# Significance of half sum of non-simple positive roots

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?

• 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. – Sam Hopkins Dec 25 '18 at 4:16
• 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 4:45
• @LSpice: Should be true in any characteristic: see here – Mikko Korhonen Dec 25 '18 at 5:33
• 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
• 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 5:42