Prologue

0.1 References

• Carter’s “Finite Groups of Lie Type”[1]

• Humphreys’ “Linear Algebraic Groups”[2]

• Course Videos: https://www.youtube.com/playlist?list=PLYKpGBt1pueCLzs_b-VYKe_bX9My-3nUt

0.2 Notation

0.3 Outline

Guide for Math 8190 Videos:

\begin{align*}9-14-20\end{align*} A survey on the representation theory of finite-dimensional algebras

\begin{align*}9-25-20\end{align*} Frobenius kernels, simple G_r-modules, Steinberg’s Tensor Product Theorem, Connections with Lusztig’s Conjecture

\begin{align*}9-28-20\end{align*} Kempf’s Vanishing Theorem, Good and Weyl filtrations, Cohomology

\begin{align*}10-2-20\end{align*} Good and Weyl Filtrations

\begin{align*}10-5-20\end{align*} Highest Weight Categories

\begin{align*}10-7-20\end{align*} Examples: Schur algebras, tilting modules, Weyl modules, Bott-Borel-Weil Theorem

\begin{align*}10-9-20\end{align*} Bott-Borel-Weil Theory

\begin{align*}10-12-20\end{align*} Bott-Borel-Weil Theorem with applications

\begin{align*}10-14-20\end{align*} Basic Properties of Characters, Weyl’s Character Formula

\begin{align*}10-16-20\end{align*} Weyl’s Character Formula

\begin{align*}10-19-20\end{align*} Linkage, Strong Linkage, Examples

\begin{align*}10-21-20\end{align*} Strong Linkage Principle, Translation Functors (intro)

\begin{align*}10-23-20\end{align*} Translation Functors

\begin{align*}10-26-20\end{align*} Translation Functors and Characters

\begin{align*}10-28-20\end{align*} Properties of Translation Functors

\begin{align*}11-2-20\end{align*} Translation, Wall Crossing, Lusztig’s Conjecture

\begin{align*}11-4-20\end{align*} Representations of G_rT and G_rB

\begin{align*}11-6-20\end{align*} Representations of G_rT and G_rB, Good (p,r)-filtrations, Donkin’s (p,r)-Filtration Conjecture

\begin{align*}11-9-20\end{align*} Strong Linkage for G_rT-modules, Extensions for G_rT-modules, Steinberg Modules

\begin{align*}11-11-20\end{align*} Filtrations and Reciprocity for G_rT and G_r Modules

\begin{align*}11-13-20\end{align*} G_rT-modules and reciprocity, Injective G_rT-modules, Humphreys-Verma Conjecture, Donkin’s Tilting Module Conjecture

\begin{align*}11-16-20\end{align*} G-structures for injective G_r-modules, Donkin’s Tilting Module Conjecture

\begin{align*}11-18-20\end{align*} Injective G_T-modules and Injective G-modules

\begin{align*}11-20-20\end{align*} Cohomology of Frobenius kernels, finite generation of cohomology

\begin{align*}11-23-20\end{align*} Cohomology of Frobenius kernels, restricted nullcone, Jantzen’s Conjecture on support varieties, conjectures for low primes

1 Friday, August 21

1.1 Intro and Definitions

Here \({\mathbb{A}}^n\mathrel{\vcenter{:}}= k^n\) with the Zariski topology, so the closed sets are varieties.

A variety \(V\) is irreducible iff \(V\) can not be written as \(V = \cup_{i=1}^n V_i\) with each \(V_i \subseteq V\) a proper subvariety.

1.2 Jordan-Chevalley Decomposition

Define \(R_u(G)\) to be the subgroup of unipotent elements in \(R(G)\).

2 Monday, August 24

2.1 Review and General Setup

• \(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) is algebraically closed
• \(G\) is a reductive algebraic group
• \(T\subseteq G\) is a maximal split torus

Split: \(T\cong \bigoplus {\mathbb{G}}_m\).

We’ll associate to this a root system, not necessarily irreducible, yielding a correspondence \begin{align*} (G, T) \iff (\Phi, W) \end{align*} with \(W\) a Weyl group.

This will be accomplished by looking at \({\mathfrak{g}}= \mathrm{Lie}(G)\). If \(G\) is simple, then \({\mathfrak{g}}\) is “simple,” and \(\Phi\) irreducible will correspond to a Dynkin diagram.

There is this a 1-to-1 correspondence \begin{align*} G \text{ simple}/\sim \iff A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2 \end{align*} where \(\sim\) denotes isogeny.

Taking the Zariski tangent space at the identity “linearizes” an algebraic group, yielding a Lie algebra.

We have the coordinate ring \(k[G] = k[x_1, \cdots, x_n] / \mathcal{I}(G)\) where \(\mathcal{I}(G)\) is the zero set. This is equal to \(\left\{{f:G\to k}\right\}\),

2.2 The Associated Lie Algebra

Let \(G\) be reductive and \(T\) be a split torus. Then \(T\) acts on \({\mathfrak{g}}\) via an adjoint action. (For \(\operatorname{GL}_n, {\operatorname{SL}}_n\), this is conjugation.)

There is a decomposition into eigenspaces for the action of \(T\), \begin{align*} {\mathfrak{g}}= \qty{\bigoplus_{\alpha\in \Phi} g_\alpha} \oplus t \end{align*} where \(t = \mathrm{Lie}(T)\) and \(g_\alpha \mathrel{\vcenter{:}}=\left\{{x\in {\mathfrak{g}}{~\mathrel{\Big|}~}t.x = \alpha(t) x\,\, \forall t\in T}\right\}\) with \(\alpha: T\to K^{\times}\) a rational function (a root).

In general, take \(\alpha\in\hom_{\text{AlgGrp}}(T, {\mathbb{G}}_m)\).

Let \(G = \operatorname{GL}(n, k)\) and \begin{align*} T = \left\{{ \begin{bmatrix} a_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & a_n \end{bmatrix} {~\mathrel{\Big|}~}a_j\in k^{\times} }\right\} .\end{align*}

Then check the following action:

which indeed acts by a rational function.

Then \begin{align*} g_\alpha = {\operatorname{span}}\left\{{ \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} }\right\} = g_{(1, -1, 0)} .\end{align*}

For \({\mathfrak{g}}= {\mathfrak{gl}}(3, k)\), we have \begin{align*} {\mathfrak{g}}= t &\oplus g_{(1, -1, 0)} \oplus g_{(-1, 1, 0)} \\ &\oplus g_{(0, 1, -1)} \oplus g_{(0, -1, 1)} \\ &\oplus g_{(1, 0, -1)} \oplus g_{(-1, 0, 1)} .\end{align*}

2.3 Representations

Let \(\rho: G\to \operatorname{GL}(V)\) be a group homomorphisms, then equivalently \(V\) is a (rational) \(G{\hbox{-}}\)module.

For \(T\subseteq G\), \(T\curvearrowright G\) semisimply, so we can simultaneously diagonalize these operators to obtain a weight space decomposition \(V = \bigoplus_{\lambda \in X(T)} V_\lambda\), where \begin{align*} V_\lambda &\mathrel{\vcenter{:}}=\left\{{v\in V{~\mathrel{\Big|}~}t.v = \lambda(t)v\,\, \forall t\in T}\right\} \\\ X(T) &\mathrel{\vcenter{:}}=\hom(T, {\mathbb{G}}_m) .\end{align*}

2.4 Classification

Let \(G\) be a simple algebraic group (ano closed, connected, normal subgroups other than \(\left\{{e}\right\}, G\)) that is nonabelian that is nonabelian.

Let \(G = {\operatorname{SL}}(3, k)\). Then \begin{align*} T = \left\{{ t = \begin{bmatrix} a_1 & 0 & 0 \\ 0 & a_1 a_2^{-1} & 0\\ 0 & 0 & a_2^{-1} \end{bmatrix} {~\mathrel{\Big|}~} a_1, a_2\in k^{\times} }\right\} \end{align*} and \begin{align*} t. \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} = a_1^2 a_2^{-1} \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} .\end{align*} and \(\alpha_1 = (2, -1)\).

Then \begin{align*} {\mathfrak{g}}= {\mathfrak{g}}_{(2, -1)} \oplus {\mathfrak{g}}_{(-2, 1)} \oplus {\mathfrak{g}}_{(-1, 2)} \oplus {\mathfrak{g}}_{(1, -2)} \oplus {\mathfrak{g}}_{(1, 1)} \oplus {\mathfrak{g}}_{(-1, -1)} .\end{align*}

Then \(\alpha_2 = (-1, 2)\) and \(\alpha_1 + \alpha_2 = ( 1, 1)\).

This gives the root space decomposition for \({\mathfrak{sl}}_3\):

Then the Weyl group will be generated by reflections through these hyperplanes.

3 Wednesday, August 26

3.1 Review

• \(G\) a reductive algebraic group over \(k\)
• \(T = \prod_{i=1}^n {\mathbb{G}}_m\) a maximal split torus
• \({\mathfrak{g}}= \mathrm{Lie}(G)\)
• There’s an induced root space decomposition \({\mathfrak{g}}= t\oplus \bigoplus_{\alpha\in \Phi}{\mathfrak{g}}_\alpha\)
• When \(G\) is simple, \(\Phi\) is an irreducible root system
  • There is a classification of these by Dynkin diagrams

• We have representations \(\rho: G\to \operatorname{GL}(V)\), i.e. \(V\) is a \(G{\hbox{-}}\)module

• For \(T\subseteq G\), we have a weight space decomposition: \(V = \bigoplus_{\lambda \in X(T)} V_\lambda\) where \(X(T) = \hom(T, {\mathbb{G}}_m)\).

  Note that \(X(T) \cong {\mathbb{Z}}^n\), the number of copies of \({\mathbb{G}}_m\) in \(T\).

3.2 Root Systems and Weights

Let \(\Phi = A_2\), then we have the following root system:

In general, we’ll have \(\Delta = \left\{{\alpha_1, \cdots, \alpha_n}\right\}\) a basis of simple roots.

For any \(\alpha\in \Phi\), let \(s_\alpha\) be the reflection across \(H_\alpha\), the hyperplane orthogonal to \(\alpha\). Then define the Weyl group \(W = \left\{{s_\alpha {~\mathrel{\Big|}~}\alpha\in \Phi}\right\}\).

Here the Weyl group is \(S_3\):

If \(G\) is simply connected, then \(X(T) = \bigoplus {\mathbb{Z}}\omega_i\).

See Jantzen for definition of simply-connected, \({\operatorname{SL}}(n+1)\) is simply connected but its adjoint \(PGL(n+1)\) is not simply connected.

3.3 Complex Semisimple Lie Algebras

When doing representation theory, we look at the Verma modules \(Z(\lambda) = U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}}^+)} \lambda \twoheadrightarrow L(\lambda)\).

Thus the representations are indexed by lattice points in a particular region:

Question 1:

Suppose \(G\) is a simple (simply connected) algebraic group. How do you parameterize irreducible representations?

For \(\rho: G\to \operatorname{GL}(V)\), \(V\) is a simple module (an irreducible representation) iff the only proper \(G{\hbox{-}}\)submodules of \(V\) are trivial.

Answer 1: They are also parameterized by \(X(T)_+\). We’ll show this using the induction functor \(\mathop{\mathrm{ind}}_B^G \lambda =H^0(G/B, \mathcal{L}(\lambda))\) (sheaf cohomology of the flag variety with coefficients in some line bundle).

We’ll define what \(B\) is later, essentially upper-triangular matrices.

Question 2: What are the dimensions of the irreducible representations for \(G\)?

Answer 2: Over \(k={\mathbb{C}}\) using Weyl’s dimension formula.

For \(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\): conjectured to be known for \(p\geq h\) (the Coxeter number), but by Williamson (2013) there are counterexamples. Current work being done!

4 Friday, August 28

4.1 Representation Theory

Review: let \({\mathfrak{g}}\) be a semisimple lie algebra \(/{\mathbb{C}}\). There is a decomposition \({\mathfrak{g}}= {\mathfrak{b}}^+ \oplus {\mathfrak{n}}^- = {\mathfrak{n}}^+ \oplus t\oplus {\mathfrak{n}}^-\), where \(t\) is a torus. We associate \(U({\mathfrak{g}})\) the universal enveloping algebra, and representations of \({\mathfrak{g}}\) correspond with representations of \(U({\mathfrak{g}})\).

Let \(\lambda \in X(T)\) be a weight, then \(\lambda\) is a \(U({\mathfrak{b}}^+){\hbox{-}}\)module. We can write \(Z(\lambda) = U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}}^+)} \lambda\).

4.1.1 Induction

Let \({\mathfrak{g}}\) be an algebraic group \(/k\) with \(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\), and let \(H \leq G\). Let \(M\) be an \(H{\hbox{-}}\)module, we’ll eventually want to produce a \(G{\hbox{-}}\)modules.

Step 1: Make \(M\) into a \(G\times H\) where the first component \((g, 1)\) acts trivially on \(M\).

Taking the coordinate algebra \(k[G]\), this is a \((G-G){\hbox{-}}\)bimodule, and thus becomes a \(G\times H{\hbox{-}}\)module. Let \(f\in k[G]\), so \(f:G\to K\), and let \(y\in G\). The explicit action is \begin{align*} [(g, h) f] (y) \mathrel{\vcenter{:}}= f(g^{-1} y h) .\end{align*}

Note that we can identify \(H\cong 1\times H \leq G\times H\). We can form \((M\otimes_k k[G])^H\), the \(H{\hbox{-}}\)fixed points.

4.1.2 Properties of Induction

1. \(({-}\otimes_k k[G]) = \hom_H(k, {-}\otimes_k k[G])\) is only left-exact, i.e. \begin{align*} \qty{0\to A\to B\to C\to 0}\mapsto \qty{0\to FA \to FB \to FC \to \cdots} .\end{align*}

2. By taking right-derived functors \(R^jF\), you can take cohomology.

Note that in this category, we won’t have enough projectives, but we will have enough injectives.

3. This functor commutes with direct sums and direct limits.

4. (Important) Frobenius Reciprocity: there is an adjoint, restriction, satisfying \begin{align*} \hom_G(N, \mathop{\mathrm{ind}}_H^G M) = \hom_H(N\downarrow_H, M) .\end{align*}

5. (Tensor Identity) If \(M\in {\mathsf{Mod}}(H)\) and additionally \(M \in {\mathsf{Mod}}(G)\), then \(\mathop{\mathrm{ind}}_H^G = M \otimes_k \mathop{\mathrm{ind}}_H^G k\).

If \(V_1, V_2 \in {\mathsf{Mod}}(G)\) then \(V_1 \otimes_k V_2 \in {\mathsf{Mod}}(G)\) with the action given by \(g(v_1\otimes v_2) = gv_1 \otimes gv_2\).

6. Another interpretation: we can write \begin{align*} \mathop{\mathrm{ind}}_H^G(M) = \left\{{f\in \mathop{\mathrm{Hom}}(G, M_a) {~\mathrel{\Big|}~} f(gh) = h^{-1} \cdot f(g) \, \forall g\in G, h\in H}\right\} \qquad M_a = M \mathrel{\vcenter{:}}={\mathbb{A}}^{\dim M} .\end{align*}

I.e., equivariant wrt the \(H{\hbox{-}}\)action.

Then \(G\) acts on \(\mathop{\mathrm{ind}}_H^G M\) by left-translation: \((gf)(y) = f(g^{-1} y)\).

7. There is an evaluation map: \begin{align*} {\varepsilon}: \mathop{\mathrm{ind}}_H^G(M) &\to M \\ f&\mapsto f(1) .\end{align*}

This is an \(H{\hbox{-}}\)module morphism. Why? We can check \begin{align*} {\varepsilon}(h.f) &\mathrel{\vcenter{:}}=(h.f)(a) \\ &= f(h^{-1} ) \\ &= hf(1) \\ &= h({\varepsilon}(f)) .\end{align*}

We can write the isomorphism in Frobenius reciprocity explicitly: \begin{align*} \hom_G(N, \mathop{\mathrm{ind}}_H^G M) &\xrightarrow{\cong} \hom_H(N, M) \\ \phi & \mapsto {\varepsilon}\circ \phi .\end{align*}

8. Transitivity of induction: for \(H\leq H' \leq G\), there is a natural transformation (?) of functors: \begin{align*} \mathop{\mathrm{ind}}_H^G({-}) = \mathop{\mathrm{ind}}_{H'}^G\qty{\mathop{\mathrm{ind}}_H^{H'}({-}) } .\end{align*}

4.2 Classification of Simple \(G{\hbox{-}}\)modules

Suppose \(G\) is a connected reductive algebraic group \(/k\) with \(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\).

Let \(G = \operatorname{GL}(n, k)\). There is a decomposition:

Step 1: Getting modules for \(U\).

Then there’s a general fact: \(U^+ T U \hookrightarrow G\) is dense in the Zariski topology for any reductive algebraic group. We can form

• \(B^+ \mathrel{\vcenter{:}}= T\rtimes U^+\), the positive borel,
• \(B^- \mathrel{\vcenter{:}}= T\rtimes U\), the negative borel,

Suppose we have a \(U{\hbox{-}}\)module, i.e. a representation \(\rho: U \to \operatorname{GL}(V)\). We can find a basis such that \(\rho(u)\) is upper triangular with ones on the diagonal. In this case, there is a composition series with 1-dimensional quotients, and the composition factors are all isomorphic to \(k\).

Moral: for unipotent groups, there are only trivial representations, i.e. the only simple \(U{\hbox{-}}\)modules are isomorphic to \(k\).

Step 2: Getting modules for \(B\).

Modules for \(B\) are solvable, in which case we can find a flag. In this case, \(\rho(b)\) embeds into upper triangular matrices, where the diagonal action may now not be trivial (i.e. diagonal is now arbitrary).

Thus simple \(B{\hbox{-}}\)modules arise by taking \(\lambda \in X(T) = \hom(T, {\mathbb{G}}_m) = \hom(T, \operatorname{GL}(1, k))\), then letting \(u\) act trivially on \(\lambda\), i.e. \(u.v = v\). Here we have \(B \to B/U = T\), so any \(T{\hbox{-}}\)module can be pulled back to a \(B{\hbox{-}}\)module.

Step 3: Getting modules for \(G\).

Let \(\lambda \in X(T)\), then \(H^0(\lambda) = \mathop{\mathrm{ind}}_B^G \lambda = \nabla(\lambda)\).

5 Monday, August 31

5.1 Review of Representation Theory of Modules

Take \(R\) a ring, then consider \(M\) an \(R{\hbox{-}}\)module to be a “vector space” over \(M\). Note that \(M\) is an \(R{\hbox{-}}\)module \(\iff\) there exists a ring morphism \(\rho: R\to \hom_{\text{AbGrp}}(M, M)\).

Now let \(G\) be a group and consider \(G{\hbox{-}}\)modules \(M\). Then a \(G{\hbox{-}}\)module will be defined by taking \(M/k\) a vector space and a \(G{\hbox{-}}\)action on \(M\). This is equivalent to having a group morphism \(\rho: G\to \operatorname{GL}(M)\).

For \(M\) a \(G{\hbox{-}}\)module, given a group action, define \begin{align*} \rho: G&\to \operatorname{GL}(M) \\ \rho(g)(m) &= g.m \end{align*} where \(\rho(h): M\to M\).

Similarly, for \(\rho: G\to \operatorname{GL}(M)\) a group morphism, define the group action \(g.m \mathrel{\vcenter{:}}=\rho(g)m\). Thus representations of \(G\) and \(G{\hbox{-}}\)modules are equivalent.

Recall from last time: we defined a functor \(\mathop{\mathrm{ind}}_H^G({-}): H{\hbox{-}}\text{mod} \to G{\hbox{-}}\text{mod}\), where \(\mathop{\mathrm{ind}}_H^G = \qty{k[G] \otimes M}^H\), the \(H{\hbox{-}}\)invariants. This functor is left-exact but not right-exact, so we have cohomology \(R^j \mathop{\mathrm{ind}}_H^G\) by taking right-derived functors.

Goal: classify simple \(G{\hbox{-}}\)modules for \(G\) a reductive connected algebraic group.

For \(G = \operatorname{GL}(n , k)\), we have a decomposition

We have

• \(B = T\rtimes U\) the negative Borel,
• \(B = T\rtimes U^+\) the Borel

For \(U{\hbox{-}}\)modules: \(k\) is the only simple \(U{\hbox{-}}\)module. Importantly, if \(V\) is a \(U{\hbox{-}}\)module, then the fixed points are never zero, i.e. \(V^U = \hom_{U{\hbox{-}}\text{Mod}}(k, V) \neq 0\).

For \(B{\hbox{-}}\)modules: let \(X(T) \mathrel{\vcenter{:}}=\hom(T, {\mathbb{G}}_m) = \hom(T, \operatorname{GL}(1, k))\). These are the simple representations for the torus \(T\). Thus \(\lambda \in X(T)\) represents a simple \(T{\hbox{-}}\)module.

We have a map \(B \to B/U = T\), so we can pullback \(T{\hbox{-}}\)representations to \(B{\hbox{-}}\)representations (“inflation”), since we have a map \(T\to \operatorname{GL}(1, k)\) and we can just compose. So \(\lambda\) is a 1-dimensional (simple) \(B{\hbox{-}}\)module where \(U\) acts trivially.

Lee’s theorem: all irreducible representations for \(B\) are one-dimensional. Thus these are the simple \(B{\hbox{-}}\)modules.

For \(G{\hbox{-}}\)modules: define \(\nabla(\lambda) \mathrel{\vcenter{:}}=\mathop{\mathrm{ind}}_B^G(\lambda) = H^0(\lambda)\).

Questions:

1. When does \(H^0(\lambda) = 0\)?
2. What is \(\dim_{k{\hbox{-}}\text{Vect}} H^0(\lambda)\)?
3. What are the composition factors of \(H^0(\lambda)\)?

Known in characteristic zero, wildly open in positive characteristic.

Goal: classify simple \(G{\hbox{-}}\)modules. Strategy: used dominant highest weights.

As opposed to Verma modules, the irreducibles will be a dual situation where they sit at the bottom of the module. Indicated by the notation \(\nabla\) pointing down!

6 Friday, September 04

Some concepts used in the proof of other theorems: Let \(G\) be a reductive algebraic group and \({\mathfrak{g}}\) its lie algebra. There is an associative algebra \(U({\mathfrak{g}})\) which reflects the representation theory of \(G\).

Fact: \({\mathfrak{g}}{\hbox{-}}\)mod \(\equiv U({\mathfrak{g}}){\hbox{-}}\)modules which are unitary, i.e. \(1.m = m\).

We can write a basis \begin{align*} {\mathfrak{g}}= \left\langle{e_\alpha, h_i, f_\beta {~\mathrel{\Big|}~}\alpha\in\Phi^+,\, \beta\in\Phi^-,\, i = 1,2,\cdots,n}\right\rangle ,\end{align*} the Chevalley basis. It turns out that the structure constants are all in \({\mathbb{Z}}\).

We want to form a \({\mathbb{Z}}{\hbox{-}}\)lattice in \(U({\mathfrak{g}})\), denoted \begin{align*} U({\mathfrak{g}})_{\mathbb{Z}} = \left\langle{ e_\alpha^{[n]} = {e_\alpha^n \over n!},\, f_\beta^{[n]} = {f_\beta^n \over n!}, {h_i \choose m} }\right\rangle .\end{align*}

We then form the distribution algebra (or hyperalgebra in earlier literature) as \(\mathrm{Dist})G) \mathrel{\vcenter{:}}= U({\mathfrak{g}})_{\mathbb{Z}}\otimes_{\mathbb{Z}}k\) for \(k\) any field (e.g. \(\operatorname{ch}(k) = p\)).

6.1 Review

Recall that \(H^0(\lambda) \mathrel{\vcenter{:}}=\mathop{\mathrm{ind}}_B^G \lambda\). Proved in last (missed) class:

Goal: We now want to classify simple \(G{\hbox{-}}\)modules. So we need an iff criterion for when \(H^0(\lambda) \neq 0\).

We look at the set of dominant weights \begin{align*} X(T)_+ &= \left\{{\lambda \in X(T) {~\mathrel{\Big|}~}{\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle}\geq 0 \forall \alpha\in\Delta}\right\} &= \left\{{\lambda \in X(T) {~\mathrel{\Big|}~}\lambda = \sum_{i=1}^\ell n_i w_i,\, n_i \geq 0}\right\} .\end{align*}

6.2 Characters of \(G{\hbox{-}}\)modules

Let \(G\) be reductive, so (importantly) it has a maximal torus \(T\). Let \(M\in G{\hbox{-}}\mathrm{mod}\), so (importantly) \(M\in T{\hbox{-}}\mathrm{mod}\).

Then there is a weight space decomposition \(M = \bigoplus_{\lambda \in X(T)} M_\lambda\). We then write the character of \(M\) as \begin{align*} \operatorname{ch}M \mathrel{\vcenter{:}}=\sum_{\lambda \in X(T)} \qty{\dim M_\lambda} e^{\lambda} \in {\mathbb{Z}}[X(T)] .\end{align*}

Next time: more characters, and Weyl’s dimension formula.

7 Wednesday, September 09

Todo

8 Wednesday, September 16

8.1 Group Schemes

If \(F\) is representable, there is a correspondence \(F(R) \cong \hom_R(A, R)\). In the above example, \begin{align*}A = k[x_{11}, x_{12}, x_{21}, x_{22}] / \left\langle{x_{11} x_{22} - x_{12}x_{21}}\right\rangle,\end{align*} which is exactly the coordinate algebra.

Suppose \(G\) is an affine group scheme, and let \(A = k[G]\) be the representing object. Then there is a correspondence \begin{align*} G{\hbox{-}}\text{modules} \iff k[G] {}^{ \vee }{\hbox{-}}\text{modules} .\end{align*}

For \(G\) reductive, the RHS is equivalent to \(\operatorname{Dist}(G){\hbox{-}}\)modules.

If \(G\) is finite, then \(A {}^{ \vee }\cong k[G] {}^{ \vee }\) is a cocommutative Hopf algebras. Thus representations for finite group schemes are equivalent to representations for finite-dimensional cocommutative Hopf algebras.

On group scheme side: see reduction, spectral sequences, conceptual arguments. On the algebra side: have bases, underlying vector space, can do concrete computations. Can take \(\operatorname{Spec}\qty{k[G]} {}^{ \vee }\) to recover a group scheme.

8.2 Hopf Algebras

For \(A\) a \({\mathsf{Alg}_{/k} }\), we have a multiplication and a unit, which can be defined in terms of diagrams. To categorically reverse arrows, we can ask for a comultiplication and a counit. \begin{align*} \Delta: A &\to A^{\otimes 2} \\ \\ \epsilon: A &\to k .\end{align*}

We’ll want another map, an antipode \begin{align*} s: A\to A .\end{align*}

The comultiplication should satisfy

The counit should satisfy

And the antipode should satisfy

8.2.1 Module Constructions

Let \(A\) be a Hopf algebra.

1. For \(A{\hbox{-}}\)modules \(M, N\), we can form the \(A{\hbox{-}}\)module \(M\otimes_k N\) with \begin{align*} \Delta(a) &= \sum a_i \otimes a_j \\ \\ a(m\otimes n) &= \sum a_1 m \otimes a_2 n .\end{align*}

2. If \(M\) is finite-dimensional over \(A\), then \(M {}^{ \vee }= \hom_k(M, k) \ni f\) is an \(A{\hbox{-}}\)module, and we can define \((af)(x) \mathrel{\vcenter{:}}= f(s(a)x)\) for \(a\in A, x\in M\).

8.3 Frobenius Kernels

Let \(G\) be an algebraic group (scheme) over \(k\), where \(\operatorname{ch}(k) = p\). Let \(F:G\to G\) be the Frobenius, where e.g. \begin{align*} F:\operatorname{GL}(n, {-}) &\to \operatorname{GL}(n, {-})\\ (x_{ij}) & \mapsto (x_{ij}^p) .\end{align*}

Then \(F\) is a map of group schemes.

Recall that \begin{align*} \operatorname{Dist}(G) = \left\langle{ {x_\alpha^n \over n!}, {y_\beta^m \over m!}, {H_i \choose k} }\right\rangle .\end{align*}

We get a chain of finite dimensional algebras \begin{align*} \operatorname{Dist}(G_1) \leq \operatorname{Dist}(G_2) \leq \cdots \leq \operatorname{Dist}(G) \end{align*} where \begin{align*} \operatorname{Dist}(G_1) = \left\langle{ {x_\alpha^n \over n!}, {y_\beta^m \over m!}, {H_i \choose k} {~\mathrel{\Big|}~}0\leq n,m,k \leq p-1 }\right\rangle ,\end{align*}

where in general \(\operatorname{Dist}(G_\ell)\) goes up to \(p^{\ell} - 1\). Recall that \(G_r\) representations were equivalent to \(\operatorname{Dist}(G_r)\) representations.

Some basic questions (Curtis, Steinberg, 1960s):

1. What are the simple modules for Frobenius kernels? I.e., what are the irreducible representations for \(G_r\)?

2. How are the representations for \(G_r\) related to those for \(G\)?

It turns out the representations for \(G_r\) will lift to representations to \(G\). Use “twisted tensor product” (Steinberg).

9 Friday, September 18

9.1 Frobenius Kernels

Let \(\operatorname{ch}(k) p > 0\) and let \(G\) be an algebraic group scheme. We have a Frobenius map \(F:G\to G\) given by \(F((x_{ij})) = (x_{ij}^p)\), which we can iterate to get \(F^r\) for \(r\in {\mathbb{N}}\). Setting \(G_r = \ker F^r\) the \(r\)th Frobenius kernel, we get a normal series of group schemes \begin{align*} G_1 {~\trianglelefteq~}G_2 {~\trianglelefteq~}\cdots {~\trianglelefteq~}G .\end{align*}

There is an associated chain of finite dimensional Hopf algebras \begin{align*} \operatorname{Dist}(G_1) \leq \operatorname{Dist}(G_2) \leq \cdots \leq \operatorname{Dist}(G) .\end{align*}

Then \(k[G] {}^{ \vee }= \operatorname{Dist}(G_r)\), and we get an equivalence of representations for \(G_r\) to representations for \(\operatorname{Dist}(G_r)\).

A special case will be when \(G\) is a reductive algebraic group scheme. We’ll start by finding a basis for \(\operatorname{Dist}(G_r)\).

Recall the PBW theorem: we have a basis for \({\mathfrak{g}}\) given by \begin{align*} \left\{{x_\alpha {~\mathrel{\Big|}~}\alpha\in \Phi^+ }\right\} &\text{ Positive root vectors} \\ \left\{{h_i {~\mathrel{\Big|}~}i=1,\cdots, n}\right\} &\text{ A basis for } t \\ \left\{{x_\alpha {~\mathrel{\Big|}~}\alpha\in \Phi^- }\right\} &\text{ Negative root vectors} \\ .\end{align*}

We can then obtain a basis for \(U({\mathfrak{g}})\): \begin{align*} U({\mathfrak{g}}) = \left\langle{ \prod_{\alpha\in\Phi^+} x_\alpha^{n(\alpha)} \prod_{i=1}^n h_i^{k_i} \prod_{\alpha\in\Phi^+} x_{-\alpha}^{m(\alpha)} }\right\rangle .\end{align*}

We can similarly obtain a basis for the distribution algebra \begin{align*} \operatorname{Dist}(G) = \left\langle{ \prod_{\alpha\in\Phi^+} { x_{\alpha}^{n(\alpha)} \over n!} \prod_{i=1}^n {h_i \choose k_i} \prod_{\alpha\in\Phi^+} { x_{-\alpha}^{n(\alpha)} \over n!} }\right\rangle ,\end{align*} and we can similar get \(\operatorname{Dist}(G_r)\) by restricting to \(0\leq n(\alpha), k_i, m(\alpha) \leq p^r - 1\). Above the \(k_i\) are allowed to be any integers. This yields a triangular decomposition \begin{align*} \operatorname{Dist}(G_r) = \operatorname{Dist}(U_r^+) \operatorname{Dist}(T_r) \operatorname{Dist}(U_r^-) ,\end{align*} where we’ll denote the first two terms \(\operatorname{Dist}(B_r^+)\) and the last two as \(\operatorname{Dist}(B_r)\).

9.2 Induced and Coinduced Modules

Goal: Classify simple \(G_r{\hbox{-}}\)modules. We know the classification of simple \(G{\hbox{-}}\)modules, so we’ll follow similar reasoning. We started by realizing \(L(\lambda) \hookrightarrow\mathop{\mathrm{ind}}_B^G \lambda\) as a submodule (the socle) of some “universal” module.

Let \(M\) be a \(B_r{\hbox{-}}\)module, we can then define \begin{align*} \mathop{\mathrm{ind}}_{B_r}^{G_r}M = \qty{k[G_r] \otimes M }^{B_r} ,\end{align*} where we’re now taking the \(B_r{\hbox{-}}\)invariants. We get a decomposition as vector spaces, \begin{align*} k[G_r] = k[U_r^+] \otimes_k k[B_r] \end{align*} and thus an isomorphism \begin{align*} \mathop{\mathrm{ind}}_{B_r}^{G_r}M = \qty{k[G_r] \otimes M }^{B_r} \cong k[U_r^+] \otimes\qty{ k[B_r] \otimes M}^{B_r} \cong k[U_r^+] \otimes M \end{align*} since \(k[B_r]\otimes M \cong \mathop{\mathrm{ind}}_{B_r}^{B_r} M \cong M\).

We then define \begin{align*} \mathop{\mathrm{coInd}}_{B_r}^{G_r} = \operatorname{Dist}(G_r) \otimes_{\operatorname{Dist}(B_r)} \otimes M ,\end{align*} which is an analog of \(U({\mathfrak{g}})\otimes_{U({\mathfrak{b}})} M\).

We have \(\operatorname{Dist}(U_r^+) \otimes\operatorname{Dist}(B_r) \cong \operatorname{Dist}(G_r)\), so

\begin{align*} \mathop{\mathrm{coInd}}_{B_r}^{G_r} = \operatorname{Dist}(G_r) \otimes_{\operatorname{Dist}(B_r)} \otimes M \cong \operatorname{Dist}(U_r^+) \otimes_k \operatorname{Dist}(B_r) \otimes_{\operatorname{Dist}(B_r)} M \cong \operatorname{Dist}(U_r^+) \otimes_k M ,\end{align*} which we’ll define as the coinduced module.

We can compute the dimension: \begin{align*} \dim \mathop{\mathrm{ind}}_{B_r}^{G_r} M = \dim \mathop{\mathrm{coInd}}_{B_r}^{G_r} M = \qty{\dim M} p^{r{\left\lvert {\Phi^+} \right\rvert}} .\end{align*}

Open: don’t know how to compute composition factors.

9.3 Verma Modules

Recall that \(W(\lambda) \mathrel{\vcenter{:}}= U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}}^+)} \lambda\) were the Verma modules for lie algebras.

Let \(\lambda \in X(T)\), we have \(T_r \leq T\) and restriction yields a map \(X(T) \to X(T_r)\). Given a weight \(\lambda\), we can write it \(p{\hbox{-}}\)adically as \begin{align*} \lambda = \lambda_0 + \lambda_1 p + \lambda_2 p^2 + \cdots + \lambda_{r-1} + \cdots .\end{align*}

This yields an exact sequence \begin{align*} 0 \to p^r X(T) \to X(T) \to X(T_r) \to 0 ,\end{align*}

and thus \(X(T) / p^r X(T) \cong X(T_r)\).

Let \(\lambda \in X(T_r)\), then \(\lambda\) becomes a \(B_r{\hbox{-}}\)module by letting \(U_r\) act trivially, since we have \begin{align*} \cdots U_r \to B_r \twoheadrightarrow T_r \to 0 .\end{align*}

Set \(Z(r) = \mathop{\mathrm{coInd}}_{B_r}^{G_r} \lambda\), and set \(Z(r)' = \mathop{\mathrm{ind}}_{B_r}^{G_r} \lambda\). Then \(\dim Z_r(\lambda) = \dim Z_r'(\lambda) = p^{r{\left\lvert {\Phi^+} \right\rvert}}\). We’ll then think of

• \(\mathop{\mathrm{coInd}}\twoheadrightarrow L_r(\lambda)\) being in the head,
• \(L_r(\lambda) \hookrightarrow\mathop{\mathrm{ind}}\) being the socle.

Note that the dimensions aren’t known, nor are the projective covers or injective hulls.

We have a form of translation invariance, namely \begin{align*} Z_r(\lambda + p^r\nu) = Z_r(\lambda) \qquad &\forall \nu \in X(T) \\ Z_r'(\lambda + p^r\nu) = Z_r'(\lambda) \qquad &\forall \nu \in X(T) .\end{align*}

10 Monday, September 21

Let \(G\) be a reductive algebraic group scheme, \(k=\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\) with \(p>0\), equipped with the Frobenius map \(F:G\to G\) with \(F^r\) its \(r{\hbox{-}}\)fold composition. We defined Frobenius kernels \(G_r \mathrel{\vcenter{:}}=\ker F^r\), which are in correspondence with the cocommutative Hopf algebras \(\operatorname{Dist}(G_r)\).

Goal: We want to classify simple \(G_r{\hbox{-}}\)modules, and to do this we’ll use socles.

We have a maximal torus \(T\subseteq G\) and thus \(T_r \subseteq G_r\) after acting by Frobenius. This yields a SES \begin{align*} 0 \to p_r X(T) \to X(T) \to X(T)/p^r X(T) = X(T_r) \to 0 .\end{align*}

How to think about this: take \(\lambda \in X(T_r)\), then we can write \(\lambda = \lambda + p^r \sigma\) in \(X(T_r)\) for some other weight \(\sigma \in X(T)\). We’ll define the “baby Verma modules” \begin{align*} Z_r(\lambda) \mathrel{\vcenter{:}}=\mathop{\mathrm{coInd}}_{B_r^+}^{G_r} \lambda \\ Z_r'(\lambda) \mathrel{\vcenter{:}}=\mathop{\mathrm{ind}}_{B_r^+}^{G_r} \lambda ,\end{align*}

and we have \(\dim Z_r(\lambda) = \dim Z_r'(\lambda) = p^{r {\left\lvert {\Phi^+} \right\rvert}}\).

Note the latter are \(T_r{\hbox{-}}\)modules, so we let \(U^+\) act trivially.

10.1 Simple \(G{\hbox{-}}\)modules

We know that after taking fixed points, \(Z_r(\lambda)^{U_r}\) and \(Z_r'(\lambda)^{U_r^+}\) are one-dimensional, and thus \begin{align*} Z_r(\lambda) / \mathop{\mathrm{Rad}}Z_r(\lambda) \cong L_r(\lambda) \qquad \mathop{\mathrm{Soc}}_{G_r} Z_r'(\lambda) = L_r(\lambda) \end{align*} following the same argument considering \(H_0(\lambda)\).

For any \(\lambda \in X(T_r)\) we have \(0\neq L_r = \mathop{\mathrm{Soc}}_{G_r} Z_r'(\lambda)\). By the one-dimensionality above, we know \begin{align*} L_r(\mu) = L_r(\lambda) \iff \lambda = \mu \in X(T_r) .\end{align*}

Letting \(N\) be a simple \(G_r{\hbox{-}}\)module, we can consider it as a \(B_r{\hbox{-}}\)module, and the simple \(B_r{\hbox{-}}\)modules are one dimensional and obtained from simple \(T_r{\hbox{-}}\)modules. We then know that for some \(\lambda \in X(T_r)\), \begin{align*} 0 \neq \hom_{B_r}(N, \lambda) \\ &= \hom_{G_r}(N, \mathop{\mathrm{ind}}_{B_r}^{G_r} \lambda) ,\end{align*} which implies that \(N\hookrightarrow\mathop{\mathrm{ind}}_{B_r}^{G_r} \lambda = Z_r'(\lambda)\) as a submodule, and thus \(N = L_r(\lambda)\).

How to think about this: restricted regions. Choose dominant weights as representatives \begin{align*} X_r(T) &= \left\{{\lambda \in X(T)_+ {~\mathrel{\Big|}~}0\leq {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} < p^r\, \forall \alpha\in \Delta }\right\} \\ &= \left\{{\lambda \in X(T)_+ {~\mathrel{\Big|}~}\lambda = \sum_{i=1}^\ell n_i w_i,\, 0\leq n_j \leq p^r-1\, \forall j}\right\} \\ .\end{align*}

Pictures:

Some facts:

If \(\lambda \in X(T)_+\), then \(L(\lambda)\) is a simple \(G{\hbox{-}}\)module.

Question 1: What happens when we restrict \(L(\lambda)\downarrow_{G_r}\)?

Answer: This remains irreducible over \(G_r\) iff \(\lambda \in X_r(T)\), i.e. if \(L(\lambda)\downarrow_{G} \cong L_r(\lambda)\) when \(\lambda \in X_r(T)\).

Question 2: Given \(L(\lambda)\) for \(\lambda \in X(T)_+\), can we express \(L(\lambda)\) in terms of simple \(G_r{\hbox{-}}\)modules?

Answer: Yes, can be formulated in terms of Steinberg’s twisted tensor product.

11 Friday, September 25

11.1 Review and Proposition

From last time: Steinberg’s tensor product.

Let \(G\) be a reductive algebraic group scheme over \(k\) with \(\operatorname{ch}(k) > 0\). We have a Frobenius \(F:G\to G\), we iterate to obtain \(F^r\) and examine the Frobenius kernels \(G_r\mathrel{\vcenter{:}}=\ker F^r\).

If we have a representation \(\rho: G\to \operatorname{GL}(M)\), we can “twist” by \(F^r\) to obtain \(\rho^{(r)}: G \to \operatorname{GL}(M^{(r)})\). We have

Here \(M^{(r)}\) has the same underlying vector space as \(M\), but a new module structure coming from \(\rho^{(r)}\). Note that \(G_r\) acts trivially on \(M^{(r)}\).

• \(\left\{{L(\lambda) {~\mathrel{\Big|}~}\lambda \in X(T)_+}\right\}\) are the simple \(G{\hbox{-}}\)modules,
• \(\left\{{L_r(\lambda) {~\mathrel{\Big|}~}\lambda \in X_r(T)_+}\right\}\) are the simple \(G_r{\hbox{-}}\)modules,

Note that \(L(\lambda)\downarrow_{G_r}\) is semisimple, equal to \(L_r(\lambda)\) for \(\lambda \in X_r(T)\).

1960’s, Curtis and Steinberg.

Recall that socle formula: letting \(M\) be a \(G{\hbox{-}}\)module, we have an isomorphism of \(G{\hbox{-}}\)modules: \begin{align*} \mathop{\mathrm{Soc}}_{G_r} \cong \bigoplus_{\lambda \in X_r(T)} L(\lambda) \otimes\hom_{G_r}(L(\lambda), M) .\end{align*}

11.2 Proof

In order to know \(\dim L(\lambda)\) for \(\lambda \in X(T)_+\), it is enough to know \(\dim L_1(\mu)\) for \(\mu \in X_1(T)\). Schematic:

11.3 Some History

Recall that simplie \(G_1{\hbox{-}}\)modules correspond to simple \(\operatorname{Dist}(G_1){\hbox{-}}\)modules, and \(\operatorname{Dist}(G_1) \cong U({\mathfrak{g}})\).

• 1980: Lusztig proved conjecture: \(\operatorname{ch}L(\lambda)\) for \(\lambda \in X_1(T)\) is given by KL polynomials, shown for \(p \geq 2(h-1)\).

• Kato showed for \(p> h\), where \(h\) is the Coxeter number satisfying \(h = {\left\langle {\rho},~{\alpha_i {}^{ \vee }} \right\rangle} + 1\) where \(\alpha_i {}^{ \vee }\) is the highest short root.

• 1990’s: A relation to representations of quantum groups \(U_q\) and affine lie algebras \(\widehat{{\mathfrak{g}}}\):

  The first map is due to Andersen-Jantzen-Soergel for \(p\gg 0\) with no effective lower bounds, and the equivalence is due to Kazhdan-Lusztig, where the L conjecture holds for \(\widehat{{\mathfrak{g}}}\).

• 2000’s: Fiebig showed the L conjecture holds for \(p>N\) where \(N\) is an effective (but large) lower bound.

• 2013: Geordie Williamson shows L conjecture is false, with infinitely many counterexamples, and no lower bounds that are linear in \(h\).

See Donkin’s Tilting Module conjecture: expected that characters may come from \(p{\hbox{-}}\)KL polynomials instead.

Let \(G= {\operatorname{SL}}(2)\), so \(\dim T =1\). Here the restricted region of weights is given by \(X_!(T) = \left\{{0,1,\cdots, p-1}\right\}\). Then \(H^0(\lambda) = S^\lambda(V)\) for \(\lambda \in X(T)_+ = {\mathbb{Z}}_{\geq 0}\) and \(L(\lambda) \subseteq H^0(\lambda)\).

\begin{align*} L(\lambda) = H^0(\lambda) {\quad \operatorname{for} \quad} \lambda \in X_1(T) .\end{align*}

\begin{align*} \dim L(\lambda) = \lambda + 1 {\quad \operatorname{for} \quad} \lambda \in X_1(T) .\end{align*}

Take \(p=3\). Then \(\dim L(0) = 1\), \(\dim L(1) = 2\) (the natural representation), and \(\dim L(2) = 3\) (the adjoint representation). Then for \(p=4\), we have to use the twisted tensor product formula. Taking the 3-adic expansion \(4 = 1\cdot 3^0 + 1\cdot 3^1\), we have \begin{align*} L(4) = L(1) \otimes L(1)^{(1)} .\end{align*}

Since \(\dim L(1) = 2\), we get \(\dim L(4) = 4\).

Similarly, considering \(7 = 1\cdot 3^0 + 2\cdot 3^1\), we get \begin{align*} L(7) \cong L(1) \otimes L(2)^{(1)} \end{align*} and so \(\dim L(7) = 6\).

Take \(p=5\), then

• \(\dim L(0) = 1\)
• \(\dim L(1) = 2\)
• \(\dim L(2) = 3\)
• \(\dim L(3) = 4\)
• \(\dim L(4) = 5\)

What is \(H^0(5)\)? We know \(L(5)\) is a submodule, and we can write the character \begin{align*} \operatorname{ch}H^0(5) = e^5 + e^3 + e^1 + e^{-1} + e^{-3} + e^{-5} .\end{align*}

We know \(\operatorname{ch}(L(1)) = e^1 + e^{-1}\) and \(L(5) = L(1)^{(1)}\), so we can write \(\operatorname{ch}L() = e^{5} + e^{-5}\). By quotienting, we have \(\operatorname{ch}H^0(5) - \operatorname{ch}L(5) = e^3 + e^1 + e^{-1} +e^{-3} = \operatorname{ch}L(3)\). Thus the composition factors of \(H^0(5)\) are \(L(5)\) and \(L(3)\).

These correspond to an action of the affine Weyl group:

There is a strong linkage principle which describes the possible composition factors of \(H^0(\lambda)\).

We can thus find the socle/head structure:

Thus \(\operatorname{Ext} _G^1(L(5), L(3)) \cong k\).

Note that in other types, we don’t know the characters of the irreducibles in the restricted region, so we don’t necessarily know the composition factors.

12 Monday, September 28

12.1 Kempf’s Theorem

Next topic: Kempf’s Vanishing Theorem. Proof in Jantzen’s book involving ampleness for sheaves.

Setup:

We have

along with the weights \(X(T)\).

We can consider derived functors of induction, yielding \(R^n \mathop{\mathrm{ind}}_B^G \lambda = \mathcal{H}^n(G/B, \mathcal{L}(\lambda)) \mathrel{\vcenter{:}}= H^n(\lambda)\) where \(\mathcal{L}(\lambda)\) is a line bundle and \(G/B\) is the flag variety.

Recall that

• \(H^0(\lambda) = \mathop{\mathrm{ind}}_B^G(\lambda)\),
• \(\lambda \not\in X(T)_+ \implies H^0(\lambda) = 0\)
• \(\lambda \in X(T)_+ \implies L(\lambda) = \mathop{\mathrm{Soc}}_G H^0(\lambda) \neq 0\).

12.2 Good Filtrations and Weyl Filtrations

We define two classes of distinguished modules for \(\lambda \in X(T)_+\):

• \(\nabla(\lambda) \mathrel{\vcenter{:}}= H^0(\lambda) = \mathop{\mathrm{ind}}_B^G \lambda\) the costandard/induced modules.
• \(\Delta(\lambda) = V(\lambda) \mathrel{\vcenter{:}}= H^0(-w_0 \lambda) = \mathop{\mathrm{ind}}_B^G \lambda\) the standard/Weyl modules
  • Here \(w_0\) is the longest element in the Weyl group

We have \begin{align*} L(\lambda) &\hookrightarrow\nabla(\lambda) \Delta(\lambda) &\twoheadrightarrow L(\lambda) .\end{align*}

We define the category \(\text{Rat}{\hbox{-}}G\) of rational \(G{\hbox{-}}\)modules. This is a highest weight category (as is e.g. Category \({\mathcal{O}}\)).

We have the following situation for type \(A_2\):

And thus a decomposition:

The picture to keep in mind is the following: 4 types of modules, all indexed by dominant weights:

12.3 Cohomological Criteria for Good Filtrations

We’ll take cohomology in the following way: let \(G\) be an algebraic group scheme, and define \begin{align*} H^n(G, M) \mathrel{\vcenter{:}}= \mathrm{Ext} G^n(k, M) \end{align*}

where to compute \(\operatorname{Ext} _G^n(M, N)\) we take an injective resolution \(N \hookrightarrow I_*\), apply \(\hom_G(M, {-})\), and take kernels mod images.

Letting \(\lambda \in {\mathbb{Z}}\Phi\) be integral, so \(\lambda_{\alpha \in \Delta} = \sum n_\alpha \alpha\), define the height \begin{align*} \operatorname{ht}(\lambda) = \sum_{\alpha\in\Delta} n_\alpha .\end{align*}

\begin{align*} k[u] \text{ an injective $B{\hbox{-}}$module} \\ k\hookrightarrow\mathop{\mathrm{ind}}_T^B k \mathrel{\vcenter{:}}= I_0 = k[u] .\end{align*}

We thus get a diagram of the form

Let \(H=B\) and take \(G=G\) itself, and let \(N = k\) the trivial module and \(M\in {\mathsf{Mod}}(G)\) be any rational \(G{\hbox{-}}\)module. We have \begin{align*} E_2^{i, j} = \operatorname{Ext} ^{i}_B(k, R^j \mathop{\mathrm{ind}}_B^G M) \implies \operatorname{Ext} ^{i+j}_B(k, M) .\end{align*}

Observations:

0. \(R^0 \mathop{\mathrm{ind}}_B^G k = \mathop{\mathrm{ind}}_B^G k = k\).

1. The tensor identity works here, i.e. \(R^j \mathop{\mathrm{ind}}_B^G M = \qty{R^j \mathop{\mathrm{ind}}_B^G k} \otimes M\).

2. \(R^j \mathop{\mathrm{ind}}_B^G k = 0\) for \(j> 0\) since we have a dominant weight.

The spectral sequence thus collapses on \(E_2\):

Thus \begin{align*} E_2^{i, 0} = \operatorname{Ext} ^i_B(k, M) = H^i(B, M) .\end{align*}

Fix a Dynkin diagram and take a subset \(J\subseteq \Delta\).

Then \(L_j\rtimes U_j = P_J = P\), and we have a decomposition like

13 Wednesday, September 30

Recall that we had a dominant weight \(\lambda \in X(T)_+\) with

where we have a module with both a good and a Weyl filtration.

If \(B\subseteq P \subseteq G\) with \(P\) parabolic and \(M\in {\mathsf{Mod}}(G)\), we have a “transfer theorem”: maps \begin{align*} H^n(G; M) \xrightarrow{\mathop{\mathrm{res}}} H^n(P; M) \xrightarrow{\mathop{\mathrm{res}}} H^n(B; M) \end{align*} induced by restrictions which are isomorphisms.

Part (a) is proved in the book, we won’t show it here.

The following is an important calculation!

We showed this in a special case. Let \(i=1\) with \(\lambda \not> \mu\), then \begin{align*} \operatorname{Ext} _G^1(L(\lambda), L(\mu)) \cong \mathop{\mathrm{Hom}}_G(L(\lambda), H^0(\mu) / \mathop{\mathrm{Soc}}_G(H^0(\mu))) .\end{align*} Thus it suffices to understand only the previous layer:

We now want criteria for when we can find the following types of lifts:

Next time: state and prove a cohomological criterion (Donkin, Scott, proved independently) for a \(G{\hbox{-}}\)module to admit a good filtration. More about when tensor products of induced modules have good filtrations.

14 Friday, October 02

Recall that good filtration is a chain \(\left\{{0}\right\} \subseteq V_1 \subseteq \cdots \subseteq V\) satisfying \(V = \cup V_i\) and \(V_i/V_{i-1} \cong H^0(\lambda_i)\) for \(\lambda_i\) some weight of \(V\).

That is, we have a lift of the following form:

\(1\implies 2\): Use the fact that \(\operatorname{Ext} ^i_G(V(\lambda), H^0(\mu)) = 0\) for all \(i>0\) and all \(\mu\).

\(2\implies 3\): Clear!

\(3\implies 1\): Choose a total ordering of weights \(\lambda_0, \lambda_1, \cdots \in X(T)\) such that if \(\lambda_i < \lambda_j\) then \(i<j\). Since \(V\neq 0\), there exists a dominant weight \(\lambda \in X(T)_+\) such that \(\hom_G(V(\lambda), V) \neq 0\), so choose \(i\) minimally in this order to produce such a \(\lambda_i\). Idea: use this to start a filtration.

Then \(\hom(L(\lambda_i), V) \neq 0\), and we have \begin{align*} V(\lambda_i) \twoheadrightarrow L(\lambda_i) \hookrightarrow V .\end{align*}

We know that \begin{align*} \hom_G(V(\mu), V) = 0 \quad \forall \mu < \lambda_i \\ \hom_G(L(\mu), V) = 0 \quad \forall \mu < \lambda_i \\ \operatorname{Ext} _G^1(L(\mu), V) = 0 \quad \forall \mu \in X(T)_+ \text{ by assumption} .\end{align*}

So the following map must be an injection, since there is no socle:

Set \(V_1 = H^0(\lambda_i)\), so \(V_1 \subseteq V\). We then have a SES \begin{align*} 0 \to V_1 \to V \to V/V_1 \to 0 .\end{align*}

Applying \(\hom(V(\lambda), {-})\) we obtain

Now iterate this process to obtain a chain \(V_1 \subseteq V_2 \subseteq \cdots \subseteq V\), and set \(V' \mathrel{\vcenter{:}}=\cup_{i>0} V_i\). Then \(\dim \hom_G(V(\lambda), V') = \dim \hom_G( V(\lambda), V )\) since \(\dim \hom_G(V(\lambda), V) < \infty\). But then taking the SES \begin{align*} 0\to V' \to V \to V/V' \to 0 \end{align*} and applying \(\mathop{\mathrm{Hom}}(V(\lambda), {-})\), we have \(\mathop{\mathrm{Hom}}(V(\lambda), V/V') = 0\) and we get an isomorphism of homs. But then \(\hom(V(\lambda), V/V') = 0\) for all \(\lambda \in X(T)_+\), forcing \(V/V'=0\) and \(V=V'\).

Note: this is likely difficult to prove without cohomology! But here we can apply the ext criterion.

Let \(\lambda \in X(T)_+\), then

For \(\lambda \in X(T)_+\), let \(I(\lambda)\) be the injective hull of \(L(\lambda)\), so we have \begin{align*} 0 \to L(\lambda) \hookrightarrow I(\lambda) .\end{align*}

15 Monday, October 05

Crelle 1988 (CPS: Cline Parshall Scott)

Let HWC denote a highest weight category.

See

1. Donkin: On generalized Schur algebras

2. Irving: BGG algebras

There is a equivalence between HWC and QHA (quasi-hereditary algebras).

Interval finite poset: we’ll have a cone \(\Lambda\) of positive weights:

See handout!

• First proofs:
  • JP Wong, Type A
  • Donkin, all but characteristic 2 and \(E_7, E_8\).
  • O. Mathieu, general proof using algebraic geometry

15.1 Polynomial Representation Theory

Let \(G = \operatorname{GL}(n, k)\), then a module for \(G\) is polynomial iff the weights \(\lambda = (\lambda_1, \cdots, \lambda_n)\) satisfy \(\lambda_j \geq 0\) for all \(j\).

Next time: look at structure of injective modules, then the theory of Bott-Borel-Weil for higher sheaf cohomology.

16 Wednesday, October 07

16.1 Schur Algebras

Let \(G = \operatorname{GL}(n, k)\), then polynomial representations of \(G\) are equivalent to \(S(n, d)\) modules for all \(d\geq 0\), where we can note that \(S(n, d) = \mathop{\mathrm{End}}_{\Sigma_d}(V^{\otimes d})\). We’ll have a correspondence \begin{align*} \left\{{\substack{L(\lambda) \text{ simple modules for } S(n,d)}}\right\} \iff \Lambda^+(n, d) \text{, partitions of $d$ with at most $n$ parts} ,\end{align*}

Good example, can see all filtrations at work, tilting modules, etc.

Consider \(S(3, 3)\) for \(p=3\), we then have the partitions \(\Lambda^+(3, 3) = \left\{{(3), (2, 1), (1,1,1)}\right\}\). We can think of these in the \({\varepsilon}\) basis as \((3) = (3,0,0), (2,1) = (2,1,0)\). Since \({\operatorname{SL}}(3, k) \subset \operatorname{GL}(3, k)\), we can find the \(SL(3, k)\) weights by taking successive differences to yield \((3, 0), (1, 1), (0, 0)\) with the corresponding picture

We can compute

• \(L(1,1,1) = H^0(1,1,1)\)
• \(L(2, 1) = H^0(2, 1)\)
• \(L(3) = H^0(3)\)

We have a form of Brauer reciprocity: \begin{align*} [I(\lambda): H^0(\mu)] = [H^0(\mu) : L(\lambda) ] .\end{align*}

We can now compute the injective hulls:

What are the tilting modules? We can use the fact that \(L(1^3) = V(1^3)\). It has a good filtration and a Weyl filtration and thus must be the tilting module for \(L(1^3)\).

Using the following fact:

We can compute the following:

16.2 Simplicity of \(H^0(\lambda)\)

1. \(k = {\mathbb{C}}\) implies \(L(\lambda) = H^0(\lambda)\) for all \(\lambda \in X(T)_+\)

2. \(k= \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\) implies \(L(\lambda) = H^0(\lambda)\) if \({\left\langle {\lambda},~{\alpha_0 {}^{ \vee }} \right\rangle} \leq 1\) where \(\alpha_0\) is the highest short root.

Such \(\lambda\) are referred to as minuscule weights.

16.3 Bott-Borel-Weil Theorem

We can consider the higher right-derived functors of \(\lambda\), given by \(H^i(\lambda) = R^i \mathop{\mathrm{ind}}_B^G \lambda\) for \(\lambda \in X(T)\). You can think of this as the higher sheaf cohomology of the flag variety, \(\mathcal{H}^i(G/B, \mathcal{L}(\lambda))\).

We have Kempf Vanishing: \(H^i(\lambda) = 0\) for all \(i>0\) when \(\lambda \in X(T)_+\) is dominant (although other things may happen for non-dominant weights). There is a correspondence \((G, T) \iff (W, \Phi)\), and since \(W\) is generated by simple reflections, we can write any \(w\in W\) as \(w=\prod s_{\alpha_i}\). A reduced expression is one in which the length can not be shortened, and any two reduced expressions necessarily have the same length (number of simple reflections).

16.3.1 Dot Action on Weights

We can let \(W\) act on \(X(T)\) by reflections by the formula \(s_\alpha \lambda = \lambda - {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle}\alpha\). We then shift the action by setting \(s_\alpha \cdot \lambda = w(\lambda+\rho)-\rho\) where \(\rho = {1\over 2} \sum_{\alpha\in \Phi^+} \alpha = \sum_{j=1}^n w_j\).

Let \(G\) be a reductive algebraic group and \(k={\mathbb{C}}\). For \(\lambda \in X(T)_+\), we can describe the sheaf cohomology: \begin{align*} \mathcal{H}^i(w\cdot \lambda) = \begin{cases} H^0(\lambda) & i=\ell(w) \\ 0 & \text{otherwise} \end{cases} .\end{align*}

Moreover, if \(\lambda \not\in X(T)_+\) and \({\left\langle {\lambda+\rho},~{\alpha {}^{ \vee }} \right\rangle} \geq 0\) for all \(\alpha \in \Delta\), then \(\mathcal{H}^i(w\cdot \lambda) = 0\) for all \(w\in W\).

Wide open in characteristic \(p\), can say some things. We’ll prove this in characteristic zero.

Recall that \(k={\mathbb{C}}\) and \(H^0(\lambda) = L(\lambda)\). We’ll want to reduce to \({\operatorname{SL}}(2, {\mathbb{C}})\) parabolics. For \(\alpha\in\Delta\), let \(P_\alpha\) be the associated parabolic \(P_\alpha = L_\alpha \rtimes U_\alpha\), which is parabolic of type \(A_1\).

Idea: \(\alpha\) generates an \({\operatorname{SL}}_2\) subgroup (the Levi factor), like the Borel but sticks out in one dimension:

Then \begin{align*} s_\alpha \cdot \lambda = s_\alpha(\lambda + \rho) - \rho \\ = \lambda + \rho - {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle}\alpha - \rho \\ = \lambda - {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle}\alpha .\end{align*}

Next time: proof of Bott-Borel-Weil and its generalization to \(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\). For \(B\subset P_\alpha \subset G\), we’ll have a spectral sequence \begin{align*} E_2^{i, j} = R^i \mathop{\mathrm{ind}}_{P_\alpha}^G R^j \mathop{\mathrm{ind}}_B^{P_\alpha} \Rightarrow R^{i+j} \mathop{\mathrm{ind}}_B^G \lambda = H^{i+j}(\lambda) .\end{align*}

17 Friday, October 09

Last time: Bott-Borel-Weil. Stated for characteristic zero, working toward a generalization.

Let \(\Delta\) be the set of simple roots, and \(\alpha\in \Delta\). We can form a Levi decomposition \(P_\alpha \mathrel{\vcenter{:}}= L_\alpha \rtimes U_\alpha\):

We have \(B \subseteq P_\alpha \subseteq G\). The dot action is given by the following: Let \(W\) be the Weyl group, then \(W\) acts on \(X(T)\) by \(w\cdot \lambda = w(\lambda + \rho) - \rho\), where \begin{align*} \rho = {1\over 2} \sum_{\alpha\in \Phi^+} \alpha = \sum_{i=1}^n w_n .\end{align*}

We obtained a formula \begin{align*} S_\alpha \cdot \lambda = \lambda - {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} \alpha .\end{align*}

17.1 Bott-Borel-Weil Theory

This gives us an analog of \(A_1\) or \({\operatorname{SL}}_2\) theory. Also note that we have Serre duality: \begin{align*} H^1(\lambda) = H^0( - (\lambda + 2\rho) ) {}^{ \vee } .\end{align*}

Let \(\alpha\in \Delta\) and \(\lambda\in X(T)\), and suppose \(\lambda\) is dominant with respect to \(\alpha\), i.e. \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} \geq 0\).

• If \(\operatorname{ch}(k) = 0\) then \(\mathop{\mathrm{ind}}_B^{P_\alpha}\lambda = R^1 \mathop{\mathrm{ind}}_B^{P_\alpha} s_\alpha \cdot \lambda\)

• If \(\operatorname{ch}(k) = p\) and if there exists an \(s, m\) with \(0<s<p\) and \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = sp^m - 1\) (Steinberg weights), then \begin{align*} \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = R^1 \mathop{\mathrm{Ind}}_B^{P_\alpha} s_\alpha \cdot \lambda .\end{align*}

The proof of this will use a Grothendieck-type spectral sequence of the form \begin{align*} E_2^{i, j} = R^i \mathop{\mathrm{ind}}_{P_\alpha}^G \qty{ R^j \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda} \Rightarrow R^{i+j} \mathop{\mathrm{ind}}_B^G \lambda .\end{align*}

We’ll have a version of Grothendieck vanishing: \begin{align*} R^j \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = 0 \qquad\text{for } j > \dim P_\alpha/B = 1 .\end{align*}

So the resulting spectral sequence will only be supported on the first two lines, and \(E_3 = E_\infty\). Note the differential will be of bidegree \({\partial}_r \leadsto (r, 1-r)\), and \(E_2\) will look like the following,

Recall that \(R^i \mathop{\mathrm{ind}}_B^G \lambda \mathrel{\vcenter{:}}= H^i(\lambda)\)

Now define \begin{align*} \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{{\mathbb{Z}}} &\mathrel{\vcenter{:}}= \left\{{ \lambda \in X(T) {~\mathrel{\Big|}~}0 \leq {\left\langle {\lambda+\rho},~{\beta {}^{ \vee }} \right\rangle} \,\forall \beta \in \Phi^+ }\right\} \qquad\text{ if } \operatorname{ch}(k) = 0 \\ &\mathrel{\vcenter{:}}= \left\{{ \lambda \in X(T) {~\mathrel{\Big|}~}0 \leq {\left\langle {\lambda+\rho},~{\beta {}^{ \vee }} \right\rangle} \leq \operatorname{ch}(k) \,\forall \beta \in \Phi^+ }\right\} \qquad\text{if } \operatorname{ch}(k) = p .\end{align*}

Idea:

Note that this covers everything in the \(\operatorname{ch}(k) = 0\) case, but only gives the following hexagon in the \(\operatorname{ch}(k) = p\) case:

18 Monday, October 12

18.1 Proof of Bott-Borel-Weil

Recall the Bott-Borel-Weil theorem: in characteristic zero, we’re looking at the closure of the region containing the fundamental region \(C_{\mathbb{Z}}\):

18.2 Serre Duality and Grothendieck Vanishing

Let \(P\) be a parabolic subgroup, i.e. \(P_J = P \mathrel{\vcenter{:}}= L_J \rtimes U_J\) for some \(J\subseteq \Delta\). Set \(n(P) = {\left\lvert {\Phi^+} \right\rvert} - {\left\lvert {\Phi^+_J} \right\rvert}\).

Let \(\Phi = A_4\), which has ten simple roots:

• \(\alpha_i, 1\leq i \leq 4\)
• \(\alpha_i + \alpha_{i+1}\), \(i=1,2,3\).
• \(\alpha_1 + \alpha_2 +\alpha_3\), \(\alpha_2 + \alpha_3 + \alpha_4\)
• \(\sum_{i=1}^4 \alpha_i\).

Then \(n(P) = 10 - 3 = 7\).

18.3 Weyl’s Character Formula

Problem: Determine \(\operatorname{ch}H^0 \lambda\) for \(\lambda \in X(T)_+\).

Solution: Let \(A(\lambda) = \sum_{w\in W} \operatorname{sgn}(w) e^{w\lambda} \in {\mathbb{Z}}[X(T)]\), where we sum over the usual Weyl group and not the affine Weyl groups, taken as a formal sum in the group algebra on the weight lattice. We can then state Weyl’s character formula: \begin{align*} \operatorname{ch}H^0(\lambda) = {A(\lambda + \rho) \over A(\rho)} \qquad \text{for }\lambda \in X(T)_+ .\end{align*} This is a formal sum, so it’s surprising that the bottom term even divides the top. But there is a great deal of cancellation, we’ll see this in examples such as \(\operatorname{GL}_3\).

18.3.1 Formal Characters

Let \(M\) be a \(T{\hbox{-}}\)module, then define the character \begin{align*} \operatorname{ch}M\mathrel{\vcenter{:}}=\sum_{\mu\in X(T)} \qty{\dim M_\mu} e^\mu \quad \in {\mathbb{Z}}[X(T)] .\end{align*}

We then define the Euler characteristic \begin{align*} \chi(M) \mathrel{\vcenter{:}}=\sum_{i\geq 0} (-1)^i \operatorname{ch}H^i(M) .\end{align*} Note that by Grothendieck vanishing, \(H^i(M) = 0\) for \(i > {\left\lvert {\Phi^+} \right\rvert} = \dim(G/B)\), so this is a finite sum. In fact, if \(M\) is a \(G{\hbox{-}}\)module, then this is \(W{\hbox{-}}\)invariant and thus in fact \(\chi(M) \in {\mathbb{Z}}[X(T)]^W\).

19 Wednesday, October 14

Today:

• Weyl’s character formula

• Strong linkage

• Translation functors

Recall that we defined \begin{align*} \operatorname{ch}(M) &\mathrel{\vcenter{:}}=\sum_{\mu \in X(T)} \qty{\dim M_\mu} e^{\mu} \in {\mathbb{Z}}[X(T)]\\ \chi(M) &\mathrel{\vcenter{:}}=\sum_{i\geq 0} (-1)^i \operatorname{ch}H^i(M) \in {\mathbb{Z}}[X(T)]^W .\end{align*}

where \(H^i(M) = R^i \mathop{\mathrm{ind}}_B^G M\), and \(H^i(M) =0\) for \(i> G/B = {\left\lvert {\Phi^+} \right\rvert}\).

Note that the Euler characteristic is additive on SESs: if \(0\to A\to B\to C\to 0\) then \(\chi(B) = \chi(A) + \chi(B)\). It is also multiplicative wrt the tensor product: \(\chi(A\otimes B) \chi(A) \chi(B)\).

19.1 Weyl’s Character Formula

For any \(\alpha\in\Delta\) and \(\lambda \in X(T)\) with \({\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} \geq 0\). We have an analog of Serre duality: \begin{align*} \operatorname{ch}\mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = \operatorname{ch}R^i \mathop{\mathrm{ind}}_B^{P_\alpha} s_\alpha \cdot \lambda ,\end{align*} i.e. the induced module coincides with the Weyl module.

By definition of the dot action, we have \begin{align*} s_\alpha \cdot \lambda = s_\alpha(\lambda + \rho) - \rho .\end{align*}

As in previous calculations, we have

\begin{align*} {\left\langle {s_\alpha\cdot\lambda},~{\alpha {}^{ \vee }} \right\rangle} = -{\left\langle {\lambda+\rho},~{\alpha {}^{ \vee }} \right\rangle} - 1 \leq - 1 .\end{align*}

As in the analysis of Bott-Borel-Weil, we have \begin{align*} H^i(s_\alpha \cdot\lambda) &= H^i( R^1 \mathop{\mathrm{ind}}_B^{P_\alpha} s_\alpha\cdot\lambda ) \\ H^i(\lambda) &= H^i( \mathop{\mathrm{ind}}_B^{P_\alpha}\lambda ) ,\end{align*} since the spectral sequence collapses. Note that the two things appearing on the RHS have the same Euler characteristics.

We can thus define define a modified Euler characteristic \begin{align*} \phi(N) = \sum_{i\geq 0} (-1)^i \operatorname{ch}R^i \mathop{\mathrm{ind}}_{P_\alpha}^G(N) .\end{align*}

and obtain \(\chi(\lambda) = -\chi(s_\alpha \cdot \lambda)\). The same argument works for \({\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} < 0\).

Proof of 1: exercise.

Note: this says that one formal sum divides another.

A corollary is an analog of Weyl’s dimension formula: :::{.corollary title=“?”} Let \(\lambda \in X(T)_+\) be a dominant weight. Then \begin{align*} \operatorname{ch}H^0(\lambda) = { A(\lambda + \rho) \over A(\rho) } .\end{align*} :::

Big question: suppose \(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\). What are \(\operatorname{ch}L(\lambda)\) and \(\lambda \in X(T)_+\)? We know this for \(p\gg 0\), but in general it’s wide open. There are expressions in terms of “\(p{\hbox{-}}\)bases,” but these are hard to compute. There are only recursive formulas, none that are closed (and these may not exist).

Next time:

• Proof of Weyl’s character formula
• Compute an example.

Idea of the proof: we’ll have some \(\chi(\lambda) = \sum_\mu a_\mu e^\mu\). Well also have \(A(\rho) \qty{ \sum_\mu a_\mu e^\mu } = A(\lambda + \rho)\). This will reduce to equating coefficients of two formal sums, which will result in a system of linear equations.

20 Friday, October 16

20.1 Example: Weyl’s Character Formula

Review: suppose the following is invariant under the Weyl group, so \(\sum a_\mu e^\mu \in {\mathbb{Z}}[X(T)]^W\). In this case, we have an equality \begin{align*} \sum a_\mu e^\mu = \sum a_\mu \chi(\mu) ,\end{align*} where \(\chi(\mu) = \sum_{i\geq 0} (-1)^i \operatorname{ch}H^i(\mu)\). We also had a relation \begin{align*} \chi(w\cdot \mu) = (-1)^{\ell(w)} \chi(\mu) = \operatorname{sgn}(w) \chi(\mu) .\end{align*}

Now let \(\lambda \in X(T) \otimes{\mathbb{Q}}\), then we defined \begin{align*} A(\lambda) = \sum_{w\in W} \operatorname{sgn}(w) e^{w\lambda} \in {\mathbb{Z}}[X(T) \otimes{\mathbb{Q}}] .\end{align*}

We obtain

1. \(w' A(\lambda) = \operatorname{sgn}(w') A(\lambda)\)

2. \(A(\mu) A(\lambda) = {\mathbb{Z}}[X(T) \otimes{\mathbb{Q}}]^W\).

20.2 Strong Linkage Principle

We’ll consider representations in characteristic zero, so we can take \(k={\mathbb{C}}\). Let \(G\) bet a complex simple group, \({\mathfrak{g}}= \operatorname{Lie}(G)\), \(t\) a maximal torus, \(X\) the weights, and \(X_+\) the dominant weights. We have a correspondence \begin{align*} \left\{{\substack{(g, t)}}\right\} \iff \left\{{\substack{(\Phi, W)}}\right\} \end{align*}

where \(\Phi\) is an irreducible root system and \(W\) is the Weyl group. We’ll have a set of simple roots \(\Delta\subseteq \Phi^+\). For \(\lambda\in X\), we have \begin{align*} Z(\lambda) = U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}}^+)} \lambda \twoheadrightarrow L(\lambda) .\end{align*}

Then \(\lambda \in X_+ \iff L(\lambda)\) is finite dimensional. We have \(W\) acting on \(X\) via reflections, which we can extend to a dot action \begin{align*} w\cdot \lambda = w(\lambda + \rho) - \rho, \hspace{4em} \rho = {1\over 2}\sum_{\alpha\in\Phi^+} \alpha .\end{align*}

We define Category \({\mathcal{O}}\) which has objects \({\mathfrak{g}}{\hbox{-}}\)modules with a weight space decomposition which is locally finite wrt \({\mathfrak{n}}^+\).

20.2.1 Linkage in Category \({\mathcal{O}}\)

Set \(Z(\lambda) = \Delta(\lambda)\), then \begin{align*} [Z(\lambda) : L(\mu)] \neq 0 \implies \lambda \in W\cdot \mu .\end{align*} The LHS is computed by evaluating certain Kazhdan-Lusztig polynomials at \(x=1\).

Let \(\Phi= A_2\), then

\({\mathcal{O}}_0\) is the principal block, and the irreducibles correspond to \(\left\{{L(w\cdot 0) {~\mathrel{\Big|}~}w\in W}\right\}\), and the number of irreducibles in given by \({\left\lvert {W} \right\rvert}\). In this case, there is only 1 finite-dimensional module in any given block of category \({\mathcal{O}}\).

For \(\Phi = A_1\), we have the following situation:

In \({\mathcal{O}}_0\), there are two irreducible representations given by the Verma modules \(L(0), L(-2)\), and we find that

In this case, the projectives are given by

21 Monday, October 19

21.1 Representations in Positive Characteristic

We have the following setup: \begin{align*} G && \text{a semisimple, simply connected algebraic group} \\ k && \text{an algebraically closed field of characteristic $p>0$} \\ T && \text{a maximal torus} \\ B && \text{a Borel (negative roots)} \\ X(T) = X && \text{weights} \\ X(T)_+ = X_+ && \text{dominant weights} \\ \Phi && \text{roots} .\end{align*}

For \(\lambda \in X_+\), we consider the induced module \(H^0(\lambda) = \mathop{\mathrm{ind}}_B^G \lambda\). Not that this is not a simple module in general, so we instead ask about its composition factors.

Question: For all \(\lambda, \mu \in X_+\), what are the multiplicities \([H^0(\lambda): L(\mu)]\).

Let \(G = {\operatorname{SL}}_2(k)\), so \(\Phi = A_1\). Then \(\lambda \in X_+ = \left\{{0,1,2,\cdots}\right\}\) as we know from standard facts in lie algebras. Define \(X_1 = \left\{{0, 1, \cdots, p-1}\right\}\), then \(\dim H^0(\lambda) = \lambda + 1\). We can write the weight \(p{\hbox{-}}\)adically as \(\lambda = \sum_{i=0}^t \lambda_i p^i\) for some \(\lambda_j\in X_1\). Thus \(L(\lambda) = L(\lambda_0) \bigotimes_{i=1}^t L(\lambda_i)^{(i)}\).

Consider \(p=3, \lambda = 7\), then \(\dim H^0(7) = 8\). We can write \(7\) 3-adically as \(7 = (1)3^0 + (2)3^1\), and so \begin{align*} L(7) \cong L(1) \otimes L(2)^{(1)} .\end{align*} The first summand is 2-dimensional, and the second is 3-dimensional, so \(L(7)\) is 6-dimensional. Note that \(L(7) \hookrightarrow H^0(7)\).

We can calculate the weights in the tensor product: the first has weights \(\left\{{\pm 1}\right\}\), we take the adjoint weights in the second factor and multiply by the twist 3 to get \(\left\{{2\cdot 3, 0\cdot 3, -2\cdot 3}\right\}\). Taking all combinations of sums from these yields \(\left\{{7,5,1,-1,-5,-7}\right\}\).

Since \(\pm 3\) are left over, we know \([H^0(7): L(3)] \neq 0\). We can continue with \(3 = (1)3^1\) and write \(L(3) = L(1)^{(1)}\). We get weights of the form \(1\cdot 3, 1\cdot -3\), so nothing is left over and we’re done. We thus get a decomposition

Note the difference to Verma modules in category \({\mathcal{O}}\): we have to consider the action of the affine Weyl group, where \(W_a \mathrel{\vcenter{:}}= W \rtimes p{\mathbb{Z}}\Phi\). Here we have hyperplanes at \(p-1, 2p-1, 3p-1\), and 7 is linked to 3 (in the same orbit) for this action:

Once characters are known, can find composition factors.

21.2 Affine Weyl Group

Letting \(a\in {\mathbb{N}}\), we have \(W_a = W\rtimes a({\mathbb{Z}}\Phi)\) where \({\mathbb{Z}}\Phi\) is the root lattice. Note that there are other variants:

• \(W_a = W\rtimes a({\mathbb{Z}}\Phi {}^{ \vee })\),
• \(W_{\text{ext}} = W \rtimes X(T)\).

So we set \(W_p = W\rtimes p({\mathbb{Z}}\Phi)\) where \(p\) is a prime. What’s in this group? We know it contains “products” of reflections with translations. We find that \(W_p\) is generated by \begin{align*} s_{\beta, np}(\lambda) = \lambda - {\left\langle {\lambda},~{\beta {}^{ \vee }} \right\rangle}\beta + np \beta .\end{align*}

It is also the case that \(W_p\) acts on \(X(T)\) and there exists a dot action \begin{align*} w\cdot \lambda = w(\lambda + \rho) - \rho .\end{align*}

Consider \(A_2\). We obtain “alcoves”:

We can get a stronger version of weak linkage, which we’ll just call linkage:

21.2.1 Ordering of Weights

There is a partial ordering on the weight lattice given by \begin{align*} \mu \leq \lambda \iff \lambda - \mu = \sum_{\alpha\in \Phi^+} n_\alpha \alpha, \quad n_\alpha \geq 0 .\end{align*}

Next time: history of strong linkage, and translation functors.

22 Wednesday, October 21

22.1 Strong Linkage

Let \(G\) be a semisimple algebraic group and \(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}_p\mkern-1.5mu}\mkern 1.5mu\). We found that the affine Weyl group \(W_p\) played an important role here.

22.2 Translation Functors

Consider the case from category \({\mathcal{O}}\), e.g. by taking \({\mathfrak{g}}= {\mathfrak{sl}}_3({\mathbb{C}})\):

For \(\lambda\) a regular weight, the principal block \(\mathcal{B}_0\) is Morita-equivalent to \(\mathcal{B}_\lambda\). If \(\mu\) is a singular weight, then by Jantzen there are translation functors

\begin{align*} T_\lambda^\mu: \mathcal{B}_\lambda &\to \mathcal{B}_\mu \\ T_\mu^\lambda: \mathcal{B}_\mu &\to \mathcal{B}_\lambda .\end{align*}

In the case where \(G\) is a semisimple algebraic group and \(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\), we have the following picture instead:

22.2.1 Blocks

Two simple modules \(S, T\) are in the same block if we have a sequence \(T_1, \cdots, T_n\) such that \(S=T_1\) and \(T_n = T\) where \(\operatorname{Ext} ^1(T_i, T_{i+1}) \neq 0\).

So the question becomes, what are the blocks of \(H\)? Let \(\lambda \in X(T)_+\), so we can define \(L(\lambda)\), and let \(b(\lambda)\) be the \(G{\hbox{-}}\)block containing \(L(\lambda)\).

We have \(b(\lambda) \in \mathcal{B}(G)\) and \(b(\lambda)\) \(\subseteq X(T)_+ \cap W_p \cdot \lambda\), i.e. we have strong linkage.

Here we refer to \(b(\lambda)\) as both the block and the weights it contains.

Recall that \begin{align*} \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\mathrel{\vcenter{:}}=\left\{{ \lambda \in X(T) {~\mathrel{\Big|}~} 0 \leq {\left\langle {\lambda + \rho},~{\beta {}^{ \vee }} \right\rangle} \leq p \,\, \forall \beta\in\Phi^+ }\right\} .\end{align*} For every \(\mu, \lambda \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\), consider \(\mu - \lambda \in X(T)\). Then there is a way to conjugate it under the ordinary \(W\) action to land in the dominant region, i.e. some unique \(\nu\) such that \(\nu \in X(T)_+ \cap W(\mu - \lambda)\).

Next time: we’ll show that \(T_\lambda^\mu\) and \(T_\mu^\lambda\) form an adjoint pair. Note that if \(\mu, \lambda\) are in the same block, these are the exact functor which product the categorical equivalence.

23 Friday, October 23

23.1 Facets

\(W_p\) has a dot action on \(E \mathrel{\vcenter{:}}= X(T) \otimes_{\mathbb{Z}}{\mathbb{R}}\).

We can write \(\Phi^+ = \Phi_0^+ \cup\Phi_1^+\), and define the facet as \begin{align*} F \mathrel{\vcenter{:}}=\left\{{ \lambda \in E {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} = n_\alpha p\,\, \forall\alpha\in \Phi_0^+(F),\,\, (n_\alpha - 1)p < {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} < n_\alpha p \,\,\forall \alpha\in \Phi_1^+(F) }\right\} .\end{align*} The first condition corresponds to being on a vertex in the following diagram, while the second corresponds to being in the interior of a triangle:

23.2 Translation Functors

Let \(\lambda, \mu\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\), and define \begin{align*} T_\lambda^\mu({-}) \mathrel{\vcenter{:}}= {\operatorname{pr}}_\mu \qty{ L(\nu_1) \otimes{\operatorname{pr}}_\lambda({-}) }\\ \text{where } \nu_1 \in X(T)_+ \cap W(\mu-\lambda) .\end{align*}

This is exact as a composition of exact functors, since we’re tensoring over a field and taking projections (which are themselves exact).

23.3 Technical Preliminaries

1. If \(\lambda \in X(T)\) and \begin{align*} \sum_\mu a(\mu) e^\mu \in {\mathbb{Z}}[X(T)]^W \end{align*} is \(W{\hbox{-}}\)invariant, then we proved that \begin{align*} \chi(\lambda) \qty{ \sum_\mu a(\mu) e^{\mu} } = \sum_\mu a(\mu) \chi(\lambda + \mu) .\end{align*}

2. If \({\operatorname{pr}}_\lambda V = V\), then we have \begin{align*} \operatorname{ch}(M\otimes V) &= \operatorname{ch}(M) \operatorname{ch}(V) \\ &= \operatorname{ch}(M) \qty{\sum_{w\in W_p} a_w \chi(w\cdot\lambda) } \\ &= \qty{ \sum_{\nu \in X(T)} \dim M_\nu e^\nu } \qty{\sum_{w\in W_p} a_w \chi(w\cdot\lambda) } .\end{align*}

Given \(\operatorname{ch}V\), one can write \(\operatorname{ch}T_\lambda^\mu V\). What will be important here are stabilizers. If \(\lambda\) is on a wall, the stabilizer fixes the corresponding hyperplane.

24 Monday, October 26

24.1 Review

Let \(V\) be a finite dimensional \(G{\hbox{-}}\)module with \({\operatorname{pr}}_\lambda V = V\), and write \(\operatorname{ch}(V) = \sum_{w\in W_p} a_w \chi(w\cdot \lambda)\) where \(a_w\in {\mathbb{Z}}\) and only finitely many are nonzero. We can then write \begin{align*} \operatorname{ch}\qty{{\operatorname{pr}}_\mu(M\otimes V)} = \sum_{w\in W_p} a_w \qty{\sum_{\substack{\nu \in X(T) \\ \lambda+\nu \in W_p\cdot \mu} } \dim M_\nu \, \chi(w\cdot(\lambda + \nu)) } ,\end{align*} where we sum over all weights linked to \(\mu\)

24.2 Characters of Translated Modules

Goal: find \(\operatorname{ch}T_\lambda^\mu V\).

We can write \begin{align*} \operatorname{ch}(T_\lambda^\mu V) &= \operatorname{ch}\qty{{\operatorname{pr}}_\mu\qty{L(\nu_1) \otimes{\operatorname{pr}}_\lambda V } } \\ &= \operatorname{ch}\qty{{\operatorname{pr}}_\mu\qty{L(\nu_1) \otimes V } } \\ &= \sum_{w\in W_p} a_w \qty{\sum_\nu \dim L(\nu_1)_\nu \chi(w\cdot(\lambda + \nu)) }\\ &= \sum_{w\in W_p} a_w \qty{ \sum_\nu \dim L(\nu_1)_\nu \chi(w\cdot\lambda + \nu) } .\end{align*}

We need \(\lambda + \nu\in W_p\cdot \mu\) and \(\nu \leq \nu_1\) to apply the technical lemma.

The last step can be written because the only contributions are \(\nu \in W\nu_1\) and \(\dim L(\nu_1)_\nu = 1\), i.e. we’re on the outer shell in the figure above.

We can apply (a) and (b) from the technical lemma to write \begin{align*} \cdots &= \sum_{w\in W_p} a_w \sum_{w_1} \chi(w(w_1\cdot \mu)) .\end{align*}

By (b), \(w_1 \in {\operatorname{Stab}}_{W_p}(\lambda)\). We don’t want duplication, so we can check that \(w_1\cdot\mu = w_2 \cdot\mu = \lambda+ w\nu_1\) implies that \(w_1 \in w_2 {\operatorname{Stab}}_{W_p}(\mu)\). Thus we need to take the coset representatives stated in the theorem.

Thus we don’t need to consider any weights in the inner shell.

24.3 Equivalence of Categories

Goal: show that a pair of functors each admit a natural transformation to the identity.

Using the adjointness of \(T_\lambda^\mu\) and \(T_\mu^\lambda\)< we can write \begin{align*} \hom_G(V, T_\mu^\lambda T_\lambda^\mu V) \equiv \hom_G(T_\lambda^\mu V, T_\lambda^\mu V) .\end{align*}

So consider the identity map on the latter \(\operatorname{id}: T_\lambda^\mu V{\circlearrowleft}\), and let \(f_V: V\to T_\mu^\lambda T_\lambda^\mu V\) be the corresponding map in the former. We can a natural transformation in the following way

It suffices to show that the \(f_V\) are isomorphism as maps of \(G{\hbox{-}}\)modules, so one proceeds by

• Showing it works for simple \(G{\hbox{-}}\)modules, and

• Applying induction to composition length, using the five lemma.

Suppose \(V\) is simple, then by the prior theorem we can write \begin{align*} \operatorname{ch}T_\mu^\lambda T_\lambda^\mu V = \sum_{w\in W_p} a_w \qty{\sum_{w_1, w_2} \chi\qty{w(w_2 w_1)\cdot \lambda } } .\end{align*} We know that

• \(w_1\in {\operatorname{Stab}}_{W_p}(\mu) / \sim\)
• \(w_2\in {\operatorname{Stab}}_{W_p}(\lambda) / \sim\)

Note that if \(\mu, \lambda\) are in the same facet, then the stabilizers are the same.

So \(\lambda, \mu\) are in the same facet, so \(w_1 = \operatorname{id}, w_2 = \operatorname{id}\) and \(f_V\) is an isomorphism. We thus obtain \(T_\mu^\lambda T_\lambda^\mu =\cong {\operatorname{pr}}_\lambda\) and \(T_\lambda^\mu T_\mu^\lambda =\cong {\operatorname{pr}}_\mu\). Thus \(B_\lambda \cong B_\mu\).

Question: What happens when translating from an alcove onto a wall? A similar formula will hold in this case: we will get either induced modules or zero, depending on the dominance of the weights. This will lead into the Lusztig conjectures.

25 Wednesday, October 28

25.1 Review of Last Time

Suppose we have two weights in the same facet, i.e. they’re in the same stabilizer under the action of the affine Weyl group:

We had a theorem: if \(\lambda, \mu\) are in the same facet, then \(\mathcal{B}_\lambda \cong \mathcal{B}_\mu\) is an equivalence of categories, where the map is via the translation functors.

25.2 Description of \(T_\lambda^\mu {H^i(w\cdot \lambda) }\)

We can write \begin{align*} T_\lambda^\mu \qty{H^i(w\cdot \lambda)} &= {\operatorname{pr}}_\mu \qty{L(\nu_1) \otimes{\operatorname{pr}}_\lambda\qty{H^i(w\cdot \lambda)} } \\ &= {\operatorname{pr}}_\mu \qty{L(\nu_1) \otimes{H^i(w\cdot \lambda)} } \\ &= {\operatorname{pr}}_\mu \qty{L(\nu_1) \otimes{R^i \mathop{\mathrm{ind}}_B^G w\cdot \lambda} } \\ &= {\operatorname{pr}}_\mu\qty{R^i \mathop{\mathrm{ind}}_B^G \qty{L(\nu_1) \otimes w\cdot \lambda } } .\end{align*}

Take a composition series by \(B{\hbox{-}}\)modules of \(L(\nu_1) \otimes w\cdot \lambda\), say \begin{align*} 0 = M_0 \subseteq M_1 \cdots \subseteq M_r = L(\nu_1) \otimes w\cdot \lambda .\end{align*} where \(M_j / M_{j-1} \cong \lambda+j + w\cdot \lambda\) and \(\lambda_j < \lambda_{j'} \implies j < j'\), i.e. we can order them in a decreasing way.

Consider the SES

where applying \({\operatorname{pr}}_\mu({-})\) induces the LES

We know that \begin{align*} {\operatorname{pr}}_\mu H^i\qty{\lambda_j + w\cdot \lambda} = \begin{cases} H^i(\lambda_j + w\cdot \lambda ) & \lambda+j + w\cdot \lambda \in W_p\cdot \mu \\ 0 & \text{else} \end{cases} ,\end{align*} i.e. this projection is the identity for weights linked to \(\mu\) and zero otherwise. We also have \begin{align*} {\operatorname{pr}}_\mu H^i(M_r) = T_\lambda^\mu H^i(w\cdot \lambda) .\end{align*}

Here consider \(H_0(\lambda) \xrightarrow{T_\lambda^\mu} H_0(\mu) = 0\), since \(\mu\) is outside of the dominant region (in orange.) We also have \(H^0(w\cdot \lambda) \to H^0(w\cdot \mu) \neq 0\), since this falls into the dominant region.

Suppose \(\lambda \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\) and \(\mu \in C_{\mathbb{Z}}\). What happens when you translate \(\lambda\) (blue) off of a wall? \(T_\lambda^\mu\qty{H^0(w\cdot \lambda)}\) has a filtration with factors \(H^0(w_1\cdot \mu)\) and \(H^0(w_2\cdot \mu)\) (shown in green).

If \(w\lambda\) is a vertex with \(\mu \in C_{\mathbb{Z}}\), then \(T_\lambda^\mu(H^0(w\cdot \lambda))\) has six factors:

Recall that \(\widehat{F}\) denotes the upper closure.

In this situation, we have \(T_\lambda^\mu(L(\lambda)) = 0\):

If instead \(\mu \in \widehat{C}_{\mathbb{Z}}\), we have \(T_\lambda^\mu( L(\lambda)) = L(\mu)\):

Big Question: What happens to \(L(w\cdot \lambda)\) when translating away from a wall?

A facet \(F\) is a wall \(\iff {\left\lvert { \Phi_0^+(F) } \right\rvert} = 1\). In this case, there exists a unique \(\alpha\) such that \({\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} = n_\alpha p\).

Note that \(s_F = s_{\beta, n_p}\) where \(n_p = {\left\langle {\lambda + \rho},~{\beta {}^{ \vee }} \right\rangle}\) acts on the wall as the identity and reflects across it:

Here \({\operatorname{Stab}}_{W_p}(\lambda) = \left\{{1, s_F}\right\}\).

Consider the following situation:

1. \([T_\mu^\lambda (L(w\cdot \mu)) : L(w\cdot \lambda)] = 2\), appearing once in the socle and once in the head.

2. \(L(w\cdot \lambda) = \mathop{\mathrm{Soc}}_G T_\mu^\lambda(L(w\cdot \mu)) = T_\mu^\lambda(L(w\cdot \mu)) / \mathop{\mathrm{Rad}}T_\mu^\lambda (L(w\cdot \mu))\).

Big Problems:

1. When is the heart semisimple?

2. Determine the composition factors in the heart?

Given these, you could compute dimensions of irreducible representations.

26 Monday, November 02

Today: Lusztig conjectures, but first some alcove geometry.

26.1 Alcove Geometry

26.1.1 Length Function for Alcoves

Let \(A, B\) be alcoves, and recall that a hyperplane is give by \begin{align*} H_{\alpha, m} \mathrel{\vcenter{:}}=\left\{{ \lambda \in X(T) {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} = mp }\right\} .\end{align*}

These are codimension 1 objects:

We can also divide these into positive and negative sides:

\begin{align*} H_{\alpha, m}^+ &\mathrel{\vcenter{:}}=\left\{{ \lambda \in X(T) {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} > mp }\right\} \\ H_{\alpha, m}^- &\mathrel{\vcenter{:}}=\left\{{ \lambda \in X(T) {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} < mp }\right\} .\end{align*}

Let \(S(A, B)\) be the set of hyperplanes separating \(A\) and \(B\).

If \(H\in S(A, B)\), then define a function \begin{align*} {\varepsilon}(H) \mathrel{\vcenter{:}}= \begin{cases} 1 & A\in H_{\alpha, m}^- \\ -1 & A\in H_{\alpha, m}^+ \end{cases} ,\end{align*} and from it construct a distance function \begin{align*} d(A, B) \mathrel{\vcenter{:}}=\sum_{H\in S(A, B)} {\varepsilon}(H) .\end{align*}

Recall that we can define \begin{align*} C_{\mathbb{Z}}= \left\{{\lambda \in X(T) {~\mathrel{\Big|}~}0 < {\left\langle {\lambda+\rho},~{\alpha {}^{ \vee }} \right\rangle}

and for \(\lambda \in C_{\mathbb{Z}}\)

\begin{align*} d(w\cdot \lambda) = d(C_{\mathbb{Z}}, wC_{\mathbb{Z}}) .\end{align*}

for \(w\in W_p\).

Now consider the following situation:

Recall that computing \(\operatorname{ch}L(\lambda)\) is equivalent to finding \([H^0(w\cdot \lambda): L(w_2\cdot \lambda)]\). We know that \begin{align*} [T_\mu^\lambda (L(w\cdot\mu)) : L(w\cdot \mu)] = 2 ,\end{align*} where the \(w\)s are now the same, and the structure of the module is

where knowing the structure of the heart of the module is an open problem.

The setup is the following: Let \(\lambda\in X(T) \cap C_{\mathbb{Z}}\) where \(\mu \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\). Since it’s on a wall, there is a nontrivial stabilizer \({\operatorname{Stab}}_{W_p}(\mu) = \left\langle{s}\right\rangle\). We can write \begin{align*} \operatorname{ch}L(w\cdot \lambda) = \sum_{w'} a_{w, w'} \chi(w' \cdot \lambda) .\end{align*}

The sum is over \(w'\in W_p\) where \(w'\cdot \lambda \in X(T)_+\), and we have strong linkage \(w'\cdot \lambda \uparrow w\cdot \lambda\). Now consider translating from \(\lambda\) to the wall \(\mu\). We can write \begin{align*} \operatorname{ch}L(w\cdot \lambda) = \sum_{w'} a_{w, w'} \chi(w' \cdot \mu) .\end{align*}

Since \(\left\langle{s}\right\rangle\) is the stabilizer, we have \begin{align*} \operatorname{ch}T_\mu^\lambda L(w\cdot \mu) &= \sum_{w'} a_{w, w'} \chi(w' \cdot \lambda) + \sum_{w'} a_{w, w'} \chi(w's \cdot \mu) \\ &= \operatorname{ch}L(w\cdot \lambda) + \sum_{w'} a_{w, w'} \chi(w's \cdot \mu) ,\end{align*}

We can now check that \begin{align*} \operatorname{ch}T_\mu^\lambda T_\lambda^\mu L(w\cdot \lambda) = \operatorname{ch}L(w\cdot \lambda) + \chi(ws\cdot \lambda ) + \sum_{w'} b_{w'} \chi(w'\cdot\lambda) .\end{align*}

Note that in the last sum, the \(b_{w'}\) with \(w'\in W_p\) satisfy \(w'\cdot \lambda \in X(T)_+\) and \(d(w'\cdot\lambda) < d(w\cdot \lambda)\). We would like to compute \begin{align*} [H^0(w_1\cdot \lambda) : L(w_2\cdot \lambda) .\end{align*} using induction on \(d(w_1\cdot \lambda)\).

Suppose that for all \(w_1\) with \(d(w_1\cdot\lambda) < d(w\cdot \lambda)\), we know all of the \(b_{w'}\). In this case, we know \(\chi(ws\cdot \lambda)\) by Weyl’s character formula. We’d then only need to know the translated character \begin{align*} T_\mu^\lambda T_\lambda^\mu L(w\cdot \lambda) .\end{align*} Note that it’s sufficient to know all of its composition factors \begin{align*} [T_\mu^\lambda T_\lambda^\mu L(w\cdot \lambda): L(w_2\cdot \lambda) ,\end{align*}

and since we know the head and the socle, it suffices to understand the composition factors in its heart:

If we know \(\operatorname{ch}L(\lambda)\) for \(\lambda \in X_1(T)\) in the restricted region, by Steinberg’s twisted tensor product formula we can compute \(L(\mu)\) for \(\mu \in X(T)_+\). So when is the restricted region contained in the Jantzen region, so \(X_1(T) \subseteq \operatorname{Jan}\)? Assume that \(\lambda \in X_1(T)\), then compute \begin{align*} {\left\langle {\lambda+\rho},~{\alpha_0 {}^{ \vee }} \right\rangle} = {\left\langle {\lambda},~{\alpha_0 {}^{ \vee }} \right\rangle} + (h-1) .\end{align*} Note that by extending scalars to \({\mathbb{Q}}\), we get an inequality in the ordering of the form \(\lambda \leq_{\mathbb{Q}}(p-1_\rho)\). We can then write

\begin{align*} {\left\langle {\lambda+\rho},~{\alpha_0 {}^{ \vee }} \right\rangle} &= {\left\langle {\lambda},~{\alpha_0 {}^{ \vee }} \right\rangle} + (h-1) \\ \leq {\left\langle {(p-1) \rho},~{\alpha_0 {}^{ \vee }} \right\rangle} + (h-1) \\ &= (p-1)(h-1) + (h-1) \\ &= p(h-1) .\end{align*} When is this less than or equal to \(p(p-h-2)\)? We can check that this happens iff \begin{align*} p(h-1) &\leq p(p-h-2) \\ \iff h-1 &\leq p-h-2 \\ \iff 2h+1 &\leq p ,\end{align*} so the conjecture (as stated in its original form) reads \begin{align*} p\geq 2h+1 \implies X_1(T) \subseteq \operatorname{Jan} .\end{align*}

Next time: we’ll define \(G_rT{\hbox{-}}\)modules. We’ll define a graded category to prevent the weights from collapsing mod \(p\). These turn out to be easier to work with than \(G_r{\hbox{-}}\)modules, and results can be pushed down.

27 Wednesday, November 04

Today: \(G_r{\hbox{-}}T\) modules.

Note that \(G_r{~\trianglelefteq~}G_r T\), with \(G_r T/G_r \cong T^{(r)}\). We consider \(G_r T{\hbox{-}}\)modules, which are \(G_r{\hbox{-}}\)modules with a \(T\) action given by \begin{align*} G_r \times M &\to M \\ (g, m) &\mapsto g\cdot m \end{align*} which are \(T{\hbox{-}}\)equivariant, i.e. \(t(g\cdot m) = (t\cdot g)(t\cdot m)\) for \(t\in T, g\in G_r\), and \(m\in M\) is a \(G_r T{\hbox{-}}\)module. This essentially induces a grading on \(G_r\).

27.1 Representations for \(G_r T\) and \(G_r B\)

Recall that we have a Frobenius map, for which we take the scheme-theoretic kernel: \begin{align*} F&: G \to G \\ F^r &\mathrel{\vcenter{:}}= F\circ F \circ \cdots \circ F \\ G_r &\mathrel{\vcenter{:}}=\ker F^r ,\end{align*} and we then define \begin{align*} G_r T \mathrel{\vcenter{:}}=(F^r)^{-1} (T) \\ G_r B \mathrel{\vcenter{:}}=(F^r)^{-1} (B) \end{align*} taken as scheme-theoretic objects.

Noting that \(B\subset G_r B\), for \(\lambda \in X(T)\) we define \begin{align*} \widehat{Z}_r' (\lambda) &\mathrel{\vcenter{:}}=\mathop{\mathrm{ind}}_B^{G_r B} \lambda \\ \widehat{Z}_r (\lambda) &\mathrel{\vcenter{:}}=\mathop{\mathrm{coInd}}_B^{G_r B} \lambda .\end{align*}

These are enhancements of the baby Verma modules, in the sense that if we take restrictions we get \begin{align*} \widehat{Z}_r' (\lambda) \downarrow_{G_r} = \mathop{\mathrm{ind}}_{B_r}^{G_r} \lambda .\end{align*}

We similarly have \begin{align*} {Z}_r' (\lambda) \downarrow_{G_r T} &= \mathop{\mathrm{ind}}_{B_r T}^{G_r T} \lambda \\ \widehat{Z}_r' (\lambda) \downarrow_{G_r T} &= \mathop{\mathrm{coInd}}_{B_r T}^{G_r T} \lambda .\end{align*}

The next theorem is related to the fact that when comparing these categories of modules, one is essentially a graded version of the other.

Note that \(G_r {~\trianglelefteq~}G_r T\), where the quotient is \(T^{(r)}\) which is twisted by Frobenius \(r\) times.

27.2 Summary: Classification of Simple \(G_r T{\hbox{-}}\)Modules

• \(\mathop{\mathrm{Soc}}_{B_r^+} \widehat{Z}_r '(\lambda) = \lambda\)
• \(\widehat{Z}_r ' (\lambda) ^{U^+} = \lambda\), where the RHS denotes the \(U^+\) invariants.
• Let \(\widehat{L}_r(\lambda) \mathrel{\vcenter{:}}=\mathop{\mathrm{Soc}}_{G_r T} \widehat{Z}_r' (\lambda)\).
• Each simple \(G_r T{\hbox{-}}\)module is isomorphic to \(\widehat{L}_r (\lambda)\) for some \(\lambda\in X(T)\).
• \(\widehat{L}_r(\lambda) \downarrow_{G_r} \cong L_r (\lambda)\) for all \(\lambda \in X_1(T)\).
• Translation invariance: \(\widehat{L}_r(\lambda + p^r\sigma) = \widehat{L}_r (\lambda) \otimes p^r\sigma\)
• \(\widehat{L}_r (\lambda + p^r \sigma) \downarrow_{G_r} = L_r(\lambda)\) for all \(\lambda \in X_r(T)\).

This essentially allows you to replace working mod \(p\) in characteristic \(p\) with working with integers instead, allowing the usual weight theory to be used.

28 Friday, November 06

28.1 Good \((p, r){\hbox{-}}\) Filtrations

Last time: \(G_r T\) and \(G_r B\) modules. We roughly know the category of \(G_r\) modules, and we think of \(G_r T\) as graded \(G_r{\hbox{-}}\)modules. We defined \begin{align*} \widehat{Z}_r' (\lambda) &\mathrel{\vcenter{:}}=\mathop{\mathrm{ind}}_{B}^{G_r B}(\lambda) \\ \widehat{Z}_r' (\lambda) &\mathrel{\vcenter{:}}=\mathop{\mathrm{coInd}}_{B^+}^{G_r B^+}(\lambda) .\end{align*}

We can use these for classification since we have a correspondence \begin{align*} \left\{{\substack{\text{Simple }G_rT{\hbox{-}}\text{modules}}}\right\} &\iff \left\{{\substack{X(T)}}\right\} \\ \widehat{L}_r(\lambda) = \widehat{L}_r(\lambda_0) \otimes p^r \lambda &\mapsfrom \lambda = \lambda_0 + p^r \lambda_1 ,\end{align*} where \(\widehat{L}_r(\lambda_0)\) is a simple \(G_r{\hbox{-}}\)module and \(\lambda_0 \in X_r(T)\).

Recall Kempf’s vanishing theorem: if \(\lambda \in X(T)_+\) is a dominant weight, then \(H^{>0}(\lambda) = 0\).

This question was open for a while, until the following was found:

Later: we’ll see how to construct these filtrations by factoring induction into intermediate inductions.

Suppose \(\widehat{L}_r(\mu_0 + p^r\mu_1)\) is a composition factor of \(\widehat{Z}_r'\). Then since we have \(G{\hbox{-}}\)modules, we can use the tensor identity to write \begin{align*} R^i \mathop{\mathrm{ind}}_{G_r B}^{G} L_r(\mu_0) \otimes p^r \mu_1 &= L_r(\mu_0) \otimes R^i \mathop{\mathrm{ind}}_{G_r B}^{G} p^r \mu_1 \\ &= L_r(\mu_0) \otimes H^i(\mu_1)^{(r)} ,\end{align*} where the last equality follows from a theorem we won’t prove here. We can set \(i=0\) to yield \begin{align*} \mathop{\mathrm{ind}}_{G_r B}^{G} L_r(\mu_0) \otimes p^r \mu_1 \cong L_r(\mu_0) \otimes H^0(\mu_1)^{(r)} .\end{align*} Recall that \(H^0(\lambda) = \mathop{\mathrm{ind}}_{G_r B}^{G} \widehat{Z}_r'(\lambda)\), so we’ll take a composition series for \(\widehat{Z}_r'(\lambda)\) and apply the induction functor to it. So let such a composition series be given by \begin{align*} 0\subseteq N_0 \subseteq N_1 \subseteq \cdots \subseteq N_s = \widehat{Z}_r'(\lambda) ,\end{align*} where \(N_i / N_{i-1} \cong L(\mu_0) \otimes p^r \mu_1\) for some \(\mu_0\in X_r(T)\) and \(\mu_1\in X(T)\). Now apply the functor \(\mathop{\mathrm{ind}}_{G_r B}^{G}({-})\) which yields \begin{align*} 0\subseteq \cdots \subset \mathop{\mathrm{ind}}_{G_r B}^{G} N_i \subseteq \cdots \subseteq H^0(\lambda) .\end{align*}

Question: is this a good \((p, r)\) filtration?

Note that if we have \begin{align*} 0 \to N_1 \to N_2 \to N_2/N_1 \to 0 \end{align*} this yields \begin{align*} 0 \to \mathop{\mathrm{ind}}N_1 \to \mathop{\mathrm{ind}}N_2 \to \mathop{\mathrm{ind}}(N_2/N_1) \to R^1 \mathop{\mathrm{ind}}N_1 \to \cdots .\end{align*}

Here we need \(\mathop{\mathrm{ind}}(N_2/N_1) \cong \mathop{\mathrm{ind}}N_2 / \mathop{\mathrm{ind}}N_1\), so we need to show \(R^1 \mathop{\mathrm{ind}}N_1 = 0\).

Using the tensor identity we can write \begin{align*} R^1 \mathop{\mathrm{ind}}N_1 &= R^1 \mathop{\mathrm{ind}}_{G_r B}^G L_r(\sigma_0) \otimes p^r \sigma_0 \\ &= L_r(\sigma_0) \otimes\qty{R^1 \mathop{\mathrm{ind}}_{G_r B}^G \sigma_1 }^{(r)} \end{align*} and \({\left\langle {\sigma_1 + \rho},~{\beta {}^{ \vee }} \right\rangle} \geq 0\), so \(R^1 \mathop{\mathrm{ind}}_{G_r B}^G \sigma_1 = 0\). Thus we can extend the region from Kempf’s vanishing slightly:

Finding composition factors for the \(\widehat{Z}_r'\) is in general a hard problem: if we had this, we’d have the characters of the irreducibles. Some combinatorics can be used here.

28.2 Strong Linkage

Note that strong linkage for \(H^0(\lambda)\) implies strong linkage for \(\widehat{Z}_r'(\lambda)\).

Next time: extensions in \({\mathsf{G_r T}{\hbox{-}}\mathsf{Mod}}\) and the Steinberg module (very important in representation theory, has some nice properties).

29 Monday, November 09

29.1 Strong Linkage

We have two categories:

• \(G_r T\), with a notion of strong linkage, and
• \(G_r\), which instead only has linkage.

We’ll restate a few theorems.

In the case of \(\Phi = A_2\), we’ll consider the two different categories.

We have the following picture for \(\widehat{Z}\):

Considering \(X_1(T)\) and \([\widehat{Z}_1(\lambda) : \widehat{L}_1(\mu)] \neq 0\), and \(\widehat{Z}_1(\lambda)\) has 6 composition factors as \(G_1T{\hbox{-}}\)modules.

On the other hand, for \(Z\), we have the following:

This again has 6 composition factors, obtained by ??

29.2 Extensions

Let \(\lambda, \mu \in X(T)\). We can use the Chevalley anti-automorphism (essentially the transpose) to obtain a form of duality for extensions: \begin{align*} \operatorname{Ext} _{G_r T}^j \qty{ \widehat{L}_r(\lambda), \widehat{L}_r(\mu) } = \operatorname{Ext} _{G_r}^j \qty{ \widehat{L}_r(\mu), \widehat{L}_r(\lambda) } \qquad \text{for }j\geq 0 .\end{align*}

We have a form of a weight space decomposition \begin{align*} \operatorname{Ext} _{G_r}^j \qty{L_r(\lambda), L_r(\mu) } = \bigoplus_{\gamma \in X(T)} \operatorname{Ext} _{G_r}^j \qty{L_r(\lambda), L_r(\mu) }_{\gamma} \end{align*} where we are taking the fixed points under the torus \(T\) action on the first factor (for which \(T_r\) acts trivially). We can write this as \begin{align*} \cdots &= \bigoplus_{\gamma \in X(T)} \operatorname{Ext} _{G_r}^j \qty{L_r(\lambda), L_r(\mu) \otimes\gamma } \\ &= \bigoplus_{\gamma \in X(T)} \operatorname{Ext} _{G_rT}^j \qty{L_r(\lambda), L_r(\mu) \otimes p^r v } \\ &= \bigoplus_{v \in X(T)} \operatorname{Ext} _{G_rT}^j \qty{ \widehat{L}_r(\lambda), \widehat{L}_r(\mu + p^r v) } .\end{align*}

So if we know extensions in the \(G_r\) category, we know them in the \(G_r T\) category.

There is an isomorphism \begin{align*} \operatorname{Ext} _{G_r T}^1 \qty{ \widehat{L}_r(\lambda), \widehat{L}_r(\mu) } \cong \mathop{\mathrm{Hom}}_{G_R T}\qty{ \mathop{\mathrm{Rad}}_{G_r T} \widehat{Z}_r(\lambda), \widehat{L}_r(\mu) } .\end{align*}

Finally, for \(\lambda, \mu \in X(T)\), if the above \(\operatorname{Ext} ^1\) vanishes, then \(\lambda \in W_p \cdot \mu\) (i.e. \(\lambda\) and \(\mu\) are linked).

29.3 The Steinberg Modules

Consider \(A_2\):

Taking the representation corresponding to \((p-1, p-1)\) yields the “first Steinberg module” \begin{align*} L(p-1, p-1) = L((p-1)\rho) \mathrel{\vcenter{:}}=\operatorname{St}_1 .\end{align*}

In this case, we have an equality of many modules:

\begin{align*} H^0((p-1) \rho) = L((p-1) \rho) = V((p-1) \rho) = T((p-1) \rho) .\end{align*}

Main difference to category \({\mathcal{O}}\): infinitely many highest weight representations?

Upcoming:

• Viewing the \(G_r T\) category as “almost” a highest weight category
• Defining standard and costandard modules \(\Delta(\lambda)\) and \(\nabla(\lambda)\).
• Injective \(G_r T{\hbox{-}}\)modules
• Results of Verma and Humphreys on the Lifting Conjecture: does every \(G_r T{\hbox{-}}\)module come from a \(G{\hbox{-}}\)module?
• Donkin’s Tilting Module Conjecture (DTilt)

30 Friday, November 13

30.1 Review

Review: we’re considering \(G_r T{\hbox{-}}\)modules, with several associated modules of interest:

• Simple modules \(\widehat{L}_r(\lambda)\) for \(\lambda \in X(T)\)
• Intermediate modules \(\nabla(\lambda) = \widehat{Z}_r'(\lambda)\) and \(\Delta(\lambda) = \widehat{Z}_r(\lambda)\).
• Injective and projective modules \(\widehat{Q}_r(\lambda)\)

30.2 Reciprocity

Consider \(\widehat{Q}_r(\lambda)\), a projective \(G_r T{\hbox{-}}\)module. Note that it also happens to be injective. We saw before that the functor \(\mathop{\mathrm{coInd}}_{B_r T}^{G_r T}({-})\) is exact, and thus \(\widehat{Q}_r(\lambda)\downarrow_{B_r T}\) being projective implies that \(\widehat{Q}_r(\lambda)\downarrow{B_r}\) is also projective. This implies that \(\widehat{Q}_r(\lambda)\) has a \(\widehat{Z}_r{\hbox{-}}\)filtration.

Thus the multiplicity can be computed as \begin{align*} [\widehat{Q}_r(\lambda) : \widehat{Z}_r(\mu)] &= [\widehat{Q}_r \downarrow{B_r T} : \widehat{Z}_r(\mu)] \\ &= \dim \mathop{\mathrm{Hom}}_{B_r T}\qty{\widehat{Q}_r(\lambda), \mu } \\ &= \dim \mathop{\mathrm{Hom}}_{B_r T}\qty{ \widehat{Q}_r(\lambda), \mathop{\mathrm{ind}}_{B_r T}^{G_r T} \mu } && \text{by Frobenius reciprocity} \\ \end{align*}

We can thus continue this computation as \begin{align*} \cdots &= [\widehat{Z}_r'(\mu) : \widehat{L}_r(\mu)] \\ &= [\widehat{Z}_r(\mu) : \widehat{L}_r(\mu)] ,\end{align*} since \(\operatorname{ch}\widehat{Z}_r (\mu) = \operatorname{ch}\widehat{Z}_r'(\mu)\).

Thus we have the following reciprocity theorem

For \(G = {\operatorname{SL}}_2\), consider \(G_1 T\) or \(G_1\) where \(\lambda = 0,1,2,\cdots, (p-1)\). We have a notion of linkage: \(\lambda, \mu\) are in the same \(G_1\) block iff \(\lambda + \mu = p-2\). Note that \(\lambda = p-1\) is in its own block.

We have \begin{align*} Z_r(\lambda) = \mathop{\mathrm{coInd}}_{B_1^+}^{G_1} \lambda \twoheadrightarrow L(\lambda) .\end{align*} If \(\lambda + \mu = p-2\), then we have the following situation:

Taking \(\lambda = p-1\), we have \(Z_r((p-1)\rho) = L(p-1) = {\mathsf{Stk}}_1\).

Applying reciprocity, we gave \begin{align*} [Q_1(0) : Q_1(\mu)] = [Q_1(\mu): L(0)] .\end{align*} Since \(Q_1(0)\) has factors \(Z_1(0)\) and \(Z_1(p-2)\), we have

We can identify the two filtrations here:

Similarly, for \(Q_1(p-2)\) we have

We have

• \(\dim \widehat{Q}_1(\lambda) = 2p\) for \(\lambda \neq p-1\)

• \(\dim \widehat{Q}_1(p-1) = p\) for \(\lambda = p-1\).

30.3 Toward Lifting Conjectures

Recall that \(G_r T \subseteq G\).

Question: Given \(\widehat{Q}_r(\lambda)\) for a restricted weight \(\lambda \in X_r(T)\), does \(\widehat{Q}_r(\lambda)\) lift to \(G\)? I.e., does there exist a \(G{\hbox{-}}\)module \(M(\lambda)\) such that \(M(\lambda)\downarrow_{G_r T} = \widehat{Q}_r(\lambda)\)?

30.3.1 Donkin’s Tilting Module Conjecture

From MSRI, 1990. Some notation first: for \(\lambda \in X_r(T)\), define \begin{align*} \widehat{\lambda} \mathrel{\vcenter{:}}= 2(p-1) \rho + w_0 \lambda .\end{align*}

Next time:

• Proof of theorem

• \(\widehat{Q}_r(\lambda) \divides {\mathsf{Stk}}_r \otimes L(\sigma)\) as \(G{\hbox{-}}\)modules, and is also projective as a \(G_r T{\hbox{-}}\)module.

• Find a \(G{\hbox{-}}\)summand \(M(\lambda)\) such that \(M(\lambda)\downarrow_{G_r T} = \widehat{Q}_r (\lambda)\).

• More with injective modules.

• Possibly something about cohomology of Frobenius kernels.

31 Monday, November 16

We’ll focus on the following theorem:

Recall Donkin’s tilting conjecture:

31.1 Existence of \(G{\hbox{-}}\)Structures: Some Preliminaries

We want to consider \(G{\hbox{-}}\)structures on injective modules. Let \(V, W\) be \(G{\hbox{-}}\)modules, then \begin{align*} \mathop{\mathrm{Hom}}_{G_r}(V, W) = [V {}^{ \vee }\otimes W]^{G_r} ,\end{align*} which is a module on which \(G_r\) acts trivially where we pull back the action using the map \(G \to G/G_r\). Moreover, there exists a \(G{\hbox{-}}\)modules \(M\) such that \(\mathop{\mathrm{Hom}}_{G_r}(V, W) = M^{(r)}\) twisted \(r\) times.

We can consider blocks, e.g. by considering \({\operatorname{pr}}_\nu M\). This is the sum of all submodules with composition factors in the same block as \(L(\nu)\). We can write \(M = \bigoplus_{\nu\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}} {\operatorname{pr}}_\nu M\).

Thus we can write \begin{align*} \mathop{\mathrm{Hom}}_{G_r}(V, M) = \bigoplus_{\nu\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}} \mathop{\mathrm{Hom}}_{G_r}^\nu(V, W) = \qty{ {\operatorname{pr}}_\nu M }^{(r)} .\end{align*}

Note that for \(p>h\) the Coxeter number, \(0\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\), since \begin{align*} 0 \leq {\left\langle {0+\rho},~{\alpha {}^{ \vee }} \right\rangle} = {\left\langle {\rho},~{\alpha_0 {}^{ \vee }} \right\rangle} = h-1 < p ,\end{align*} where \(h = {\left\langle {\rho},~{\alpha_0 {}^{ \vee }} \right\rangle}\).

We can then write \begin{align*} \mathop{\mathrm{Hom}}_G(V, W) = \mathop{\mathrm{Hom}}_{G_r}(V, W)^{G_r} \subseteq \mathop{\mathrm{Hom}}_{G_r}^0(V, W) ,\end{align*} where the middle term involves trivial modules.

31.1.1 Sketch of Proof

In other words, if we restrict this down to \(G_r T{\hbox{-}}\)modules, we get a \(G_r T{\hbox{-}}\)summand. Note that if DFilt holds, \(M\) is a Tilting module.

Recall that \(M\) is a projective and injective \(G_r T{\hbox{-}}\) module. It suffices to show that \(\widehat{L}_r(\lambda)\in \mathop{\mathrm{Soc}}_{G_r T} M\), for which we look at the hom space \begin{align*} \mathop{\mathrm{Hom}}_{G_r T} (\widehat{L}_r(\lambda), {\mathsf{Stk}}_r \otimes L( (p^r-1)\rho + w_0 \lambda ) ) &= \mathop{\mathrm{Hom}}_{G_r T} (\widehat{L}_r(\lambda) \otimes{\mathsf{Stk}}_r, L( (p^r-1)\rho + w_0 \lambda ) ) \\ &= \mathop{\mathrm{Hom}}_{G_r T} ({\mathsf{Stk}}_r, \widehat{L}_r(-w_0 \lambda) \otimes L( (p^r-1)\rho + w_0 \lambda ) ) ,\end{align*}

Let \(Q_\lambda\) be the injective hull of \(L_\lambda\) as a \(G{\hbox{-}}\)module. This yields an injection \(L(\lambda)\hookrightarrow M\). Since we also have a map into the injective hull, we can extend:

Moreover, \(\phi:\widehat{Q}_r(\lambda)\to Q_\lambda\) is an injective map since \(L(\lambda) \subseteq \mathop{\mathrm{Soc}}_{G_r}(\lambda)\). Thus the image \(\phi\qty{\widehat{Q}_r(\lambda) } \cong \widehat{Q}_r(\lambda)\) and is a \(G_r T\) summand of \(\phi(M)\) since \(\widehat{Q}_r(\lambda)\) is an injective \(G_r T{\hbox{-}}\)module. We have a split exact sequence \begin{align*} 0 \to \phi(\widehat{Q}_r(\lambda)) \hookrightarrow\phi(M) \to ? \to 0 .\end{align*}

The idea is now that \(\phi(M)\) is a \(G{\hbox{-}}\)module, so we’ll show \(\phi(M)\) is indecomposable as both a \(G\) and a \(G_r T\) module. In this case, we’ll have \begin{align*} \phi(M) = \phi(\widehat{Q}_r(\lambda)) \cong \widehat{Q}_r(\lambda) .\end{align*}

It suffices to prove that \(\mathop{\mathrm{Soc}}_{G_r} \phi(M) = L_r(\lambda)\). Note that that it suffices to show there’s only one summand, since the socle can’t be decomposed further, which will yield irreducibility. We have \begin{align*} \mathop{\mathrm{Soc}}_G \phi(M) = L(\lambda) .\end{align*}

We also know that \(\mathop{\mathrm{Soc}}_G \phi(M) \subseteq \mathop{\mathrm{Soc}}_{G_r} \phi(M)\). But if \(L(\mu) \subseteq \mathop{\mathrm{Soc}}_{G_r} \phi(M)\), then it is also contained in \(\mathop{\mathrm{Soc}}_G \phi(M)\). Hence if \(\mathop{\mathrm{Soc}}_{G_r}\phi(M)\) is isotypic of type \(L(\lambda)\), we have a decomposition \begin{align*} \mathop{\mathrm{Soc}}_{G_r}\phi(M) &= \bigoplus_{\sigma\in X_r(T)} L(\sigma) \otimes\mathop{\mathrm{Hom}}_{G_r}(L(\sigma), \phi(M)) \\ &= L(\lambda) \otimes\mathop{\mathrm{Hom}}_{G_r}(L(\lambda), \phi(M)) \\ &= \bigoplus_{\nu\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}} L(\lambda) \otimes\mathop{\mathrm{Hom}}_{G_r}^\nu (L(\lambda), \phi(M)) ,\end{align*} where we’ve broken this up into blocks. Note that \(\phi(M)\) is an indecomposable \(G{\hbox{-}}\)module, thus using linkage we can conclude that there is only one summand. By the previous statement, we have \begin{align*} k = \mathop{\mathrm{Hom}}_{G}(L(\lambda), \phi(M)) \subseteq \mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M)) ,\end{align*} and hence

\[\begin{equation} \mathop{\mathrm{Soc}}_{G_r} \phi(M) = L(\lambda) \otimes\mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M)) .\end{equation}\]

The critical equation to show:

\[\begin{equation} \mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M)) = k .\end{equation}\]

Suppose that \(\mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M)) \neq k\). Then there exists a \(\nu\in X(T)_+ \cap W_p \cdot 0\) with \(\nu\neq 0\) such that \(L(\nu)^{(r)} = L(p^r \nu)\) is a composition factor of \(\mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M))\). This would imply that \(L(\lambda) \otimes L(\nu)^{(r)} = L(\lambda+ p^r\nu)\) is a composition factor of \({\mathsf{Stk}}_r \otimes H^0( (p^r-1)\rho + w_0 \lambda)\).

The idea now is to check the weights in this module either by Weyl’s character formula or weight estimates. The upshot: for \(p>2h-2\), \(\lambda+ p^r \nu\) is a weight if \(\nu = 0\). \(\hfill\blacksquare\)

How might this bound be lowered? Maybe the condition of the hom space being trivial is too strong, since being indecomposable isn’t equivalent to having a simple socle.

31.2 Steinberg Tensor Product Theorem for Injective \(G_r T{\hbox{-}}\)Modules

Next time: we’ll complete injective modules, and if we have time, we’ll talk about cohomology.

32 Wednesday, November 18

32.1 Steinberg Tensor Product Formula

Recall the Steinberg tensor product formula: Let \(\lambda \in X(T_+)\) with \(\operatorname{ch}(k) = p\), then write \begin{align*} \lambda = \sum_{j=1}^s p^j \lambda_j \end{align*} with \(\lambda_j \in X_1(T)\). Then \begin{align*} L(\lambda) = L(\lambda_0) \otimes\bigotimes_{j=1}^s L(\lambda_j)^{(j)} .\end{align*}

The goal is to find a formula resembling this for \(G\) and \(G_r{\hbox{-}}\)modules.

Let \(L(\lambda)\) be an irreducible representation for \(G\), and set \(Q_\lambda\) to be the injective hull of \(L(\lambda)\), so we have an exact sequence \(0 \to L(\lambda) \hookrightarrow Q(\lambda)\).

Next time: cohomology of Frobenius kernels and relation to the nilpotent cones, related to ABG equivalence, computing the cohomology ring for \(G_1\).

33 Friday, November 20

33.1 Cohomology of Frobenius Kernels

These predate cohomology for the small quantum group.

Letting \(G\) be a reductive algebra over \(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}_p\mkern-1.5mu}\mkern 1.5mu\), recall that we have the \(r\)th Frobenius \(F^r:G\to G\), where \(G_r \mathrel{\vcenter{:}}=\ker F^r\) and \(A \mathrel{\vcenter{:}}=\operatorname{Dist}(G_r)\). We define \(R = H^\cdot(G_r, k) = H^\cdot(A, K) = \operatorname{Ext} _A(k, k)\). How do we compute this ring?

We have

We then concatenate these in \(\operatorname{Ext} _A^{s+t}(k, k)\).

The best answers occur when \(r=1\) and \(p>h\), due to results by Friendlander-Parhsll for \(p\geq 3h-3\), and by Andersen-Jantzen for \(p>h\).

33.2 Finite Generation of Cohomology

A finite group scheme is essentially the same as a cocommutative Hopf algebra.

34 Monday, November 23

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

34.1 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 \(p<h\)?

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

For \(\lambda\in X(T)\), set \begin{align*} \Phi_\lambda \mathrel{\vcenter{:}}=\left\{{ \alpha\in\Phi {~\mathrel{\Big|}~}{\left\langle {\lambda+\rho},~{\alpha {}^{ \vee }} \right\rangle} \in p{\mathbb{Z}}}\right\} .\end{align*}

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

34.2 Support Varieties

Very common in modern mathematics. Define \(R \mathrel{\vcenter{:}}= H^{-}(G_1; k)\), which is a finitely generated algebra. Note that \(R\) acts on \(\operatorname{Ext} _{G_1}^{-}(M, M)\), which is finitely generated over \(R\). So what is the support of this module? Define \(V_{G_1}(M) \mathrel{\vcenter{:}}=\operatorname{mSpec}\qty{R/J_M}\), where \(J_M \mathrel{\vcenter{:}}=\operatorname{Ann}_R \operatorname{Ext} _{G_1}^{-}(M, M)\).

35 Bibliography