# Characterization of restricted weights of representations of real semisimple Lie groups

I need to use the following theorem:

Let $\mathfrak{g}$ be a semisimple real Lie algebra, $\Sigma$ a set of restricted roots for $\mathfrak{g}$. Let $\rho$ be any finite-dimensional representation of $\mathfrak{g}$. Then any restricted weight $\lambda$ of $\rho$ satisfies $$\forall \alpha \in \Sigma,\quad 2\frac{\langle \lambda, \alpha \rangle}{\langle \alpha, \alpha \rangle} \in \mathbb{Z}.$$

The case of non-restricted weight and roots is of course well-known and easily found in the literature. For the general result, the closest things I found are Propositions II.4.21 to II.4.23 in S. Helgason, Geometric Analysis on Symmetric Spaces; and the last page of the proof of Theorem 8.49 in A.W. Knapp, Lie Groups Beyond an Introduction. But neither of these two passages gives quite exactly the result I am looking for.

I am not asking for a proof: I already have one (it is not very hard). But it looks like such a basic thing that it should already be written somewhere. If anyone knows any reference it would be very much appreciated!

• Here is a wider context (maybe of no interest to you). A semisimple $\mathfrak{g}$ over a field $k$ of characteristic 0 is ${\rm{Lie}}(G)$ for a Zariski-connected semisimple linear algebraic group $G$ over $k$ (Cor. 7.9 Ch. II of Borel's book "Linear Algebraic Groups"), and we can arrange that linear representations of $\mathfrak{g}$ and $G$ coincide. Maximal split $k$-tori $S$ of $G$ are $G(k)$-conjugate (Thm. 20.9(ii) in LAG), and for $k=\mathbf{R}$ their Lie algebras are the maximally non-compact Cartan subalgebras of $\mathfrak{g}$. Then Thm. 21.6 of LAG is the result (in vast generality). – nfdc23 Mar 6 '16 at 22:41
• Also, Corollary 5.8 in Borel-Tits paper on reductive groups in IHES 27 is the same integrality result for connected semisimple groups (over any field), in characteristic 0 equivalent to your question in terms of Lie algebras. Unfortunately there is a gap in that proof (the error is that the appeal to 5.7 there is insufficient if the weight $a$ is divisible). I don't know if Thm. 21.6 in Borel's LAG textbook (mentioned in my previous comment) has the same error (as I didn't learn this material from Borel's book, so I never read that part), so I don't know a simple reference. Sorry! – nfdc23 Mar 6 '16 at 22:51
• Yes, I mean a real Lie algebra, thank you. Sorry about that! – Ilia Smilga Mar 7 '16 at 2:17

As nfdc23 indicates in comments, the basic source is $\S5$ of the 1965 IHES paper on reductive groups by Borel and Tits here, in particular 5.7-5.8. Note however the essential list of corrections in $\S5$ of their 1972 IHES paper here, following various complements to the earlier paper. Here they clarify the formulation of 5.7 needed for their conclusion in 5.8 that the restricted roots (relative to a maximal $k$-split torus) form a "root system" in the Bourbaki sense. (If I recall correctly, what Borel does in the expanded second edition GTM 126 of his older Benjamin lecture notes takes this correction into account.)
This is all placed in the broader context of reductive groups (emphasizing those with a nontrivial semisimple derived group) defined but non-isotropic over an arbitrary field $k$. As a special case one gets the semisimple (or reductive) real Lie groups which occur as groups of rational points of linear algebraic groups but are non-compact. The history is of course hard to disentangle, since the parallel study of the structure of Lie groups was going on throughout that period. But the Bourbaki notion of "root system" (possibly non-reduced, allowing roots and their doubles to occur) was critical for the way Borel and Tits formulated their ideas. [Note that their paper refers to an earlier version of what ultimately became Chapter VI in the 1968 volume of Bourbaki's treatise on Lie groups and Lie algebras containing Chapters IV-VI. Suitable updates to their 1965 references are given in the 1972 paper.]