We develop a symmetry-adapted multipolar $\mathbf{k}·\mathbf{p}$ theory close to the bulk $\mathrm{\Gamma }$ point for time-reversal-symmetric, noncentrosymmetric $C_{3v}$ crystals in the strong atomic spin-orbit-coupling ($jj$-coupling) limit. Using a $j\in \{1/2,3/2,5/2\}$ multiplet basis appropriate for heavy-element $p$ and $d$ bands, we systematically construct all symmetry-allowed spin-orbit-coupling terms up to fifth order in momentum and generalize the usual spin texture to a total-angular-momentum texture. For $j>1/2$, multipolar spin-orbit coupling qualitatively reshapes Fermi surfaces and makes the topology of Bloch states band dependent. This leads to anisotropic high-$j$ textures that go beyond a single Rashba helix. We classify these textures by their total-angular-momentum vorticity $W_n$ for every energy band and identify distinct $|W_n|=1,2,5$ phases. We show that their crossovers generate enhanced and nonmonotonic current-induced spin-polarization responses, namely, the Edelstein effect, upon tuning the chemical potential. Our results provide a symmetry-based framework for analyzing and predicting multipolar spin-orbit coupling, total-angular-momentum textures, and spintronic responses in heavy-element materials without an inversion center.

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• Electronic structure
• Rashba coupling
• Spin polarization
• Spin texture
• Spin-orbit coupling
• Spin-orbit torque
• Spintronics
• Symmetries
• Band structure methods
• Boltzmann theory
• Continuous symmetries in condensed matter
• Crystal-field theory
• Discrete symmetries in condensed matter
• Group theory
• Symmetries in condensed matter
• k dot p method