Quasinormal Modes of Slowly Rotating Black Holes in Hernquist Halos: Redshift and Angular Momentum

We investigate axial quasinormal modes of black holes embedded in Hernquist halos, using an established analytic static geometry with an inner vacuum cavity extending to $4M_{\mathrm{BH}}$ and solving for frame dragging to first order in rotation. We compute the fundamental and first overtone branches for mass ratios $ε\equiv M_{\mathrm{BH}}/M_{\mathrm{halo}}$ of $0.1$ and $0.01$, and halo compactness $\mathcal C\equiv M_{\mathrm{halo}}/a_0$ in the range $10^{-4}\leq\mathcal C\leq0.3$. The cavity geometry gives a local Kerr estimate in which redshift rescales the vacuum frequencies, horizon angular momentum sets the rotational splitting, and uniform frame dragging adds a real frequency shift. At zero spin, the cavity redshift reproduces the complex $\ell=2$ frequencies with relative residuals below $0.04\%$ for the fundamental and $0.31\%$ for the first overtone. Its leading weak compactness behavior, $1-\mathcal C$, explains the weak mass ratio dependence in dilute halos; at larger compactness, the exact redshift accounts for most of the separation between the two mass ratio sequences. At fixed total angular momentum, halos with vanishing local angular momentum and those with vanishing coordinate angular velocity exhibit opposite compactness trends in the real spin coefficient. At $\mathcal C=0.3$, their real fundamental mode spin coefficients differ by approximately $38\%$ and $54\%$ for the two mass ratios. Fixing the horizon angular momentum reduces these differences below $1\%$ for both overtones, showing that angular momentum partition accounts for most of the configuration dependence.