## About the intrinsic definition of the Weyl group of complex semisimple Lie algebras

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The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h}$ and we have the root space decomposition. The Weyl group now is the group generated by the reflections according to roots.

Naively this definition depends on the choices of the Cartan subalgebra $\mathfrak{h}$. Of course we can prove that for different choices the resulting Weyl groups are isomorphic.

My question is: can we define the Weyl group intrinsically such that we don't need the do check the unambiguity.

One thought is: we have the abstract Cartan subalgebra $\mathfrak{H}:=\mathfrak{b}/[\mathfrak{b},\mathfrak{b}]$ of $\mathfrak{g}$ (which is in fact not a subalgebra of $\mathfrak{g}$). Can we define the Weyl group along this way? Again is there any references for this?

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Don't you end up proving the result does not depend on the Borel you picked? – Mariano Suárez-Alvarez Aug 2 at 3:00
@ Mariano You are right. It can be proved that for two Borel subalgebra $\mathfrak{b}$ and $\mathfrak{b}'$ , the resulting quotient $\mathfrak{b}/[\mathfrak{b}, \mathfrak{b}]$ and $\mathfrak{b}'/[\mathfrak{b}', \mathfrak{b}']$ are canknically isomorphic. It's in Representation Theory and Complex Geometry Chapter 3 by Chriss/Ginzburg. Oop, still there are choices. – Zhaoting Wei Aug 3 at 6:12

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@Zhaoting: I think all the approaches suggested in the answers tend to show the impossibility of giving the desired intrinsic definition "such that we don't need to check the unambiguity". Under the surface the conjugacy theorems and other scaffolding are concealed. – Jim Humphreys Aug 3 at 14:18
@Jim: I see. Thank you very much! – Zhaoting Wei Aug 4 at 5:25

## 4 Answers

Probably the earliest intrinsic definition of Weyl group occurs in section 1.2 of the groundbreaking paper "Representations of Reductive Groups Over Finite Fields" by Deligne and Lusztig (Ann. of Math. 103, 1976, available at JSTOR). This is done elegantly in the closely related but more general setting of a reductive algebraic group `$G$` over an arbitrary algebraically closed field (though their interest is mainly in prime characteristic). Letting `$X$` denote the set of all Borel subgroups of `$G$`, the set of `$G$`-orbits on `$X \times X$` provides a natural model for a universal Weyl group of `$G$` (or its Lie algebra).

[ADDED] In the algebraic group setting, this intrinsic definition depends just on knowing what a connected reductive (or semisimple) group is and what a Borel subgroup is (maximal closed connected solvable subgroup). But obviously one can't exploit the "Weyl group" without knowing more of the structure theory: conjugacy theorems, Bruhat decomposition. (Is it a group? finite?) In the easier characteristic 0 Lie algebra theory, where `$X$` becomes the set of Borel subalgebras (whose definition requires some theory) with conjugation action by the adjoint group, this abstract notion of "Weyl group" similarly needs unpacking. But the Deligne-Lusztig definition is a good conceptual one for their purposes and sneaks in the underlying set `$X$` of the flag variety of `$G$`. Any intrinsic definition of the Weyl group needs serious background in Lie theory.

In the treatment by Chriss and Ginzburg, even when one is primarily interested in the Lie algebra picture, the group in the background tends to play an important role. Indeed, in the early work of Borel and Chevalley on semisimple algebraic groups, the Weyl group appears most naturally in the guise of the finite quotient `$W_G(T) :=N_G(T)/T$` for a fixed maximal torus `$T$`. Then one sees `$W$` as generated by reflections relative to roots, etc. As in the parallel Lie algebra setting in characteristic 0, the maximal tori (or Cartan subalgebras) are all conjugate under the adjoint group action, but this falls short of giving an intrinsic definition of the sort provided by Deligne-Lusztig.

[Weyl himself gave the group an awkward name, but was mainly concerned with its use in the context of a compact Lie group. The notion basically originates earlier in the work of Cartan, but it took a while to see the root system and Weyl group as combinatorial objects including the Coxeter presentation of the group as a reflection group (carried over by Witt to Lie algebras).]

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Of course, this elegant "intrinsic definition" rests on all of the usual conjugacy results, though it isn't clear what the OP is really seeking by asking for a definition which avoids the need to check "the unambiguity". – quasi-coherent Aug 2 at 12:34
@quasi (if I may call you that): See my added paragraph, where I emphasize that the definition itself uses no more than basic definitions. The price for that is having to figure out what it actually means in concrete terms; then you do need more theory. – Jim Humphreys Aug 2 at 17:36
@Jim, a related approach defining an abstract "Weyl group" for probably the most general category of groups that should have Weyl groups, appeared in the recent work by Bader and Furman related to Margulis' superrigidity, see for example homepages.math.uic.edu/~furman/preprints/…, and after your introduction, I see it inspired from the Deligne-Lusztig definition. – Asaf Aug 2 at 18:14

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@Jim Maybe we can look at the set of $G$ -orbits of $X \times X$ and say that "this is the Weyl group". But can we define a multiplication just on this set of $G$-orbits? If we can, then this is what I am seeking for: an intrinsic definition of Weyl group. – Zhaoting Wei Aug 2 at 23:10

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@Zhaoting: This is all worked out carefully by Deligne-Lusztig in their section 1.2. But I'd emphasize that it uses most of the deep structure theory (including conjugation theorems and Bruhat decomposition in the group version) to reach the intrinsic formulation. – Jim Humphreys Aug 3 at 14:14
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Let $g_r$ be the set of regular semi-simple elements of the Lie algebra, and $\tilde g_r$ be the set of these elements with a choice of Borel containing it. The Weyl group is the group of deck transformations of the cover $\tilde {g}_r\to {g}_r$.

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This is a great point! Maybe the remaining problem to me is that can we relate this deck transformation with the set of $G$-orbits on $X \times X$, as Jim Humphreys pointed out in his answer. – Zhaoting Wei Aug 2 at 23:13

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@Zhaoting: Like the other intrinsic descriptions, this requires a lot of the structure theory to relate it to the concrete Weyl group attached to a Cartan subalgebra (or maximal torus) of a semisimple Lie algebra (or group). Here you also need regular elements (Kostant/Steinberg; cf. Bourbaki Ch. 7-8): regular semisimple elements are dense and each lies in exactly `$|W|$` Borel subalgebras (or subgroups), corresponding to positive systems of roots or Weyl chambers. – Jim Humphreys Aug 3 at 14:07
@Ben: This is a very nice topological viewpoint, which I guess goes back to work on the topology of compact Lie groups (Adams, Bott, Samelson, ...)? Is there a good reference for the translation to semisimple Lie algebras and Borel subalgebras? – Jim Humphreys Aug 3 at 14:11

Yes: this is the approach to defining the 'abstract Weyl group' introduced in "Representation Theory and Complex Geometry" by Chriss/Ginzburg on p. 135 (2nd Edition, Birkhauser).

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I have heard that, originally, the Weyl group was designed (and worked out e.g. by Chevalley) as some Galois group which are then intrinsic.

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