  Recent theoretical work has established the presence of hidden spin and
orbital textures in non-magnetic materials with inversion symmetry. Here, we
propose that these textures can be detected by nuclear magnetic resonance (NMR)
measurements carried out in the presence of an electric field. In crystals with
hidden polarizations, a uniform electric field produces a staggered magnetic
field that points to opposite directions at atomic sites related by spatial
inversion. As a result, the NMR resonance peak corresponding to inversion
partner nuclei is split into two peaks. The magnitude of the splitting is
proportional to the electric field and depends on the orientation of the
electric field with respect to the crystallographic axes and the external
magnetic field. As a case study, we present a theory of electric-field-induced
splitting of NMR peaks for $^{77}$Se, $^{125}$Te and $^{209}$Bi in Bi$_2$Se$_3$
and Bi$_2$Te$_3$. In conducting samples with current densities of $\simeq
10^6\, {\rm A/cm}^2$, the splitting for Bi can reach $100\, {\rm kHz}$, which
is comparable to or larger than the intrinsic width of the NMR lines. In order
to observe the effect experimentally, the peak splitting must also exceed the
linewidth produced by the Oersted field. In Bi$_2$Se$_3$, this requires narrow
wires of radius $\lesssim 1\, \mu{\rm m}$. We also discuss other potentially
more promising candidate materials, such as SrRuO$_3$ and BaIr$_2$Ge$_2$, whose
crystal symmetry enables strategies to suppress the linewidth produced by the
Oersted field.
