Highest Weights and Rational Representations of Split Reductive Groups — Examples

With the identification $X(T_2)=\mathbb{Z}\chi$ of [F1], the fundamental weight is $\omega =\chi$ and $X(T_2{)}^{+}=\{m\chi :m\ge 0\}$, so [F2] gives the simple modules $L(m)=V(m\chi )$ for $m\ge 0$. The trivial representation is one-dimensional hence simple with highest weight $0$, so $L(0)$ is the trivial representation. The standard representation $k^2$ is simple: for any nonzero $v=(a,b)$ one has $\left(\genfrac{}{}{0px}{}{a0}{b{a}^{-1}}\right)e_1=v$ when $a\mathrm{\ne }0$ and $\left(\genfrac{}{}{0px}{}{0-b^{-1}}{b0}\right)e_1=v$ when $a=0$, so the submodule generated by any nonzero vector is all of $k^2$; moreover $e_1$ is fixed by $U^{+}$ and is a $T_2$-eigenvector of weight $\chi$, so $e_1$ is primitive of weight $\chi =\omega$ and $k^2\cong L(1)$ by [F2].

Fix $m\ge 0$ and let $v\in L(m)$ be a primitive vector of weight $m\chi$, which exists and is unique up to scalar by [F3]. By [F4] $L(m)$ is generated as a $U^{-}$-module by $v$, and $U^{-}=U_{-\alpha }\cong {\mathbf{G}}_a$. By [F5] write $u_{-\alpha }(c)\cdot v={\sum }_{i\ge 0}{c}^i{v}_i$, where $v_0=v$, $v_i\in L(m{)}_{(m-2i)\chi }$, and only finitely many $v_i$ are nonzero. Let $W={\mathrm{span}}_k\{v_i:i\ge 0\}$. For every $k$-algebra $R$ and $a\in R$, the group law gives $u_{-\alpha }(a)\cdot (u_{-\alpha }(c)\cdot v)=u_{-\alpha }(a+c)\cdot v$; comparing coefficients of $c^j$ in these polynomial expressions yields $u_{-\alpha }(a)\cdot v_j={\sum }_{i\ge j}\left(\genfrac{}{}{0px}{}{i}{j}\right)a^{i-j}{v}_i$, so $W_R$ is stable under every $U^{-}(R)$-point. Thus $W$ is a $U^{-}$-submodule containing $v$, and [F4] implies $L(m)=W$. The weights $(m-2i)\chi$ are distinct, so this is a direct sum of one-dimensional spaces $k{v}_i$ for the nonzero coefficients; hence every weight space of $L(m)$ has dimension at most one.

Assume now $p=\mathrm{char}(k)>0$ and put $W=k{e}_1^p\oplus k{e}_2^p\subseteq S^p(k^2)$. For $g\in {\mathrm{S}\mathrm{L}}_2(R)$ with $g{e}_1=a{e}_1+c{e}_2$, the multinomial expansion in characteristic $p$ gives $g\cdot e_1^p=(a{e}_1+c{e}_2{)}^p=a^p{e}_1^p+c^p{e}_2^p\in W_R$, and likewise $g\cdot e_2^p=(b{e}_1+d{e}_2{)}^p=b^p{e}_1^p+d^p{e}_2^p\in W_R$; thus $W$ is a $G$-submodule of $S^p(k^2)$, of dimension two because $e_1^p,e_2^p$ are distinct basis monomials. The same computation is the identity $g\cdot e_i^p=(g\cdot e_i{)}^p$.

If $v_i\mathrm{\ne }0$, then $(m-2i)\chi$ is a weight of $L(m)$, so by the Weyl-group stability of [F3] and $s_{\alpha }(\chi )=-\chi$ of [F1], the character $(2i-m)\chi$ is again a weight; by [F3] applied to $L(m)$ it has the form $(m-2j)\chi$ with $j\ge 0$, so $2i-m=m-2j$, that is $i+j=m$ and $0\le i\le m$. Hence at most $m+1$ of the vectors $v_i$ are nonzero and ${\dim }_{k}L(m)\le m+1$; every weight other than $m\chi$ lies in $\{(m-2)\chi ,\dots ,-m\chi \}$ with multiplicity at most one.

The submodule $W$ is $T_2$-stable with weight spaces $k{e}_1^p$ of weight $p\chi$ and $k{e}_2^p$ of weight $-p\chi$. Let $S\subseteq W$ be a nonzero submodule; by [F6] $S$ contains a nonzero weight vector, hence a nonzero multiple of $e_1^p$ or of $e_2^p$. If $e_1^p\in S$, then $u_{-\alpha }(1)\cdot e_1^p=e_1^p+e_2^p\in S$ and hence $e_2^p\in S$; if $e_2^p\in S$, then $u_{\alpha }(1)\cdot e_2^p=e_2^p+e_1^p\in S$ and hence $e_1^p\in S$. In both cases $S=W$, so $W$ is simple.

Finally $e_1^p$ is fixed by $U^{+}$ and is a $T_2$-eigenvector of weight $p\chi$, so it is a primitive vector of weight $p\chi$ in the simple module $W$; the submodule it generates is contained in $W$ by step 1.3 and contains it by step 2.2, so it is all of $W$, and $p\chi$ is dominant. By the classification [F2] the simple module with highest weight $p\chi =p\omega$ is $L(p)$, whence $W\cong L(p)$. In the basis $e_1^p,e_2^p$ of $W$, step 1.3 gives the entrywise $p$-th powers of the standard representation matrices. These are the action matrices of the Frobenius twist $L(1{)}^{(p)}$. The $k$-linear map sending its two basis vectors to $e_1^p,e_2^p$ is therefore a $G$-module isomorphism onto $W$, over every field $k$ of characteristic $p$.

The basis monomials $e_1^a{e}_2^{p-a}$ of $V$ are $T_2$-eigenvectors of weights $(2a-p)\chi$ for $a=0,\dots ,p$, i.e. of the characters $-p,-p+2,\dots ,p$; each weight space of $V$ is therefore one-dimensional and spanned by a single monomial, and $T_2$ acts through the characters $p,p-2,\dots ,-p$ as claimed.

The primitive vectors of $V$ are exactly the nonzero multiples of $e_1^p$. Indeed, a primitive vector is a nonzero $T_2$-eigenvector fixed by $U^{+}$ (Unipotent algebraic groups and unipotent representations, Primitive vectors for a Borel pair), hence by step 1.1 is a nonzero multiple of some monomial $m_a=e_1^a{e}_2^{p-a}$; and $u_{\alpha }(t)\cdot m_a=e_1^a(e_2+t{e}_1{)}^{p-a}$ has, as the coefficient of $e_1^p$, the term $t^{p-a}$ (the $i=p-a$ summand, with binomial coefficient $1$). If $a<p$, the original monomial has zero coefficient of $e_1^p$, whereas the translated coefficient is the nonzero polynomial $t^{p-a}$ in $k[t]$. Thus fixedness over every base algebra excludes $a<p$. For $a=p$, $u_{\alpha }(t)\cdot e_1^p=e_1^p$ for all $t$, so $e_1^p$ is fixed.

Let $S\subseteq V$ be a nonzero submodule; if $S$ is simple, then by [F2] it contains a primitive vector $v$, which by step 2.1 is a nonzero multiple of $e_1^p$; hence $e_1^p\in S$. By [F1] the $G$-submodule generated by $e_1^p$ is $W$ (it is nonzero and contained in the simple module $W$), so $W\subseteq S$, and simplicity of $S$ gives $S=W$. Therefore $W$ is the unique simple submodule of $V$ and the socle of $V$ is $W$. Since ${\dim }_{k}V=p+1$ while ${\dim }_{k}W=2$, a semisimple $V$ would be the sum of its simple submodules, namely $W$, which is impossible; hence $V$ is not semisimple.

The representation $V=S^p(k^2)$ is finite-dimensional and rational, $G={\mathrm{S}\mathrm{L}}_2$ is a split reductive group of characteristic $p$, and $V$ is not semisimple by step 3.1, while its simple submodules and those of every rational $G$-module are still classified by the dominant weights by [F4]. This refutes the claim that every finite-dimensional rational representation of a split reductive group is semisimple.