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 Rasmussen - Layer structure of irreducible Lie algebra modules. 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?

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    $\begingroup$ Disclaimer: this is pure speculation. But, in e.g. the theory of cluster algebras the “almost positive roots” play a distinguished role: these are the positive roots together with the negative simple roots (see arxiv.org/abs/1707.00340). Your rho-prime could also be thought of as the half-sum of the almost positive roots. $\endgroup$ Dec 25, 2018 at 4:16
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    $\begingroup$ Just an observation: if $B$ is a Borel subgroup with maximal torus $T$ and unipotent radical $U$, then $2\rho$ is the sum of the weights of $T$ on $\operatorname{Lie}(U)$, whereas $2\rho'$ is the sum of the weights of $T$ on $\operatorname{Lie}([U, U])$ (at least away from small-characteristic issues). $\endgroup$
    – LSpice
    Dec 25, 2018 at 4:45
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    $\begingroup$ @LSpice: Should be true in any characteristic: see here $\endgroup$ Dec 25, 2018 at 5:33
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    $\begingroup$ One comment is that there seems to be no analogue for $\rho'$ for the expression $\rho = \sum \varpi_i$, where $\varpi_i$ is a fundamental weight. (Also, note that everything here depends on working with a fixed system of simple roots.) $\endgroup$ Dec 25, 2018 at 22:37
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    $\begingroup$ Following the comment of Jim Humphreys, in type $A_l$: $\rho' = \frac{1}{2} \varpi_1 + ( \sum_{i = 2}^{l-1} \varpi_i ) + \frac{1}{2}\varpi_l$, in type $B_l$: $\rho' = \frac{1}{2} \varpi_1 + \sum_{i=2}^l \varpi_i$, in type $C_l$: $\rho' = \frac{1}{2} \varpi_1 + ( \sum_{i = 2}^{l-2} \varpi_i ) + \frac{3}{2} \varpi_{l-1} + \frac{1}{2} \varpi_l$, in type $D_l$: $\rho' = \frac{1}{2} \varpi_1 + ( \sum_{i = 2}^{l-3} \varpi_i ) + \frac{3}{2} \varpi_{l-2}+\frac{1}{2} \varpi_{l-1} + \frac{1}{2} \varpi_{l}$. Not sure what an answer to this question could be, are there any other references using $\rho'$? $\endgroup$
    – spin
    Dec 29, 2018 at 5:42


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