In the discussion of connections between logic and cylindric algebras (section 4.3) in Henkin, Monk, and Tarski's Cylindric Algebras, Part II, they introduce the notion of a "restricted" formula (p. 152), shown in the included picture.
Here, the symbol $\rho$ denotes their rank function, which assigns an ordinal $\leq \alpha$ to each relation symbol.
The crucial bit I'm not quite getting is what is going on with the displayed formula, where the variables are of the form, e.g., $v_{\kappa 0}$. Restricted formulas differ by dropping $\kappa$, but I'm not sure what $\kappa$ is doing in the first place. Is $\kappa \eta$ just supposed to be read as multiplication? Something else?
I believe that $^{\rho\xi}\alpha$ denotes the functions mapping $\rho\xi$ to $\alpha$. If that's correct, $\kappa\eta$ is function application. My belief is based on similar conventions being used by Don Monk in his Mathematical Logic and Introduction to Set Theory.