Global SU(N) Symmetry and Complex Scalar Invariance

[isospin]

If a complex scalar (e.g. a complex Lagrangian) is invariant under a global SU(N) symmetry, the complex conjugate of this scalar is also invariant, right?
That is, the real and imaginary part of the complex scalar are both invariances with respect to the SU(N) independently.
I am not quite sure if it, is the statement correct?

Thanks.

 

[genneth]

A scalar, by definition, would be invariant under a global SU(N) symmetry for N>1. There is no way to act sensibly on a scalar with a matrix to get a scalar back. The invariance of the conjugate follows.