Williamson's Counterexample

Tilting modules for $G$

For $\lambda\in\mathbb{X}^ {\ast}(T)_ {+}$,

• standard objects $M(\lambda)$,
• costandard objects $N(\lambda)$,
• tilting objects $T(\lambda)$ (Koszul dual to simples), exists by general theory of highest weight categories.

Finkelberg-Mirkovic conjecture: understandings of $\mathrm{Rep}(G)$ very explicitly in terms of geometry.

Goal: understand why these modules are relevant to computing characters of simples (result of Jantzen).

Frobenius

Representations of the group scheme $G_ {1}$ over a field $\mathbb{k}$, with $\text{char}(\mathbb{k})=p>0$. For any $\mathbb{k}$-scheme $X$, the Frob twist of $X$ is the fiber product $$X^{(1)}:=\operatorname{Spec}(\mathbb{k})\times_ {\operatorname{Spec}\mathbb{k}}X,$$ where $\operatorname{Spec}(\mathbb{k})\rightarrow \operatorname{Spec}\mathbb{k}$ is induced by $x\mapsto x^{p}$.

If $X = \operatorname{Spec}A$ for some $\mathbb{k}$-algebra $A$, then $X^{(1)}=\operatorname{Spec}A$ but $A$ is a $\mathbb{k}$-algebra with $\lambda\cdot a : = \lambda^{\frac{1}{p}} a$. We have a morphism of $\mathbb{k}$-schemes $\operatorname{Fr}: X\rightarrow X^{(1)}$.

Consider $G^{(1)}\supseteq B^{(1)} \supseteq T^{(1)}$ and $\operatorname{Fr}_ {F}:G\rightarrow G^{(1)}.$ Given $V\in\mathrm{Rep}(G^{(1)})$, we can consider $\mathrm{Fr}^{\ast}_ {G} V\in \operatorname{Rep}(G)$. We also have $$\operatorname{Fr}_ {T}^{\ast} : \mathbb{X}^{\ast}(T^{(1)})\rightarrow \mathbb{X}^{\ast}.$$ The classification of simples holds for $G^{(1)}$. For $\lambda\in\mathbb{X}^{\ast}(T^{(1)})$, we write $L^{(1)}(\lambda)$ the corresponding simple $G^{(1)}$-module. Set $$\mathbb{X}_ {\text{res}}^{+}:=\{\lambda\in\mathbb{X}: \forall \alpha\in R^{s}, 0\leq \langle \lambda,\alpha^{\vee}\rangle < p \}.$$

Usually fix isomorphism of $\mathbb{k}$-algebraic groups $G\cong G^{(1)}$, identifying $B\cong B^{(1)}$, $T^{(1)}\cong T$, so that $\operatorname{Fr}_ {G}^{\ast}$ is identified with multiplication by $p$. Then the theorem becomes $$L(\lambda+p\mu)\cong L(\lambda)\otimes \mathrm{Fr}^{\ast}_ {G}(L(\mu)).$$

Representations of $G_ {1}$

Then $G_ {1}$ is a finite affine group scheme over $\mathbb{k}$, i.e. $\mathscr{O}_ {G}$ is a finite dimensional Hopf algebra over $\mathbb{k}$, which has a concrete description. Let $\mathfrak{g}$ be the lie algebra of $G$, then $X\mapsto X^{[p]}$ nonlinear maps $\mathfrak{g}\rightarrow \mathfrak{g}$. Inside $\mathrm{U}\mathfrak{g}$, elements of the form $X^{p}-X^{[p]}$ with $X\in\mathfrak{g}$ are central, which generate a subalgebra $Z_ {\mathrm{Fr}}$ canonically isomorphic to $\mathscr{O}((\mathfrak{g}^{\ast})^{(1)})$.

The restricted universal enveloping algebra $\mathrm{U}_ {0}\mathfrak{g}$ of $\mathfrak{g}$ is the quotient of $\mathrm{U}\mathfrak{g}$ by the ideal generated by elements of the form $X^{p}-X^{[p]}$ with $X\in\mathfrak{g}$. Then $\mathrm{U}_ {0}\mathfrak{g}$ is a finite dimensional algebra of dimension $p^{\text{dim}G}$.

Jantzen proved that $$ \mathscr{O}(G)_ {1}\cong (\mathrm{U}_ {0}\mathfrak{g})^{\ast}. $$

Let $\mathrm{Rep}(G_ {1})$ be the category of finite dimensional $G_ {1}$-modules, which is same as modules over $\mathrm{U}_ {0} \mathfrak{g}$.

Let -$\mathrm{soc}_ {G_ {1}}(M)$ the largest semi-simple submodule, -$\mathrm{top}_ {G_ {1}}(M)$ the largest semi-simple quotient. Each simple $N$ admits an injective hull, i.e. the unique injective $I_ {N}$ such that $\mathrm{soc}(I_ {N})\cong N$. Similarly there is a unique projective hull $P_ {N}$, such that $\mathrm{top}(P_ {N})\cong N$.

Since $G_ {1}\subset G$, there is a restriction functor $$\mathrm{Rep}(G)\rightarrow \mathrm{Rep}(G_ {1}).$$ Since $\operatorname{Fr}: G/G_ {1}\xrightarrow{\cong} G^{(1)}$, a $G$-module is of the form $\operatorname{Fr}^{\ast}_ {G}V$ for some $G^{(1)}$-module $V$, if and only if its restriction to $G_ {1}$ is trivial.

How do we classify simple $G_ {1}$-modules?

Slogan is that Weyl modules corresponds to baby Verma modules.

Let $B^{+}$ be the opposite Borel subgroup to $B$ with respect to $T$, and $\mathfrak{b}^{+}$ be its Lie algebra. Then there is an injection $$\mathrm{U}_ {0}\mathfrak{b}^{+}\hookrightarrow \mathrm{U}_ {0}\mathfrak{g}.$$ For any $\lambda\in\mathbb{X}^{\ast}$, the 1-dim $B^{+}$-module $\mathbb{k}(\lambda)$ gives 1-dimensional $\mathrm{U}_ {0}\mathfrak{b}^{+}$-module, which depends only on $\overline{\lambda}\in \mathbb{X}^{\ast}/p\mathbb{X}^{\ast}$.

So this is very different from category $\mathcal{O}$.

For $\lambda\in\mathbb{X}^{\ast}$, we weite $Q(\lambda)$ for the injective hull of $L_ {1}(\lambda)$. A general result on finite group schemes implies that $\mathscr{O}_ {G_ {1}}\cong \mathscr{O}_ {G_ {1}}^{\ast}$, and hence a $G_ {1}$-module is injective if and only if it is projective.

Therefore, $Q(\lambda)$ is also the projective cover of $L_ {1}(\lambda)$.

Representations of $G_ {1}T$.

Let $G_ {1}T$ be the subgroup scheme generated by $G_ {1}$ and $T$. The datum of a $G_ {1}T$-module structure on a $\mathbb{k}$-vector space $V$ is equivalent to $\mathrm{U}_ {0}\mathfrak{g}$-module strict with a $T$-module structure (an $\mathbb{X}^{\ast}$-grading) such that $\mathrm{U}_ {0}\mathfrak{t}$ acts on $V_ {\lambda}$ by the chacter $\mathrm{U}_ {0}\mathfrak{t}\rightarrow \mathbb{k}$ by differential of $\lambda\in\mathbb{X}^{\ast}$. In particular, each $G_ {1}T$-module has an action of $T$ so its $T$-weights make sense.

We have restriction functors $$\mathrm{Rep}(G)\rightarrow \mathrm{Rep}(G_ {1}T)\rightarrow \mathrm{Rep}(G_ {1})$$ where the second functor is forgetting $\mathbb{X}^{\ast}$-grading.

We have $\mathrm{soc}_ {G_ {1}}(M|_ {G_ {1}})\cong \mathrm{soc}_ {G_ {1}T}(M)|_ {G_ {1}}$ for any $G_ {1}T$-module $M$.

For any $\lambda\in \mathbb{X}^{\ast}$, the baby Verma module can be lifted to a $G_ {1}T$-module $$\hat{Z}(\lambda):=\mathrm{U}_ {0}\mathfrak{g}\otimes_ {\mathrm{U}_ {0}\mathfrak{b}^{+}}\mathbb{k}(\lambda)$$ where $T$ acts on $\mathbb{k}(\lambda)$ by $\lambda$. Now $\hat{Z}(\lambda)$ really depends on $\lambda$. In fact, for any $\lambda\in\mathbb{X}^{\ast}$, and $\mu\in\mathbb{X}^{\ast}$, there is a canonical isomorphism $$ \hat{Z}(\lambda+\operatorname{Fr}^{\ast}_ {T}(\mu))\cong \hat{Z}(\lambda)\otimes \mathbb{k}_ {T^{(1)}}(\mu),$$ where $G_ {1}T$ acts on the second tensor factor by $G_ {1}T\rightarrow T^{(1)}$.

For any $\lambda\in\mathbb{X}^{\ast}$, we write $\hat{Q}(\lambda)$ for the injective hull of $\hat{L}(\lambda)$ in $\mathrm{Rep}(G_ {1}T)$,

• $\hat{Q}(\lambda)_ {G_ {1}}\cong Q(\lambda)$.
• $\hat{Q}(\lambda+\operatorname{Fr}^{\ast}_ {T}\mu)\cong \hat{Q}(\lambda)\otimes \mathbb{k}_ {T^{(1)}}(\mu)$. $\hat{Q}(\lambda)$ is the projective cover of $\hat{L}(\lambda)$.