When we first discussed Weyl chambers, we defined the fundamental Weyl chamber associated to a base as the collection of all the vectors satisfying for all simple roots . Today, I want to discuss the closure of this set — allowing — and show that it’s a fundamental domain for the action of the Weyl group .
To be more explicit, saying that the fundamental Weyl chamber is a fundamental domain means that each vector is in the orbit of exactly one vector in . That is, there is a unique so that for some .
First, to existence. Given a vector , we consider its orbit — the collection of all the as runs over all elements of . We have to find a vector in this orbit which lies in the fundamental Weyl chamber . To do this, we’ll temporarily extend our partial order to all of by saying that if is a nonnegative -linear combination of simple roots. Relative to this order, pick a maximal vector ; that is, one so that for any we never have . There may well be more than one such maximal vector, given what we’ve said so far, but there will always be at least one.
I say that this is actually in the fundamental Weyl chamber. Indeed, if it weren’t then there would be some simple root so that . But then we could look at the vector . We calculate
which is a positive -linear combination of simple roots. Thus , which is impossible by assumption. In fact, this gives us a method for constructing a maximal vector in the orbit. Just start with and form its inner product with all the simple roots. If we find one for which the inner product is negative, reflect the vector through the plane perpendicular to that simple root. Eventually, you’ll end up with a vector in the fundamental Weyl chamber!
Now for uniqueness: if there are two vectors and in the orbit that lie within the fundamental Weyl chamber, then we must have for some . What I’ll show is that if we have for two vectors in the fundamental Weyl chamber, then must be the product of simple reflections which leave fixed, and thus .
We’ll prove this by induction on the length of the Weyl group element . If , then is the identity and the statement is obvious. If then (by the result we proved last time) must send some positive root to a negative root. In particular, cannot send all simple roots to positive roots. So let’s say that is a simple root for which . Then we observe
since and are both in the fundamental Weyl domain. Thus it is forced that , that , and then that . But sends fewer positive roots to negative ones than does, so and we can invoke the inductive hypothesis to finish the job.
The upshot of all this is that we know what the space of orbits of looks like! It has one point for each vector . If is in the interior of this fundamental domain, then the orbit looks just like a copy of . On the other hand, if lies on one of the boundary hyperplanes the orbit looks like “half” of the Weyl group. That is, if then , so both of the corresponding group elements “collapse” into one point in this orbit. As lies on more and more of the boundary hyperplanes, more and more of the orbit “folds up”, until finally at we have an orbit consisting of exactly one point.