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The Weyl character formula is a central result in the finite dimensional representation theory of semisimple Lie groups, algebraic groups, Lie algebras. Related questions on MO include these here and here. But there is another, mainly notational, question which I'm still curious about:

What is the origin of symbols such as $\rho$ and $\delta$ used to denote the half-sum of positive roots in Lie theory, or equivalently the sum of fundamental dominant weights, relative to a given simple system of roots?

I haven't made an exhaustive search of the early literature, but here is what I've noticed:

1) Apparently Weyl himself (Math. Z., 1925-26) avoided using a symbol for this purpose, but instead wrote his character formula in terms of weight coordinates. He denoted what we would call the fundamental dominant weights by $\varphi_1, \dots, \varphi_n$. (Brauer's 1937 note in C.R. Acad. Sci. Paris on tensor product decompositions follows Weyl, also avoiding a specific symbol. )

Weyl lectured on Lie algebras (previously called 'infinitesimal groups") and Lie groups at IAS in 1933-35, with notes written up by Jacobson and Brauer respectively during their visits. (I don't recall what's in the mimeographed notes, but I recall seeing a copy years ago in the IAS library. Are those notes available online somewhere?)

2) A number of people used the notation g, including Freudenthal ((nag. Math., 1954-56), then Kostant (Trans. Amer. Math. Soc., 1959), Possibly the German word Gewicht for "weight'" influenced Freudenthal's choice of $g$.

3) In work influenced by the formulas of Weyl and Kostant, both Steinberg and Cartier used instead the notation $\phi$ in their short notes published in Bull. Amer. Math. Soc., 1961.

4) Jacobson wrote essentially the first textbook treatment of Lie algebras (1962), where (perhaps influenced by Weyl's lectures?) he chose the symbol $\delta$ for the half-sum of positive roots. This convention was followed in the 1969 lecture notes Topics in Lie Algebras by Samelson and the 1974 text by Varadarjan (both published in later editions by Springer).

5) As far as I know, the first use of $\rho$ occurs in Serre's 1965 Algiers lectures, written up in a short but highly influential set of lecture notes and published in typescript in 1966 for the W.A. Benjamin Lecture Notes series with the title Algebres de Lie semi-simples; see page VII-17. (Around 1987 an English translation was published by Springer.) Of course, Serre was then active in the Bourbaki group. Their Chap. IV-VI of the treatise Groupes et algebres de Lie appeared in 1968 but must have been in development for some years, since references were made to Chap. VI in the 1965 Borel-Tits IHES paper on reductive groups (with tentative numbering later changed in the published chapters). Borel was also active in Bourbaki, but he and Tits did little then with representation theory and went their own way on root system notation.

In my 1972 Springer GTM I still used $\delta$, probably because I had picked up some of the theory from Jacobson's book while at Cornell in 1962. After buying a copy of Serre's lectures in 1967, I based my supplementary lectures on structure and classification of semisimple Lie algebras at a summer school in Maine in 1968 on his approach but didn't treat representation theory. These lectures increased my interest in writing a textbook, which I got more inspired to do by the 1971 BGG paper. (In 1969 I did buy a copy of Bourbaki but was already used to writing $\delta$ and using other root system notation.)

ADDED: Notation is essential in mathematics but often problematic. For me the term "origin" combines the interrelated notions of "first use" and "rationale". To correct what I wrote above, another look at Harish-Chandra's early work shows that he actually used the symbol $\rho$ for the half-sum of positive roots as early as 1951. on page 69 of his important paper here. Like many other authors, he was not totally consistent in his choices; he also used $\rho$ for "representation" in the same paper.

Concerning one comment by Francois, I did consider asking some of the (retired) people involved in this history, but didn't want to bother them with a relatively minor question. I'm also not sure how reliable anyone's recollection (including mine) would be after half a century or more. I regret now that I never raised this question with people like Jacobson and Serre long ago when I had informal contact with them.

One other remark is that the really odd notation $\phi$ used in the two 1961 notes (based in part on Kostant's 1959 paper) in Bull. Amer. Math. Soc. by Steinberg and Cartier might possibly have been a printer's misreading of their handwritten symbols $\varrho$ or such. To complicate matters, one reviewer substitutes Kostant's symbol $g$ for $\phi$.

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Here on page 383 Weyl asserts "that not just the sum $\sum^+ \rho$ of the positive roots, but already the half of it" is an integral weight, and in the proof he goes on to call an arbitrary root $\alpha$, a positive root $\rho$. Maybe whoever later decided that he needed a symbol was influenced by this passage and just took the $\rho$? –  Torsten Schoeneberg Jan 22 at 10:24

1 Answer 1

Re 5): Cartier already uses $\rho$ in Séminaire «Sophus Lie» (1955): exposé 19, p. 1, and in Weyl's character and dimension formulas: exposé 21, pp. 7-9.

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That's quite interesting, though Cartier used $\phi$ later in his short paper in 1961. I haven't looked at the Sophus Lie seminar for a long time, but that is a suggestive source for Serre's choice. Still, it's a puzzle because $\rho$ is so often used (in several languages) to denote a representation. (And $\delta$ is still mysterious, though it avoids the usual letters $\alpha, \beta, \gamma$ for roots.) –  Jim Humphreys Jan 20 at 0:04
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Cartier did it despite the conflict that he was also using $\rho$ for his representations -- so I guess he must have had a damn good reason. You should ask him... –  Francois Ziegler Jan 20 at 1:39
    
See my edited version above which makes a long story even longer. –  Jim Humphreys Jan 20 at 14:54

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