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No textbook treatment I've seen of the Chevalley restriction theorem

$$\mathbb R[\mathfrak g]^G \cong \mathbb R[\mathfrak t]^W$$

cites a specific source for it, and the proofs, where attributed, are credited to Steinberg, without further references. In particular, no one ever seems to cite anything by Chevalley himself, but perhaps I just haven't searched diligently enough. So ...

Where does this result first appear in the literature?

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    $\begingroup$ The original appearance is probably Chevalley's 1950 ICM address mathunion.org/ICM/ICM1950.2/Main/icm1950.2.0021.0024.ocr.pdf which mentions the statement without proof in the 2nd paragraph of section IV (for connected compact Lie groups, but this implies it for all connected reductive groups in char. 0). The unique 1955 paper by both Borel and Chevalley alludes to the result but offers no proof. The unique 1963 paper of Kostant in the American Journal of Math gives credit to Chevalley for it but provides no reference or proof. Probably Steinberg was the first to publish a proof. $\endgroup$
    – nfdc23
    Commented Jun 3, 2017 at 12:03
  • $\begingroup$ The Chevalley address is actually something I skimmed before asking this question, and I even remember that he says he has "proposed" to call N/H the Weyl group, but somehow I missed this. Thank you. Do you have a reference to Steinberg's proof too? $\endgroup$
    – jdc
    Commented Jun 4, 2017 at 9:19
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    $\begingroup$ A reference for Steinberg's proof in the fully Lie algebra case with $\mathfrak{g}$ a semisimple Lie algebra over $\mathbf{R}$ (avoiding reference to $G$) is Theorem 2.1.5.1(i) in Warner's Harmonic Analysis on Semisimple Lie Groups I; this settles your question for connected reductive groups in char. 0. The fully algebraic group version for connected semisimple groups over any field $k$ (using $k[G]$ & $k[T]$, reduces to alg. closed case) is 6.2+6.4 in Steinberg's Regular elements of semisimple groups in Publ. Math. IHES 25. For your hybrid version, did you try Steinberg's Collected Works? $\endgroup$
    – nfdc23
    Commented Jun 4, 2017 at 16:40
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    $\begingroup$ By the way, Steinberg's IHES paper is freely available online at numdam.org/article/PMIHES_1965__25__49_0.pdf; here he works with a connected semisimple algebraic group $G$ over any algebraically closed field, identifying the group with its rational points over that field. In any case, I'm unclear about the emphasis on the field of real numbers in the question, since Chevalley's result is essentially an algebraic statement about split semisimple Lie algebras in characteristic 0. $\endgroup$
    – Jim Humphreys
    Commented Jun 6, 2017 at 16:19
  • $\begingroup$ Thank you! I searched for it after nfdc23 suggested it. (The nearest copy of the book containing the Borel–Chevalley paper, on the other hand, seems to be a few hours' drive. Google books shows the first few pages.) I stated it this way just because it was the version I needed and I didn't know what the original version was. $\endgroup$
    – jdc
    Commented Jun 9, 2017 at 5:29

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