The rationality of elements of prime order in compact connected simple Lie groups (Q1180661)

Let \(G\) be a compact connected simple Lie group of rank \(\ell\). The authors study conjugacy classes in \(G\) of elements of prime order whose character values generate a given extension field of \(\mathbb{Q}\). Let \(x\in G\) of prime order \(p\) and let \(A(x)\) be the field over \(\mathbb{Q}\) generated by the character values \(\text{tr}_ V\rho(x)\) as \((\rho,V)\) varies over all finite dimensional complex representations of \(G\). The field degree of \(\mathbb{Q}(\xi)/A(x)\), \(\xi=e^{i2\pi/p}\), is called the depth of \(x\). The authors consider \(A\)-polynomials \(A_ n(t)={1\over| W|}\sum^ \ell_{k=0}s_ k(n)(t^ k-1)\), where \(s_ k(n)\) is the number of elements of the Weyl group \(W\) that admit a primitive \(n\)th root of unity as an eigenvalue of multiplicity \(k\). Then they construct \(B\)-polynomials \(B_{d,D}(t)=\sum_{d| n| D}\mu({n\over d})A_ n(t)\), where \(\mu\) is the Möbius function. They prove that these \(B\)- polynomials are the counting polynomials for their conjugacy classes. Namely, let \(e(W)\) denote the exponent of \(W\) and let \(p\) be a prime number not dividing \(| W|\) such that the greatest common divisor of \(p-1\) and \(e(W)\) is \(D\). Then the number of conjugacy classes of elements of \(G\) of order \(p\) and depth \(d\) is obtained by \(B_{d,D}(p)\). They apply this result to get a simple criterion for the existence of such a conjugacy class and also some other consequences [cf. \textit{C. L. Morgan}, Can. J. Math. 31, 252-254 (1987; Zbl 0374.20058)].