Let $F$ be a local field of characteristic zero and $\chi$ be a character of $GL_1(F)$.
Let $\pi$ be the normalized parabolic induction from the character $\chi |\cdot|^{\frac{n-1}{2}} \times \chi |\cdot|^{\frac{n-1}{2}-1} \times \cdots \times \chi |\cdot|^{\frac{1-n}{2}}$ of $GL_1(F)^n$ to $GL_n(F)$.
Then I don't know why $\chi (det)$ should be the (unique) irreducible quotient of $\pi$.
Could you anyone explain this to me?