Classification of Simple Modules

Chevalley’ theorem

Theorem 1 (Chevalley). For any $\lambda\in\mathbb{X}^{\ast}(T)_ {+}$, the induced module $N(\lambda)$ contains a unique simple submodule $L(\lambda)$. The assignment $\lambda\mapsto L(\lambda)$ induces an bijection $$\mathbb{X}^{+}\xrightarrow{\cong}\{\text{simple $G$-modules}\}/\cong.$$

Exercise 2.

  1. For any $\lambda\in\mathbb{X}^{+}$, $N(\lambda)^{U^{+}} = N(\lambda)_ {\lambda}.$
  2. If $V\subseteq N(\lambda)$ is any non-zero submodule, then $V^{U^{+}}\neq 0$.

Remark 3. $w_ {0}(\lambda)$ is the unique minimal element in $\text{wt}(L(\lambda))$ so for any $\lambda\in\mathbb{X}_ {+}$, we have that $L(\lambda)^{\ast}\cong L(-w_ {0}\lambda)$. Therefore, $L(\lambda)$ is the unique simple quotient of $M(\lambda)$.

Weyl character formula

Definition 4. Given $M\in\mathrm{Rep}(G)$ and $\lambda\in\mathbb{X}^{+}$, let $[M:L(\lambda)]$ be the multiplicity of $L(\lambda) in M$.

For any $M\in\mathrm{Rep}(G)$, one has that $$[M]=\sum_ {\lambda\in\mathbb{X}_ {+}}[M:L_ {\lambda}][L_ {\lambda}].$$ For any $0\rightarrow M’\rightarrow M\rightarrow M’‘\rightarrow 0$ exact in $\mathrm{Rep}(G)$, and for any $\mu\in\mathbb{X}$, one has $0\rightarrow M’_ {\mu}\rightarrow M_ {\mu}\rightarrow M’’_ {\mu}\rightarrow 0$ exact. Therefore, $M\mapsto \text{ch}(M)$ gives a map $$K_ {0}(\mathrm{Rep}(G))\rightarrow \mathbb{Z}[\mathbb{X}]^{W}.$$

Proposition 5. The morphism $\text{ch}:K_ {0}(\mathrm{Rep}(G))\rightarrow \mathbb{Z}[\mathbb{X}]^{W}$ is an isomorphism.

Set $\rho:=\frac{1}{2}\sum_ {\alpha\in R^{+}}\alpha\in \mathbb{Q}\otimes_ {\mathbb{Z}}\mathbb{X}$. Given $w\in W,\lambda\in\mathbb{X}$, we set $w\cdot \lambda:=w(\lambda+\rho)-\rho$.

Theorem 6 (Weyl’s character formula). For any $\lambda\in\mathbb{X}_ {+}$, one has $\text{ch}(N(\lambda))=\text{ch}(M(\lambda))=\frac{\sum_ {w\in W}(-1)^{\ell(w)}e^{w\cdot \lambda}}{\sum_ {w\in W}(-1)^{\ell(w)}e^{w\cdot 0}}$

Corollary 7. We have that $[M(\lambda)]=[N(\lambda)]$.

Corollary 8. We have $\text{dim}(N(\lambda))=\frac{\prod_ {\alpha\in R_ {+}}{\langle \lambda+\rho,\alpha^{\vee} \rangle}}{\prod_ {\alpha\in R_ {+}}{\langle \rho,\alpha^{\vee} \rangle}}$.

Central decomposition

Let $Z(G)$ be the center of $G$, then $Z(G)\subseteq T$ is the Cartier dual of $\mathbb{X}^{\ast}(T)/\mathbb{Z}R$. This group scheme may not be smooth if $p>0$.

But its representation is simple: $\mathrm{Rep}(Z(G))$ is semi-simple, and we have a bijection $$\{\text{simple objects in }\mathrm{Rep}(Z(G)) \xrightarrow{\cong} \mathbb{X}^{\ast}(T)/\mathbb{Z}R\}$$ and any simple object is 1-dimensional.

Then we have a central decomposition $$\mathrm{Rep}(G)=\oplus_ {\chi\in \mathbb{X}^{\ast}(T)/\mathbb{Z}R} \mathrm{Rep}(G)^{Z(G)=\chi},$$ and $$\mathrm{Rep}^{\infty}(G)=\oplus_ {\chi\in \mathbb{X}^{\ast}(T)/\mathbb{Z}R} \mathrm{Rep}^{\infty}(G)^{Z(G)=\chi}.$$

Linkage principle

  • The affine Weyl group $W_ {\text{aff}}:=W\ltimes \mathbb{Z}R$, and for $\lambda\in \mathbb{Z}R$, we weite $t_ {\lambda}\in W_ {\text{aff}}$ the corresponding element.
  • $p$-dot action of $W_ {\text{aff}}$ on $\mathbb{X}$: $wt_ {\lambda}\cdot_ {p}\mu:=w(\mu+\rho+p\lambda)-\rho$.
  • $W_ {\text{ext}}:=W\ltimes \mathbb{X}^{\ast}(T)$.

Theorem 9 (Humphreys, Jantzen, Anderson). For $\lambda,\mu\in \mathbb{X}_ {+}$, if $\mathrm{Ext}^{1}(L(\lambda),L(\mu))\neq 0$, then $W_ {\text{aff}}\cdot_ {p}\lambda = W_ {\text{aff}}\cdot_ {p} \mu$.

Corollary 10. The assignment $(M_ {c})_ {c\in \mathbb{X}/(W_ {\text{aff}},\cdot_ {p})}\mapsto \bigoplus_ {c}M_ {c}$ gives an equivalence of categories $$\prod_ {c\in \mathbb{X}/(W_ {\text{aff}},\cdot_ {p})}\mathrm{Rep}^{\infty}(G)_ {c}\xrightarrow{\cong}\mathrm{Rep}^{\infty}(G).$$