In this note, for $s \in \mathbb R$ and $1 \leq p, r \leq \infty$, we introduce and study Sobolev-Fourier-Lorentz spaces $\dot{ H}^s_{ \mathcal{ L}^{ p, r}}(\mathbb{ R}^d)$. In the family spaces $\dot{ H}^s_{ \mathcal{ L}^{ p, r}}(\mathbb{ R}^d)$, the critical invariant spaces for the Navier-Stokes equations correspond to the value $s = \frac{ d}{ p} - 1$. When the initial datum belongs to the critical spaces $\dot{ H}^{ \frac{ d}{ p} - 1}_{ \mathcal{ L}^{ p,r}}(\mathbb{ R}^d)$ with $d \geq 2, 1 \leq p <\infty$, and $1 \leq r < \infty$, we establish the existence of local mild solutions to the Cauchy problem for the Navier-Stokes equations in spaces $L^\infty([0, T]; \dot{ H}^{ \frac{ d}{ p} - 1}_{ \mathcal{ L}^{ p, r}}(\mathbb{ R}^d))$ with arbitrary initial value, and existence of global mild solutions in spaces $L^\infty([0, \infty); \dot{ H}^{ \frac{ d}{ p} - 1}_{ \mathcal{ L}^{ p, r}}(\mathbb{ R}^d))$ when the norm of the initial value in the Besov spaces $\dot{ B}^{ \frac{ d}{ \tilde p} - 1, \infty}_{ \mathcal{ L} ^{ \tilde p,\infty}}(\mathbb{ R}^d)$ is small enough, where $\tilde p$ may take some suitable values.
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