We prove a new uncertainty principle for square-integrable irreducible unitary representations of connected Lie groups. The concentration of the matrix coefficients is measured in terms of weighted $L^p$ norms, with weights in the local Muckenhoupt class $A_{ \infty,{ \rm loc}}$ associated with a subRiemannian left-invariant metric and a relatively invariant measure. The result is reminiscent of the Fefferman-Phong uncertainty principle, and is new even for the Schr\"odinger representation of the reduced Heisenberg group, which corresponds to the short-time Fourier transform. As an application, we give an optimal estimate of the order of magnitude of the bottom of the spectrum and of the essential spectrum of semiclassical anti-Wick operators in $\mathbb{ R}^d$ with a nonnegative symbol $a$ in the class $A_{ \infty}$ (in particular, for polynomial symbols). Precisely, we show that the infimum $\inf _{ (x_0,\omega_0)\in{ \mathbb{ R}^{ 2d}}} -\!\!\!\!\!\int_{ B((x_0,\omega_0),\sqrt{ h})} a(x,\omega)\, dx\,d\omega$ represents (up to multiplicative constants) both a lower bound and an upper bound for the bottom of the spectrum, uniformly with respect to $h>0$. Similarly the quantity $$\liminf_{ (x_0,\omega_0)\to\infty} -\!\!\!\!\!\!\int_{ B((x_0,\omega_0),\sqrt{ h})} a(x,\omega)\, dx\,d\omega$$ represents both a lower bound and an upper bound for the bottom of the essential spectrum, uniformly with respect to $h>0$. Similar results are proved for semiclassical symbol classes.
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