Stability of rapidly-rotating charged black holes in AdS₅ times S⁵

We study the stability of charged rotating black holes in a consistent truncation of Type $IIB$ Supergravity on $AdS_5 \times S^5$ that degenerate to extremal black holes with zero entropy. These black holes have scaling properties between charge and angular momentum similar to those of Fermi surface-like operators in a subsector of ${\cal N}=4$ SYM. By solving the equation of motion for a massless scalar field in this background, using matched asymptotic expansion followed by a numerical solution scheme, we are able to compute its Quasi-Normal modes, and analyze it's regime of (in)stability. We find that the black hole is unstable when its angular velocity with respect to the horizon exceeds 1 (in units of $1/l_{AdS}$). A study of the relevant thermodynamic Hessian reveals a local thermodynamic instability which occurs at the same region of parameter space. We comment on the endpoints of this instability.