# Tuesday, April 05 ## Lie Theory :::{.remark} Setup: $k = \kbar$, $\characteristic k = p > 0$, $\lieg$ a restricted Lie algebra (e.g. $\lieg = \Lie(G)$ for $G\in\Aff\Alg\Grp\slice k$). Write $A^{\ceiling{p} } = AA\cdots A$ and set $A = u(\lieg) = U(\lieg)/ J$ where $J = \gens{x\tensorpower{k}{p} - x^{\ceiling p}}$ which is an ideal generated by central elements. Note that $A$ is a finite-dimensional Hopf algebra. Proved last time: $H^0(A; k) \in \kalg^\fg$, using a spectral sequence argument. From the spectral sequence, there is a finite morphism \[ \Phi: S(\lieg^+)^{(1)} \to H^0(A; k) ,\] making $H^0(A; k)$ an integral extension of $\im \Phi$. This induces a map \[ \Phi: \mspec H^0(A; k) \injects \lieg .\] ::: :::{.theorem title="Jantzen"} \[ \mspec H^0(A; k) \cong \mcn_p \da \ts{x\in \lieg \st x^{\ceiling p}} .\] ::: :::{.example title="?"} For $\lieg = \liegl_n$, $\mcn_p \leq \mcn$ is a subvariety of the nilpotent cone. Moreover $\mcn_p$ is stable under $G = \GL_n$, and there are only finitely many orbits. There is a decomposition into finitely many irreducible orbit closures \[ \mcn_p = \Union_i \bar{Gx_i} .\] This corresponds to Jordan decompositions with blocks of size at most $p$. ::: :::{.remark} Using spectral sequences one can show that if $M, N \in \amod$ then $\Ext^0_A(M, N)$ is finitely-generated as a module over $R\da H^0(A; k)$. So one can define support varieties $V_{\lieg}(M) = \mspec R/J_M$ where $I_M = \Ann_R \Ext^0_A(M, M)$. Some facts: - $V_{\lieg}(M) \subseteq \mcn_p \subseteq \lieg$ - If $M$ is a $G\dash$module in addition to being a $\lieg\dash$module, then $V_G(M)$ is a $G\dash$stable closed subvariety of $\mcn_p$. ::: :::{.theorem title="Friedlander-Parshall (Inventiones 86)"} Given $M\in \mods{u(\lieg)}$, \[ V_{\lieg}(M) \cong \ts{x\in \lieg \st x^{[p]} = 0, M \downarrow_{U(\gens x)} \text{ is not free over } u(\gens x) \leq u(\lieg) } \union \ts{0} ,\] which is similar to the rank variety for finite groups, concretely realize the support variety. ::: :::{.remark} Here $\gens{x} = kx$ is a 1-dimensional Lie algebra, and if $x^{[p]} = 0$ then $u(\gens x) = k[x] / \gens{x^p}$ is a PID. We know how to classify modules over a PID: there are only finitely many indecomposable such modules. ::: ## Reductive algebraic groups :::{.example title="?"} For type $A_n \sim \GL_{n+1}$, $\alpha_0 = \tilde \alpha_n = \sum_{1\leq i \leq n} \alpha_i$ and $h=n+1$. For $\G_2$, $\tilde \alpha_n = 3\alpha_1 + 2\alpha_2$ and $h=6$. ::: :::{.fact} If $p\geq h$ then $\mcn_p(\lieg) = \mcn$. ::: :::{.definition title="Good and bad primes"} A prime is *bad* if it divides any coefficient of the highest weight. By type: | Type | Bad primes | |------- |------------ | | $A_n$ | None | | $B_n$ | 2 | | $C_n$ | 2 | | $D_n$ | 2 | | $E_6$ | 2,3 | | $E_7$ | 2,3 | | $E_8$ | 2,3,5 | | $F_4$ | 2,3 | | $G_2$ | 2,3 | ::: :::{.theorem title="Carlson-Lin-Nakano-Parshall (good primes), UGA VIGRE (bad primes)"} $\mcn_p = \bar{\mco}$ is an orbit closure, where $\mco$ is a $G\dash$orbit in $\mcn$. Hence $\mcn_p(\lieg)$ is an irreducible variety. ::: :::{.remark} Let $X = X(T)$ be the weight lattice and let $\lambda \in X$, then \[ \Phi_\lambda \da \ts{ \alpha\in \Phi \st \inp{\lambda + \rho}{\alpha\dual} \in p\ZZ } .\] Under the action of the affine Weyl group, this is empty when $\lambda$ is on a wall (non-regular) and otherwise contains some roots for regular weights. When $p$ is a good prime, there exists a $w\in W$ with $w(\Phi_\lambda) = \Phi_J$ for $J \subseteq \Delta$ a subsystem of simple roots. In this case, there is a **Levi decomposition** \[ \lieg = u_J \oplus \ell_J \oplus u_J^+ .\] ::: :::{.remark} On Levis: consider type $A_5 \sim \GL_6$ with simple roots $\alpha_i$. ![](figures/2022-04-05_10-34-18.png) ::: :::{.remark} Consider induced/costandard modules $H^0( \lambda) = \ind_B^G \lambda = \nabla(\lambda)$, which are nonzero only when $\lambda \in X_+$ is a dominant weight. Their characters are given by Weyl's character formula, and their duals are essentially *Weyl modules* which admit Weyl filtrations. What are their support varieties? ::: :::{.theorem title="Nakano-Parshall-Vella, 2008"} Let \( \lambda\in X_+\) and let $p$ be a good prime, and let $w\in W$ such that $w(\Phi_\lambda ) = \Phi_J$ for $J \subseteq \Delta$. Then \[ V_{\lieg} H^0( \lambda) = G\cdot u_J = \bar{\OO} \] is the closure of a "Richardson orbit". ::: :::{.remark} \envlist - This theorem was conjectured by Jantzen in 87, proved for type $A$. - For bad primes, $H^0(\lambda)$ is computed in one of seven VIGRE papers (2007). These still yield orbit closures that are irreducible, but need not be Richardson orbits. Natural progression: what about tilting modules (good filtrations with costandard sections and good + Weyl filtrations)? We're aiming for the Humphreys conjecture. ::: :::{.remark} Let $T( \lambda)$ be a tilting module for $\lambda \in X_+$. A conjecture of Humphreys: $V_{\lieg} T( \lambda)$ arises from considering 2-sided cells of the affine Weyl group, which biject with nilpotent orbits. ::: :::{.example title="?"} In type $A_2$: ![](figures/2022-04-05_10-39-31.png) There are three nilpotent orbits corresponding to Jordan blocks of type $X\alpha_1: (1,0)$ and $X_\reg: (1,1)$ in $\liegl_3$. Three cases: - $V_{\lieg} T( \lambda) = \mcn = \bar{G X_\reg}$ - $V_{\lieg} T( \lambda) = \bar{G X_{ \alpha_1}}$ - $V_{\lieg} T( \lambda) = \ts{0}$ ![](figures/2022-04-05_10-44-03.png) ::: :::{.remark} The computation of $V_G T( \lambda)$ is still open. Some recent work: - $p=2, A_n$: done by B. Cooper, - $p > n+1, A_n$ by W. Hardesty, - $p \gg 1$, Achar, Hardesty, Riche. ::: :::{.remark} What about simple $G\dash$modules? Recall $L(\lambda) = \soc_G \nabla( \lambda) \subseteq \nabla( \lambda)$ -- computing $V_G L( \lambda)$ is open. ::: :::{.theorem title="Drupieski-N-Parshall"} Let $p > h$ and $w( \Phi_ \lambda) = \Phi_J$, then \[ V_{u_q(\lieg)} L( \lambda) = G u_J ,\] i.e. the support varieties in the quantum case are known. This uses that the Lusztig character formula is know for $u_q( \lieg)$. :::