Fundamental invariants for root subsystems

Question

Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d_1, \cdots, d_\ell$ canonically attached to $\Phi$.

Now suppose $\Psi$ is a closed subsystem of $\Phi$ and suppose $\Psi$ is generated by a subset of simple roots of $\Phi$. (In the language of Lie algebras, $\Psi$ defines a Levi subalgebra of the simple Lie algebra corresponding to $\Phi$.)

Question 1: Is it true that every fundamental invariant of $\Psi$ is a divisor of a fundamental invariant of $\Phi$?

Now suppose I drop the condition that $\Psi$ is generated by a subset of simple roots of $\Phi$. For example, $G_2$ contains $A_2$ as a closed subsystem, but this subsystem is not generated by simple roots of $G_2$. (In the language of Lie algebra, this is saying that $\Psi$ defines a generalized Levi subalgebra. These arise in the study of parahoric subgroups of the loop group.)

Question 2: What is the relationship between fundamental invariants of $\Psi$ and $\Phi$?

Answer 1

I will give the following positive answer:

Let H be a split reductive subgroup of the split reductive group G of the same rank. Let W_H and W_G be their Weyl groups. Then every fundamental invariant of W_H divides a fundamental invariant of W_G.

Unsatisfactionally, this proof will not give a bijection between fundamental invariants satisfying a divisibility condition.

This includes all the usual examples of Levi subgroups of reductive groups, as well as some funkier examples such as Spin(9)⊂F₄.

Let P_G(t) be the Poincaire polynomial $$P_G(t)=\sum_{w\in W_G} t^{\ell(w)}$$ and similarly for P_H(t).

Since H(𝔽_q) is a subgroup of G(𝔽_q), Lagrange's theorem, together with the formula for the order of a reductive group over a finite field, gives us the divisibility relation $$P_H(q)\mid q^NP_G(q)$$ for some positive integer N. (I've already cancelled the common factor of (q-1)^r since G and H both have rank r).

And as P_H has constant term 1, we can strengthen this to P_H(q) dividing P_G(q).

Once we have this divisibility relation for infinitely many q, this implies that P_H(t) divides P_G(t) in ℤ[t]. This may not be immediately obvious so here is the proof: Since P_H is monic, you can use the division algorithm to write P_G=qP_H+r with the degree of r less than the degree of P_H. For infinitely many prime powers q, |r(q)|<|P_H(q)|. Thus the divisibility relation implies that for such q, r(q)=0. r now has infinitely many roots, so is the zero polynomial.

Now let d be a fundamental invariant of W_H and let ζ be a primitive d-th root of 1.Then P_H(ζ)=0. The divisibility result we've just established implies that P_G(ζ)=0 as well. Since P_G(t) divides the product of t^e-1 as e runs over all fundamental invariants of W_G, it must be that d divides one of these fundamental invariants. QED.

Answer 2

[EDIT] Maybe it's useful after all this time to give a more complete and uniform answer to both of the questions asked, by referring to Theorem 3.4(i) in Springer's 1974 paper on regular elements of finite reflection groups here. In your situation, a Weyl group or other finite Coxeter group (= finite real reflection group) acts on a real euclidean space, which can be complexified to deal with eigenvalues. Springer denotes an arbitrary complex reflection group by $G$, which includes your groups.

The answer "yes" to Question 1 (or similarly the answer to Question 2) is clear from the classification of finite Coxeter groups (which include the Weyl groups along with non-crystallographic groups of type $ I_2(m), H_3, H_4$ of ranks 2, 3, 4). The numbers you call "fundamental invariants" are usually just called the degrees and were worked out long ago by Coxeter and others. (There is an exposition in Chapter 3 of my 1990 text on reflection groups, along with a table of degrees.) None of this concerns root systems or Lie algebras directly, just the theory of finite Coxeter groups. I've added to your tags accordingly.

But Springer's result provides a quick uniform proof without case-by-case verification. Here you consider a "parabolic subgroup" of $G$ (conjugate to a standard parabolic subgroup generated by some of the chosen simple reflections) or more generally a reflection subgroup corresponding in the Weyl group case to a pseudo-Levi subgroup of an algebraic or Lie group, e.g., $A_2$ inside $G_2$. Call the subgroup $H$ and note that the complexification of its natural reflection representation embeds naturally in that of $G$. So an eigenvalue for an element of $H$ (giving a nonzero eigenspace) is automatically an eigenvalue for the same element in $G$. Any such eigenvalue is a $d$th root of 1 for some $d \geq 1$.

Now Springer shows (applying some fairly elementary algebraic geometry to the hypersurfaces defined by basic invariant polynomials) that a given $d$th root $\zeta$ occurs as an eigenvalue for some element of $G$ if and only if $d$ divides one of the degrees of $G$.