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][1] 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][2]. 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? 


  [1]: https://mathoverflow.net/questions/80150/what-is-significant-about-the-half-sum-of-positive-roots
  [2]: https://arxiv.org/pdf/1803.06592.pdf