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a) we may take the direct product of the vector with some covariant vector (1-form obtained through metric tensor) and contract. The result is scalar.

b) we may take it's product with an axial vector (built with Levi-Civita symbol from antisymmetric tensor). The result is pseudoscalar.

I wonder if our vector may be acted on with some object giving result, that is neither scalar nor pseudoscalar, but having the following property:

with inversion of coordinates (in 3D) it acquires factor of e^{\imath \phi} ?

Does it imply necessarily that the metric should be complex?