  In the theory of algebraic groups, parabolic subgroups form a crucial
building block in the structural studies. In the case of general linear groups
over a finite field $F_q$, given a sequence of positive integers $n_1, ...,
n_k$, where $n=n_1+...+n_k$, a parabolic subgroup of parameter $(n_1, ...,
n_k)$ in $GL_n(F_q)$ is a conjugate of the subgroup consisting of block lower
triangular matrices where the $i$th block is of size $n_i$. Our main result is
a quantum algorithm of time polynomial in $\log q$ and $n$ for solving the
hidden subgroup problem in $GL_n(F_q)$, when the hidden subgroup is promised to
be a parabolic subgroup. Our algorithm works with no prior knowledge of the
parameter of the hidden parabolic subgroup. Prior to this work, such an
efficient quantum algorithm was only known for the case $n=2$ (A. Denney, C.
Moore, and A. Russell (2010), Quantum Inf. Comput., Vol. 10, pp. 282-291), and
for minimal parabolic subgroups (Borel subgroups), for the case when $q$ is not
much smaller than $n$ (G. Ivanyos: Quantum Inf. Comput., Vol. 12, pp. 661-669).
