  We give an new arithmetic algorithm to compute the generalized Discrete
Fourier Transform (DFT) over finite groups $G$. The new algorithm uses
$O(|G|^{\omega/2 + o(1)})$ operations to compute the generalized DFT over
finite groups of Lie type, including the linear, orthogonal, and symplectic
families and their variants, as well as all finite simple groups of Lie type.
Here $\omega$ is the exponent of matrix multiplication, so the exponent
$\omega/2$ is optimal if $\omega = 2$. Previously, "exponent one" algorithms
were known for supersolvable groups and the symmetric and alternating groups.
No exponent one algorithms were known (even under the assumption $\omega = 2$)
for families of linear groups of fixed dimension, and indeed the previous
best-known algorithm for $SL_2(F_q)$ had exponent $4/3$ despite being the focus
of significant effort. We unconditionally achieve exponent at most $1.19$ for
this group, and exponent one if $\omega = 2$. Our algorithm also yields an
improved exponent for computing the generalized DFT over general finite groups
$G$, which beats the longstanding previous best upper bound, for any $\omega$.
In particular, assuming $\omega = 2$, we achieve exponent $\sqrt{2}$, while the
previous best was $3/2$.
