I refer to the following paper: Reflection Positivity and Phase Transitions in Lattice Spin Models
Denote $\mathbb{T}_L$ as the discrete torus.
My problem is the following, in page $19$ expression $(3.18)$, the author defines the function $G_L(x,y)$ which is a certain type o Discrete Fourier Transform, taking two values $x,y \in \mathbb{Z}^d$ and gives a complex value.
Later, in page $42$, expression $(5.26)$, the author computes $G^{-1}_L(\eta)$, and $G_L(\eta)$, where $\eta \in (\mathbb{R}^\nu)^{\mathbb{T}_L}$.
I do not understand what value is referred to by those expressions ($G_L(\eta)$ and $G^{-1}_L(\eta))$. I suppose is some kind of common notation that I am not familiarized with.
Let $\mathbb{T}_L = \mathbb{Z}^d/L\mathbb{Z}^d$ denote the torus of side $L$. Although $G_L$ is defined as a matrix indexed by $\mathbb{T}_L$ (i.e. a map $G_L : \mathbb{T}_L \times \mathbb{T}_L \to \mathbb{R}$), we can view it (like any finite matrix) as a linear map $G_L : (\mathbb{R}^\nu)^{\mathbb{T}_L} \to (\mathbb{R}^\nu)^{\mathbb{T}_L}$. Precisely, for $\eta \in (\mathbb{R}^\nu)^{\mathbb{T}_L}$, the entries $(G_L \eta)^i_x$ of $G_L \eta \in (\mathbb{R}^\nu)^{\mathbb{T}_L}$ are given by (componentwise) matrix multiplication: $$(G_L \eta)^i_x = \sum_{y\in\mathbb{T}_L} G_L(x, y) \eta^i_y, \qquad x \in \mathbb{T}_L, \quad i = 1, \ldots, \nu.$$ On the right-hand side, $G_L$ is being used as in (3.18). On the left-hand side, it is being used as in (5.26), where $G_L^{-1}$ is used to denote the inverse map to $G_L$ (when well-defined).