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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?

LSpice
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Dr. Evil
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    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 https://arxiv.org/abs/1707.00340). Your rho-prime could also be thought of as the half-sum of the almost positive roots. – Sam Hopkins Dec 25 '18 at 04:16
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    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). – LSpice Dec 25 '18 at 04:45
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    @LSpice: Should be true in any characteristic: see here – Mikko Korhonen Dec 25 '18 at 05:33
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    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.) – Jim Humphreys Dec 25 '18 at 22:37
  • Note that ρ′ can be compujted case-by-case for simple Lie types A - G by using item (VII) in each type. These data are found at the end of Chapter VI in Bourbaki's treatise Groupes et algebres de Lie (1968),English translation published by Springer. – Jim Humphreys Dec 25 '18 at 22:44
  • @MikkoKorhonen, thanks. That link seems to be working over $\overline{\mathbb F_p}$; does it work for any field? – LSpice Dec 25 '18 at 23:55
  • @LSpice: The answer I gave there works over any algebraically closed field. For arbitrary fields I guess $[U,U] = \prod_{\alpha \in \Phi^+ - \Delta} U_\alpha$ should still be true if the characteristic is not too small. But it is not true for example for $\operatorname{Sp}_4(\mathbb{F}_2)$. – Mikko Korhonen Dec 27 '18 at 04:50
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    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 $\rho'$? – spin Dec 29 '18 at 05:42

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