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Let $H,X,Y$ be the standard basis of the Lie algebra $\mathfrak g = sl(2,\mathbb C)$. Up to a factor, the most common definition of the Casimir operator of a $\mathfrak g$-module is $c = h^2+2h+4yx$. (I write small letters for the corresponding operators on the $\mathfrak g$-module) Now, in several books (e.g. Varadarajan,Introduction to harmonic Analysis on Semisimple Lie groups) I also saw the definition $c = (h+1)^2 + 4 yx$ This divergence of course leads to other scalars with which these operators act on the representations of interest. So I wonder what the connection between the two definition is and what is it about?

I just realized that it is completely trivial that if an element $c$ is in the center of an algebra, then so is the element $c+1$ (or for that matter any $c+c'$ with $c'$ another central element. So my question is probably just why some authors prefer to use the second definition above. The first definition seems to be the canonical one if one uses the general definition of a Casimir element and the Cartan Killing form as nondegenerate bilinear form as in https://en.wikipedia.org/wiki/Casimir_element (if I compute correctly one actually gets $c = 1/8 h^2 + 1/4 h + 1/2 yx$ but everyone tends to multiply by $8$ here).

Mekanik
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  • You're completely right in observing the discrepancies, and their essential meaninglessness. Non-canonical choices sometimes have in their favor that some formula turns out better later, of course, similar to starting an analysis argument with $\varepsilon/3$ rather than $\epsilon$: no genuine advantage, but perhaps a "more perfect" formulaic ending, if that's the goal. Indeed, the "optimal" description of Harish-Chandra's isomorphism suggests a shift (by the half-sum of positive roots) to make a prettier formula... Maybe that was Varadarajan's motivation. – paul garrett Oct 03 '16 at 23:26
  • I understand. Thank you! So i guess in general one could also shift the canonical definition of the Casimir by an application of the map described in Knapp's beyond-book on page 303 (which amount in the case of sl2 to h -> h-1). I guess i will proceed by studying and digesting the story of the Harish-Chandra isomorphism then... – Mekanik Oct 04 '16 at 23:06
  • I think that course would best explain what you've encountered, yes. – paul garrett Oct 04 '16 at 23:46

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