Fourier transform of a set of L fermions operators I have a set of L fermion creation and annihilation operators:
$\lbrace{\hat{C}^+_1,...,\hat{C}^+_L\rbrace}$ and $\lbrace{\hat{C}^-_1,...,\hat{C}^-_L\rbrace}$.
Every $\hat{C}^+_l,\hat{C}^-_l$ operator respecttively creates/annihilates a fermion in position $l$ in a direct lattice.
The book "Equilibrium Statistical Physics (3rd Edition)-M. Plischke & B. Bergesen" at page 190 performs that Fourier transformation on those operators:
$$\hat{a}^+_q=\frac{1}{\sqrt{L}}\Sigma_{l=1}^{L}e^{iql}\hat{C}^+_l$$
(Physically I interpreted the action of $\hat{a}^+_q$ as creating a fermion with momentum $q$ in the dual lattice) and the inverse is:
$$\hat{C}^+_l=\frac{1}{\sqrt{L}}\Sigma_{q}e^{-iql}\hat{a}^+_q$$
saying that $q\in\lbrace{-\pi,-\pi\frac{L-1}{L},-\pi\frac{L-2}{L},\dots,0, \dots, \pi\frac{L-1}{L}, \pi\rbrace}$
clearly this set contains $2L+1$ values but from the theory of DFT I would expect this set to contain only $L$ values.
Why is that set of q's so big?
 A: You are correct in assuming that there should be no more than $L$ Fourier modes. In book you specified considered two different sets of boundary conditions.
For periodic boundary conditions it is the usual problem and you get multiples of $2\pi/L$ which they wrote as $q=j\pi/L$ with $j=-L,\dots,-2,0,2,\dots,L$ (note that only even values of $j$ appear).
For antiperiodic boundary conditions ($c_{L+1}=-c_1$), you can imagine copying the system once such that the resulting super-system is periodic with period $2L$ once again. Thus the spacing of the momenta is reduced to $\pi/L$. However, recall that for an odd function only the odd plain waves (the sines) contribute. Analogously, we can argue here that the $L$-anti-periodicity implies that all the even multiples of $\pi/L$ do not contribute such that the set of momenta consists of $q=j\pi/L$ with $j=-(L-1),\dots,-1,1,\dots,L-1$.
If you are into statistical physics, you will find this reminiscent of the Matsubara frequencies where you take even multiples of $\pi/\beta$ for bosons and odd multiples for fermions because the correlations functions in imaginary time are $\beta$-periodic for bosons and $\beta$-anti-periodic for fermions.
