Non-faithful irreducible representations of simple Lie groups

Let $G_{sc}$ be as in the answer by Victor Protsak and let $\varpi_1$, $\ldots$, $\varpi_l$ be the fundamental dominant weights.

Let $\lambda$ be a dominant weight and write $\lambda = \sum_{i = 1}^l a_i \varpi_i$ for $a_i \in \mathbb{Z}_{\geq 0}$.

Then the irreducible representation of $G_{sc}$ with highest weight $\lambda$ is faithful precisely in the following cases:

• Type $A_{l}$ ($l \geq 1$): $\gcd(l+1, a_1+2a_2+\cdots+la_l) = 1$.
• Type $B_l$ ($l \geq 2$): $a_l$ is odd.
• Type $C_l$ ($l \geq 2$): $a_1 + a_3 + a_5 + \cdots$ is odd.
• Type $D_l$ ($l \geq 4$): $l$ is odd and $a_{l-1} + a_l$ is odd
• Type $G_2$: always
• Type $F_4$: always
• Type $E_6$: $a_1 - a_3 + a_5 - a_6$ is not divisible by $3$.
• Type $E_7$: $a_2 + a_5 + a_7$ is odd
• Type $E_8$: always

This can be determined by a direct computation. The kernel of any irreducible representation of $G_{sc}$ lies in $Z(G_{sc})$, which is finite. Furthermore, you can describe $Z(G_{sc})$ explicitly and compute the action of any $z \in Z(G_{sc})$ in an irreducible representation (it is of course always multiplication by some scalar). See for example Chapter 3, Lemma 28 in "Lectures on Chevalley Groups" by Steinberg.

Note above that in types $G_2$, $F_4$ and $E_8$ the center of $Z(G_{sc})$ is trivial so every irreducible representation is faithful.

Also, for type $D_{2l}$ the center is not cyclic so no irreducible representation is faithful.

By the highest weight theory, finite-dimensional irreducible representations of a finite-dimensional simple Lie algebra $\mathfrak{g}$ are parametrized by the dominant weights $\lambda$ in the weight lattice $Q$. On the other hand, the analogous representations of the adjoint form $G_{\rm ad}$ of the corresponding simple Lie group are parametrized by the dominant weights in the root lattice $P$. The difference is measured by the finite abelian group $Q/P$, which may be identifed with the dual of the center $Z$ of the simply-connected form $G_{\rm sc}$ (in fact, $Z\simeq\! P^\vee/Q^\vee$). A representation of $G_{\rm sc}$ is faithful if and only if its restriction to $Z$ is faithful if and only if $\lambda$ is not contained in any proper sublattice of $Q$.

This explained and tabulated in terms of the corresponding root systems in nearly all introductory Lie theory textbooks, but I am partial to exposition in Goto and Grosshans, where I first learned it.