Regular and singular diagonal elements of sl_n

The roots are ${\epsilon }_i-{\epsilon }_j$ with $i\mathrm{\ne }j$, so $({\epsilon }_i-{\epsilon }_j)(H)=x_i-x_j$; hence a root vanishes at $H$ exactly when $x_i=x_j$ for the corresponding pair.

By Centralizer dimension from vanishing roots the centralizer of $H$ is $\mathfrak{h}\oplus {⨁}_{\alpha (H)=0}{\mathfrak{g}}_{\alpha }$, and each root space is one-dimensional, so $\dim {\mathfrak{s}\mathfrak{l}}_n(\mathbb{C}{)}^H=(n-1)+\mathrm{\#}\{i\mathrm{\ne }j:x_i=x_j\}$.

Therefore $H$ is regular in $\mathfrak{h}$, equivalently ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{C}{)}^H=\mathfrak{h}$, exactly when no root vanishes at $H$, that is, when all the $x_i$ are distinct; this matches the general description of Regular elements form a dense Zariski-open subset of a Cartan subalgebra, whose regular set is the complement of the hyperplanes $x_i=x_j$.

The eigenvalue condition is intrinsic to the diagonal matrix: $\mathrm{diag}(x_1,\dots ,x_n)$ has the $x_i$ as eigenvalues with multiplicity, so pairwise distinct coordinates are exactly pairwise distinct eigenvalues. The stated dimension formula and the identification of the regular case with centralizer dimension $n-1$ follow.