Find all conjugations on \(C^1\) and \(C^2\).
Solution
1. All conjugations on \(\mathbb C^1\) are\[ \boxed{J_u(z)=u\overline z,\qquad |u|=1.} \]
Let \(J\) be a conjugation of \(\mathbb C\). Put \(u=J(1)\). Conjugate homogeneity gives \[ J(z)=J(z\cdot1)=\overline z\,u. \] The condition \(J^2=I\) gives \[ 1=J^2(1)=J(u)=\overline u\,u=|u|^2, \] so \(|u|=1\).
Conversely, if \(|u|=1\), then \(J_u(z)=u\overline z\) is additive and conjugate linear, and \[ J_u^2(z) =u\overline{u\overline z} =u\overline u\,z =z. \] Thus every such \(J_u\) is a conjugation, and the list is complete.
2. All conjugations on \(\mathbb C^2\) have the form\[ \boxed{J_P(z)=P\,\overline{P^{-1}z},\qquad P\in GL_2(\mathbb C).} \]
Fix \(P\in GL_2(\mathbb C)\). The map \(J_P\) is additive and conjugate linear, and \[ J_P^2(z) =P\,\overline{P^{-1}P\,\overline{P^{-1}z}} =P\,\overline{\overline{P^{-1}z}} =z. \] So \(J_P\) is a conjugation.
Conversely, let \(J\) be any conjugation. By Exercise 13, its fixed-point set \(W\) is a real form of \(\mathbb C^2\): every vector is uniquely \(\beta+i\gamma\) with \(\beta,\gamma\in W\) [@HK71]. Thus \(\dim_{\mathbb R}W=2\). Choose a real basis \(w_1,w_2\) of \(W\) and let \(P\) be the complex matrix whose columns are \(w_1,w_2\). Exercise 16, or the same uniqueness argument, shows that \(w_1,w_2\) are a complex basis, so \(P\in GL_2(\mathbb C)\).
If \(z=P\xi\), with \(\xi\in\mathbb C^2\), then writing the coordinates of \(\xi\) into real and imaginary parts shows \[ J(z)=P\overline\xi =P\,\overline{P^{-1}z} =J_P(z). \] Hence every conjugation has the displayed form.
3. The parameterization by \(P\) is unique up to a real change of basis.
The fixed-point set of \(J_P\) is \[ P\mathbb R^2. \] Therefore \(J_P=J_Q\) if and only if \[ P\mathbb R^2=Q\mathbb R^2. \] This holds exactly when \[ Q=PR \] for some \(R\in GL_2(\mathbb R)\): such an \(R\) merely changes the real basis of the same fixed real form. Thus the family above lists all conjugations without ambiguity once one identifies \(P\) modulo right multiplication by \(GL_2(\mathbb R)\).