Exercise 16, § 4

Let $r\geqslant 1$ be an integer and G a compact connected subgroup of $\mathbf{U}(\mathbf{C}^r)$ acting irreducibly on $\mathbf{C}^r$. Let D denote the subgroup of diagonal matrices in $\mathbf{U}(\mathbf{C}^r)$.

a) Let A be a subgroup of D contained in the normalizer of G in $\mathbf{U}(\mathbf{C}^r)$. Let S be the subset of $\widehat{A}$ formed by the diagonal coefficients $a= (a_{i,j})\mapsto a_{i,i}$, where $1\leqslant i\leqslant r$. Suppose that S contains $r$ elements and that the equation

with $\chi_i\in$ S has only solutions of the form $(\chi_1, \chi_1, \chi_3, \chi_3)$ and $(\chi_1, \chi_2, \chi_1, \chi_2)$ (one then says that S is a Sidon set).

If G is semisimple, prove that $G =\mathbf{S}\mathbf{U}(\mathbf{C}^r)$. (Consider the representation of A on End($\mathbf{C}^r$), and deduce from the assumption that the complexified Lie algebra of G is stable under conjugation by D.)

b) The group G contains $\mathbf{S}\mathbf{U}(\mathbf{C}^r)$ if and only if there exists an element $g\in G$ such that the eigenvalues of $g$ are distinct and form a Sidon set in $\mathbf{C}^*$.