Giant Gravitons, Harish-Chandra integrals, and BPS states in symplectic and orthogonal $\mathcal{N}$= 4 SYM

We find generating functions for half BPS correlators in $\mathcal{N}=4$ SYM theories with gauge groups $Sp(2N)$, $SO(2N+1)$, and $SO(2N)$ by computing the norms of a class of BPS coherent states. These coherent states are built from operators involving Harish-Chandra integrals. Such operators have an interpretation as localized giant gravitons in the bulk of anti-de-Sitter space. This extends the analysis of \cite{Berenstein:2022srd} to $Sp(2N)$, $SO(2N+1)$, and $SO(2N)$ gauge theories. We show that we may use ordinary Schur functions as a basis for the sector of states with no cross-caps in these theories. This is consistent with the construction of these theories as orientifold projections of an $SU(2N)$ theory. We make note of some relations between the symmetric functions that appear in the expansion of these coherent states and symplectic Schur functions. We also comment on some connections to Schubert calculus and Gromov-Witten invariants, which suggest that the Harish-Chandra integral may be extended to such problems.