Return to Answer

$\newcommand\Z{\mathbb Z}$Point 1 has already been answered by @spin, and by @nfdc23 at Centralizers of subtori in reductive groups, derived subgroups.

In light of point 1, point 2 is the same as asking whether there are a group $G$ with derived group $G' = \operatorname{SL}_2$, a maximal torus $T$ in $G$, and a root $\alpha$ of $T$ in $\operatorname{Lie}(G)$ such that $\lambda = \frac1 2\alpha^\vee$ lies in the cocharacter lattice of $T' = G' \cap T$. This cannot happen.

Indeed, suppose it did. Note that the adjoint quotient $G_\text{ad} = G/\operatorname Z(G)$ is $\operatorname{PGL}_2$. Since the composition $\operatorname{GL}_1 \xrightarrow\lambda T' \to T_\text{ad} \mathrel{:=} T/\operatorname Z(G) \xrightarrow\alpha \operatorname{GL}_1$ is the identity, we find that $\operatorname{im} \lambda$ is a subtorus of $T'$ that projects isomorphically to $T_\text{ad}$. Since $T'$ is $1$-dimensional, actually it equals $\operatorname{im} \lambda$; that is, the adjoint quotient restricts to an isomorphism $T' \to T_\text{ad}$. However, the adjoint-quotient map $G' \cong \operatorname{SL}_2 \to G_\text{ad} \cong \operatorname{PGL}_2$ has a non-trivial kernel on every maximal torus.

$\newcommand\Z{\mathbb Z}$Point 1 has already been answered by @spin, and by @nfdc23 at Centralizers of subtori in reductive groups, derived subgroups. It is also Proposition 2.1(i) of Kac and Weisfeiler - Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic $p$.

In light of point 1, point 2 is the same as asking whether there are a group $G$ with derived group $G' = \operatorname{SL}_2$, a maximal torus $T$ in $G$, and a root $\alpha$ of $T$ in $\operatorname{Lie}(G)$ such that $\lambda = \frac1 2\alpha^\vee$ lies in the cocharacter lattice of $T' = G' \cap T$. This cannot happen.

Indeed, suppose it did. Note that the adjoint quotient $G_\text{ad} = G/\operatorname Z(G)$ is $\operatorname{PGL}_2$. Since the composition $\operatorname{GL}_1 \xrightarrow\lambda T' \to T_\text{ad} \mathrel{:=} T/\operatorname Z(G) \xrightarrow\alpha \operatorname{GL}_1$ is the identity, we find that $\operatorname{im} \lambda$ is a subtorus of $T'$ that projects isomorphically to $T_\text{ad}$. Since $T'$ is $1$-dimensional, actually it equals $\operatorname{im} \lambda$; that is, the adjoint quotient restricts to an isomorphism $T' \to T_\text{ad}$. However, the adjoint-quotient map $G' \cong \operatorname{SL}_2 \to G_\text{ad} \cong \operatorname{PGL}_2$ has a non-trivial kernel on every maximal torus.

$\newcommand\Z{\mathbb Z}$Point 1 has already been answered by @spin, and by @YCor at Centralizers of subtori in reductive groups, derived subgroups.

In light of point 1, point 2 is the same as asking whether there are a group $G$ with derived group $G' = \operatorname{SL}_2$, a maximal torus $T$ in $G$, and a root $\alpha$ of $T$ in $\operatorname{Lie}(G)$ such that $\lambda = \frac1 2\alpha^\vee$ lies in the cocharacter lattice of $T' = G' \cap T$. This cannot happen.

Indeed, suppose it did. Note that the adjoint quotient $G_\text{ad} = G/\operatorname Z(G)$ is $\operatorname{PGL}_2$. Since the composition $\operatorname{GL}_1 \xrightarrow\lambda T' \to T_\text{ad} \mathrel{:=} T/\operatorname Z(G) \xrightarrow\alpha \operatorname{GL}_1$ is the identity, we find that $\operatorname{im} \lambda$ is a subtorus of $T'$ that projects isomorphically to $T_\text{ad}$. Since $T'$ is $1$-dimensional, actually it equals $\operatorname{im} \lambda$; that is, the adjoint quotient restricts to an isomorphism $T' \to T_\text{ad}$. However, the adjoint-quotient map $G' \cong \operatorname{SL}_2 \to G_\text{ad} \cong \operatorname{PGL}_2$ has a non-trivial kernel on every maximal torus.

$\newcommand\Z{\mathbb Z}$Point 1 has already been answered by @spin, and by @nfdc23 at Centralizers of subtori in reductive groups, derived subgroups.

In light of point 1, point 2 is the same as asking whether there are a group $G$ with derived group $G' = \operatorname{SL}_2$, a maximal torus $T$ in $G$, and a root $\alpha$ of $T$ in $\operatorname{Lie}(G)$ such that $\lambda = \frac1 2\alpha^\vee$ lies in the cocharacter lattice of $T' = G' \cap T$. This cannot happen.

Indeed, suppose it did. Note that the adjoint quotient $G_\text{ad} = G/\operatorname Z(G)$ is $\operatorname{PGL}_2$. Since the composition $\operatorname{GL}_1 \xrightarrow\lambda T' \to T_\text{ad} \mathrel{:=} T/\operatorname Z(G) \xrightarrow\alpha \operatorname{GL}_1$ is the identity, we find that $\operatorname{im} \lambda$ is a subtorus of $T'$ that projects isomorphically to $T_\text{ad}$. Since $T'$ is $1$-dimensional, actually it equals $\operatorname{im} \lambda$; that is, the adjoint quotient restricts to an isomorphism $T' \to T_\text{ad}$. However, the adjoint-quotient map $G' \cong \operatorname{SL}_2 \to G_\text{ad} \cong \operatorname{PGL}_2$ has a non-trivial kernel on every maximal torus.

$\newcommand\Z{\mathbb Z}$Point 1 has already been answered by @spin, and by @YCor at Centralizers of subtori in reductive groups, derived subgroups.

In light of point 1, point 2 is the same as asking whether there are a group $G$ with derived group $G' = \operatorname{SL}_2$, a maximal torus $T$ in $G$, and a root $\alpha$ of $T$ in $\operatorname{Lie}(G)$ such that $\lambda = \frac1 2\alpha^\vee$ lies in the cocharacter lattice of $T' = G' \cap T$. This cannot happen.

Indeed, suppose it did. Note that the adjoint quotient $G_\text{ad} = G/\operatorname Z(G)$ is $\operatorname{PGL}_2$. Since the composition $\operatorname{GL}_1 \xrightarrow\lambda T' \to T_\text{ad} \mathrel{:=} T/\operatorname Z(G) \xrightarrow\alpha \operatorname{GL}_1$ is an isomorphism, we find that $\operatorname{im} \lambda$ is a subtorus of $T'$ that projects isomorphically to $T_\text{ad}$. Since $T'$ is $1$-dimensional, actually it equals $\operatorname{im} \lambda$; that is, the adjoint quotient restricts to an isomorphism $T' \to T_\text{ad}$. However, the adjoint-quotient map $G' \cong \operatorname{SL}_2 \to G_\text{ad} \cong \operatorname{PGL}_2$ has a non-trivial kernel on every maximal torus.

$\newcommand\Z{\mathbb Z}$Point 1 has already been answered by @spin, and by @YCor at Centralizers of subtori in reductive groups, derived subgroups.

In light of point 1, point 2 is the same as asking whether there are a group $G$ with derived group $G' = \operatorname{SL}_2$, a maximal torus $T$ in $G$, and a root $\alpha$ of $T$ in $\operatorname{Lie}(G)$ such that $\lambda = \frac1 2\alpha^\vee$ lies in the cocharacter lattice of $T' = G' \cap T$. This cannot happen.

Indeed, suppose it did. Note that the adjoint quotient $G_\text{ad} = G/\operatorname Z(G)$ is $\operatorname{PGL}_2$. Since the composition $\operatorname{GL}_1 \xrightarrow\lambda T' \to T_\text{ad} \mathrel{:=} T/\operatorname Z(G) \xrightarrow\alpha \operatorname{GL}_1$ is the identity, we find that $\operatorname{im} \lambda$ is a subtorus of $T'$ that projects isomorphically to $T_\text{ad}$. Since $T'$ is $1$-dimensional, actually it equals $\operatorname{im} \lambda$; that is, the adjoint quotient restricts to an isomorphism $T' \to T_\text{ad}$. However, the adjoint-quotient map $G' \cong \operatorname{SL}_2 \to G_\text{ad} \cong \operatorname{PGL}_2$ has a non-trivial kernel on every maximal torus.