Elements of compact connected simple Lie groups with prime power order and given field of characters
Let \(G\) be a compact connected simple Lie group of rank \(l\). Every finite-dimensional complex linear representation \(\rho : G \to \text{GL}(V)\) yields a character \(\chi = \text{tr}_ V \circ \rho\). If \(x \in G\) has finite order \(N\), then \(\chi(x)\) belongs to the cyclotomic field \(\mathbb{Q}(\zeta_ n)\) for every character \(\chi\), where \(\zeta_ N\) is a primitive \(N\)-th root of unity. The authors define \(A(x)\), the subfield of \(\mathbb{Q}(\zeta_ N)\) that is generated by all character values at \(x\). They concentrate on the case where \(N\) is a power of an odd prime, whence the extension \(\mathbb{Q}(\zeta_ N) /\mathbb{Q}\) is cyclic. In this case, \(A(x)\) is fully described by the degree of the extension \(\mathbb{Q}(\zeta_ N)/ A(x)\), which is called the depth \(b(x)\) of \(x\). Theorem 1 gives a polynomial formula for the number of conjugacy classes of elements of order \(N\) and depth \(b\). These polynomials may be expressed simply by means of the exponents of the Weyl group \(W\) of \(G\). For the proof, the authors employ a bijection between the set of conjugacy classes of elements of order \(N\) and certain Weyl group orbits. (Here \(W\) acts on the \(\mathbb{Z}\)-module \(\Gamma /N\Gamma\), where the lattice \(\Gamma\) is defined as the kernel of the exponential function for some maximal torus of \(G\).) The depth \(b(x)\) is then interpreted as the length of the orbit of the corresponding \(z \in \Gamma /N\Gamma\) under the stabilizer of the cyclic submodule \(\mathbb{Z} z\). Finally, Theorem 2 gives necessary and sufficient conditions for the existence of elements of depth \(b\) and order \(N = p^ k\).
finite-dimensional complex linear representations
scientific article; zbMATH DE number 166880
scientific article