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There is an argument presented in Fu and Kane's paper on inversion symmetric topological insulator which I have not yet convinced myself. Just below Eq.(3.6), the authors said that because of inversion symmetry: $[\mathcal{H},P]=0$, the Bloch state $\lvert \psi_{n,\Gamma_i} \rangle$ is a parity eigenstate, where $\Gamma_i$ is some time-reversal invariant momentum (TRIM) point. And then together with time reversal symmetry, members of the Kramers pair have the same parity, as claimed below Eq.(3.8).

My question is how do we know that $\lvert \psi_{n,\Gamma_i}\rangle$ is a parity eigenstate? To me this does not follow immediately from the commutation of the Hamiltonian and the inversion operator because the spectrum is degenerate at the TRIM points due to Kramers degeneracy. So it might be some non-trivial mixing of the Kramers states that diagonalise $\mathcal{H}$ and $P$ simultaneously. I also cannot convince myself that members of a Kramer pair have the same parity.

I would greatly appreciate it if someone can give a detailed explanation on the above points!

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The time reverse operator takes $k\rightarrow-k$, and the inversion operator also does it. So a time-reversal-invariant momentum (TRIM) point is also inversion-invariant momentum point.

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