Left and right index in the exponent

Question

What is the difference between $$\chi$$ from the left and from the right here on the page $3$, item $(c)$:

Answer 1

The author writes ' $\lambda < \chi $, $\chi$ regular ' case at page 2. That means that the work it presents will be aware of the distinction when $\lambda > \chi $ or when $\lambda \leq \chi$. So in your equality, the left exponent is just there to indicate whether $\lambda$ is in the case of $ \lambda \leq \chi$ or $\lambda > \chi$.

It is important to put it at right not to be confuse with $\lambda^\chi$ that denote the set of applications of $\chi$ to $\lambda$.