Local theta correspondence between supercuspidal representations, Ann. Sci. Éc. Norm. Supér, 2018

Corollary 5.11, \Gamma+\mathfrak g_{x,r^+} should be \Gamma+\mathfrak g_{x,-r^+}. The proof was correct.

Invariants and K-spectrums of local theta lifts

The condition (\dagger) (on page 180) should also include the case when (G,G') = (\mathrm{Sp}_{2n}(\mathbb C), \mathrm{O}_{4n}(\mathbb C)) and \rho is the trivial representation. In fact, in this case \Theta(\rho) has two irreducible constituents (See “Lee Soo Teck, Zhu Chenbo, Degenerate principal series and local theta correspondence III”).