In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the significance.

More recently, a certain variant of $\rho$-shift has appeared in the literature; see, e.g., the introduction and Equation (2.13) of this article. Namely, one defines $\rho'$ to be the half sum of non-simple positive roots. This quantity appears in mysterious ways in computations of character formulas in the abovementioned reference.

Question: What is the significance of $\rho'$? Are there other places (aside from the quoted references) where this shows up naturally?