Lowness and highness

For a language $A\in NP$ and an integer $k\ge 0$, define ${\mathrm{L}\mathrm{o}\mathrm{w}}_k=\{A\in NP:{\mathrm{\Sigma }}_k^{p,A}={\mathrm{\Sigma }}_k^p\},{\mathrm{H}\mathrm{i}\mathrm{g}\mathrm{h}}_k=\{A\in NP:{\mathrm{\Sigma }}_k^{p,A}={\mathrm{\Sigma }}_k^{p,SAT}\}\mathrm{.}$ The relativized levels and ${\mathrm{\Sigma }}_0^{p,A}=P^A$ use Relativized complexity class; SAT denotes satisfiability of general Boolean formulas, the first-level complete language supplied in Quantifier and oracle characterizations of ph. At $k=0$ the highness benchmark is $P^{SAT}$; no identification of that class with $NP$ is assumed. At positive levels the oracle characterization identifies ${\mathrm{\Sigma }}_k^{p,SAT}$ with ${\mathrm{\Sigma }}_{k+1}^p$. More generally, lowness for a specified oracle machine class $\mathcal{C}$ means ${\mathcal{C}}^A=\mathcal{C}$. Highness here concerns polynomial-time oracle access, not many-one completeness or computability-theoretic jumps.