $P_1$ is the mirabolic subgroup of $GL_n$.
I am wondering what is the modulus character of $P_i$. For example, let $Q_i=GL(X_{n-i}) \times N_i$, where $N_i$ is the unipotent radical of the Borel subgroup og $GL_i$.
I guess that the modulus character of $P_i$ restricted to $Q_i$ should be $|det_{GL_{n-i}}|^s \times 1$ for some integer $s$. If this is right, can you describe $s$ in terms of $i$?
There are explicit formulas for modulus character of $GL_n$ but it seems that there is no formula for like mirabolic groups.
I appreciate if you answer to me. No doubtly, it helps many people like me.
Thank you in advance.