# Comparing a Chevalley basis with the canonical basis of the adjoint module?

First some background: Given a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, fix a Cartan decomposition $\mathfrak{g} = \mathfrak{h} \oplus \sum_\alpha \mathfrak{g}_\alpha$. Here $\mathfrak{h}$ is a Cartan subalgebra and the sum runs over the roots $\alpha$ of $\mathfrak{g}$ relative to $\mathfrak{h}$. One can further fix a set of simple roots, leading to positive and negative roots. Then a basis of $\mathfrak{h}$ can be chosen to consist of coroots of simple roots. All of this is determined up to conjugacy under the adjoint group and leads to various compatible choices for a basis of the adjoint module $\mathfrak{g}$, including a basis vector for each 1-dimensional root space $\mathfrak{g}_\alpha$. It was realized fairly early that a careful choice would lead to structure constants in $\mathbb{Q}$.

Chevalley's 1955 paper Sur certains groupes simples went a step further, showing in a uniform way how to produce a basis over $\mathbb{Z}$ whose structure constants are uniquely determined up to sign by the root data. (Some questions on MO deal with such a Chevalley basis, for instance here.) Notation of course differs a lot in the literature.

In a 1966 paper Tits here created an algorithm for consistent sign choices. Here he realized that the conventional commutation product $[X_\alpha X_{-\alpha}] = H_\alpha$ works better if written with reversed factors. His paper was contemporaneous with SGA 3. (More recently Demazure revisited Tits' paper here along with Remark 6.7 in his Exp. XXIII of SGA 3.)

The adjoint module for $\mathfrak{g}$ is a simple finite dimensional module. Each such module is characterized up to isomorphism by its highest weight (once a system of simple roots is fixed), which in this case is the unique highest root. Since a typical simple module has much more complicated weight space structure than the adjoint module, it was long thought that no "canonical" choice of basis (even up to signs) was possible. But in 1990 Lusztig here showed the existence of a canonical basis by an indirect procedure which specializes from a quantized enveloping algebra. At first this is done just for simply-laced types, but later in general. The idea is to show the existence of a canonical basis in the part of the quantized enveloping algebra corresponding to positive (or equally well negative) roots, then apply this to a lowest (or highest) weight vector in the module. As Lusztig indicates in his Example 3.4, direct computation of his canonical basis is possible in a few low rank cases but will get extremely difficult in general.

Soon afterward Kashiwara discovered another method based on the use of crystal bases (shown by Lusztig to be equivalent to his own method); this is exposed by Jantzen in his 1996 AMS book Lectures on Quantum Groups.

I've been unable to find in the literature an explicit discussion of what happens in the special case of the adjoint module, though the proof of Jantzen's Lemma 9.6b) is suggestive. It does seem to be understood by experts that the canonical basis produces a Chevalley basis up to signs. My basic question is:

Is there a published comparison of Lusztig's canonical basis of the adjoint module (once the choices above have been made) with a Chevalley basis?

• I really like this question. May I ask you to recall the definition of the adjoint module? I am not familiar with this terminology. Also, are you fine with a partial answer only for some Lie algebras, or is it important that the answer be as general as possible? Feb 6, 2015 at 17:27
• I use "module" and "representation" in much the same sense here, so "adjoint module" just means the adjoint representation of $\mathfrak{g}$ on itself. When $\mathfrak{g}$ is simple, the module is simple (i.e., the representation is irreducible) with highest weight equal to the highest root. I haven't seen this example discussed explicitly in the literature on canonical bases, though Jantzen's lemma and Lusztig's low rank examples may help. The canonical basis is a $\mathbb{Z}$ basis of $\mathfrak{g}$. So is a Chevalley basis (which has nice structure constants for the adjoint action). Feb 6, 2015 at 20:36