# Monday, November 23

Today: last lecture. 
Note that (some) videos will be available online, see Youtube channel.

## Cohomology of Frobenius Kernels

Let $r=1$, and consider $H^0(G_1; k)$.
For $p >  h$, we know $H^{2\cdot} = k[\mathcal{N}]$ and zero otherwise.
What is known for $p<h$?

:::{.definition title="The Restricted Nillcone"}
For an arbitrary $p$, we have the following: define 
\[  
\mathcal{N}_1(\lieg) = \ts{x\in \lieg \st x^{[p]} = 0 }
,\]
where $\lieg = \Lie(G)$.
:::

Note that $p>h$, we have $\mathcal{N}_1(\lieg) = \mathcal{N}$.
Taking $\liegl_n$, this is matrices whose $p$th power is zero.
Note that the Frobenius map is still a derivation in characteristic $p$.

:::{.theorem title="Jantzen, 1986"}
\[  
\mspec(H^\cdot(G_1; k) ) \cong \mathcal{N}_1(\lieg)
.\]
:::

:::{.theorem title="Carlson-Lin-Nakano-Parshall, UGA VIGRE"}
For some $G\dash$orbit $\mathcal{O}$ in $\mathcal{N}$,
\[  
\mathcal{N}_1(\lieg) = \bar{\mathcal{O}}
,\]
i.e. it is the closure of some $G\dash$orbit.
In particular, $\mathcal{N}_1(\lieg)$ is irreducible.
:::

For $\lambda\in X(T)$, set
\[  
\Phi_\lambda \da \ts{ \alpha\in\Phi \st \inner{\lambda+\rho}{\alpha\dual} \in p\ZZ }
.\]

When $p$ is good, there exists $w\in W$ such that $w(\Phi_\lambda) = \Phi_J$ for some $J \subset\Delta$.

\todo[inline]{Something about being on or off a wall, and conjugating?}

:::{.example title="?"}
We can determine which $p$ are good for each type:

- $A_n$: $p$ is always good.
- $B_n$: $p\neq 2$
- $C_n$: $p\neq 2$
- $D_n$: $p\neq 2$
- $E_6$: $p\neq 2$
- $E_7$: $p\neq 2, 3$
- $E_8$: $p\neq 2, 3,5$
- $F_4$: $p\neq 2, 3$
- $G_2$: $p\neq 2, 3$

:::

:::{.conjecture}
Let $p$ be a good and $w(\Phi_0) = \Phi_J$ for some $J \subseteq \Delta$.
By NPV, $\mathcal{N}_1(\lieg) = G \cdot \mu_J$.
Assuming that $p\notdivides (X(T): \ZZ \Phi)$, then

- $H^{2\cdot}(G; k) = k[\mathcal{N}_1(\lieg)]$
- $H^{2\cdot + 1}(G; k) = 0$.

:::

:::{.remark}
This is a natural generalization of $p>h$.
There is some situational evidence for this to be true:

1. For $p=h-1$, we have $\mathcal{N}_1(\lieg) = \bar{\OO_{\reg}}$ and the conjecture is true.
2. For quantum groups, $H^\cdot(U_q(\lieg); \CC )$ and it is again true.

\todo[inline]{Something about BNPP}

:::

:::{.remark}
Some key points:

1. By the Gravert-Riemenschneider theorem, for $k= \bar{\FF_p}$,
\[  
R^{i>0} \ind_{P_J}^{C_r} S^\cdot(u_J^*) = 0
.\]

2. Normality of $\mathcal{N}_1(\lieg)$.


A fact about something from earlier: the ring of regular functions on springer resolution equal to those on the nilpotent cone.
:::

## Support Varieties

Very common in modern mathematics.
Define $R \da H^\wait(G_1; k)$, which is a finitely generated algebra.
Note that $R$ acts on $\ext_{G_1}^\wait(M, M)$, which is finitely generated over $R$.
So what is the support of this module?
Define $V_{G_1}(M) \da \mspec\qty{R/J_M}$, where $J_M \da \Ann_R \ext_{G_1}^\wait(M, M)$.

:::{.conjecture title="Jantzen, 1986"}
Let $p$ be good and $\lambda \in X(T)_+$ be a dominant weight.
Consider $\Phi_\lambda$ ad $w\in W$ such that $w(\Phi_\lambda) = \Phi_J$ for some $J \subseteq \Delta$. Then
\[  
V_{G_1} (H^0(\lambda)) = G\cdot \mu_J
.\]
:::

:::{.remark}
Now proved!
Jantzen proved for type $A_n$, and Nakano-Parshall-Vella proved it in general in 2002.
:::

:::{.remark}
For tilting modules, there is a conjecture for $V_G(T(\lambda))$ (Humphreys conjecture).
Type $A_n$ with $p>h+1$ was verified by W. Harelstry, and $p=2$ by Cooper.
This is still open, but is known for quantum groups and unknown for general algebraic groups.
:::