Prologue

0.1 References

0.2 Notation

1 Friday, August 21

1.1 Intro and Definitions

Let \(k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) be algebraically closed (e.g. \(k = {\mathbb{C}}, \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}_p\mkern-1.5mu}\mkern 1.5mu\)). A variety \(V\subseteq k^n\) is an affine \(k{\hbox{-}}\)variety iff \(V\) is the zero set of a collection of polynomials in \(k[x_1, \cdots, x_n]\).

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

An affine algebraic \(k{\hbox{-}}\)group is an affine variety with the structure of a group, where the multiplication and inversion maps \begin{align*} \mu: G\times G &\to G \\ \iota: G&\to G \end{align*} are continuous.

\(G = {\mathbb{G}}_a \subseteq k\) the additive group of \(k\) is defined as \({\mathbb{G}}_a \mathrel{\vcenter{:}}=(k, +)\). We then have a coordinate ring \(k[{\mathbb{G}}_a] = k[x] / I = k[x]\).

\(G = \operatorname{GL}(n, k)\), which has coordinate ring \(k[x_{ij}, T] / \left\langle{\det(x_{ij})\cdot T = 1}\right\rangle\).

Setting \(n=1\) above, we have \({\mathbb{G}}_m \mathrel{\vcenter{:}}=\operatorname{GL}(1, k) = (k^{\times}, \cdot)\). Here the coordinate ring is \(k[x, T] / \left\langle{xT = 1}\right\rangle\).

\(G = {\operatorname{SL}}(n, k) \leq \operatorname{GL}(n, k)\), which has coordinate ring \(k[G] = k[x_{ij}] / \left\langle{\det(x_{ij}) = 1}\right\rangle\).

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.

Reducible vs Irreducible

There exists a unique irreducible component of \(G\) containing the identity \(e\). Notation: \(G^0\).

\(G\) is the union of translates of \(G^0\), i.e. there is a decomposition \begin{align*} G = {\coprod}_{g\in \Gamma} \, g\cdot G^0 ,\end{align*} where we let \(G\) act on itself by left-translation and define \(\Gamma\) to be a set of representatives of distinct orbits.

One can define solvable and nilpotent algebraic groups in the same way as they are defined for finite groups, i.e. as having a terminating derived or lower central series respectively.

1.2 Jordan-Chevalley Decomposition

There is a maximal connected normal solvable subgroup \(R(G)\), denoted the radical of \(G\).

An element \(u\) is unipotent \(\iff\) \(u = 1+n\) where \(n\) is nilpotent \(\iff\) its the only eigenvalue is \(\lambda = 1\).

For any \(G\), there exists a closed embedding \(G\hookrightarrow\operatorname{GL}(V) = \operatorname{GL}(n , k)\) and for each \(x\in G\) a unique decomposition \(x=su\) where \(s\) is semisimple (diagonalizable) and \(u\) is unipotent.

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

Suppose \(G\) is connected, so \(G = G^0\), and nontrivial, so \(G\neq \left\{{e}\right\}\). Then

\(G = \operatorname{GL}(n, k)\), then \(R(G) = Z(G) = kI\) the scalar matrices, and \(R_u(G) = \left\{{e}\right\}\). So \(G\) is reductive and semisimple.

\(G = {\operatorname{SL}}(n , k)\), then \(R(G) = \left\{{I}\right\}\).

Is this semisimple? Reductive? What is \(R_u(G)\)?

A torus \(T\subseteq G\) in \(G\) an algebraic group is a commutative algebraic subgroup consisting of semisimple elements.

Let \begin{align*} T \mathrel{\vcenter{:}}= \left\langle{ \begin{bmatrix} a_1 & & \mathbf 0\\ & \ddots & \\ \mathbf 0 & & a_n \end{bmatrix} \subseteq \operatorname{GL}(n ,k) }\right\rangle .\end{align*}

Why are torii useful? For \(g = \mathrm{Lie}(G)\), we obtain a root space decomposition \begin{align*} g = \qty{\bigoplus_{\alpha \in \Phi_- }g_\alpha} \oplus t \oplus \qty{\bigoplus_{\alpha \in \Phi_+ }g_\alpha} .\end{align*}

When \(G\) is a simple algebraic group, there is a classification/correspondence: \begin{align*} (G, T) \iff (\Phi, W) .\end{align*} where \(\Phi\) is an irreducible root system and \(W\) is a Weyl group.

2 Monday, August 24

2.1 Review and General Setup

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.

Image

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

Define left translation is \begin{align*} \lambda_x: k[G] &\to k[G] \\ y &\mapsto f(x^{-1} y) .\end{align*}

Define derivations as \begin{align*} \mathrm{Der} ~k[G] = \left\{{D: k[G] \to k[G] {~\mathrel{\Big|}~}D(fg) = D(f) g + f D(g) }\right\} .\end{align*}

We can then realize the Lie algebra as \begin{align*} {\mathfrak{g}}= \mathrm{Lie}(G) = \left\{{D\in \mathrm{Der} k[G] {~\mathrel{\Big|}~}\lambda_x \circ D = D\circ \lambda_x}\right\} ,\end{align*} the left-invariant derivations.

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:

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*}

Let \(G = \operatorname{GL}(n, k)\) and \(V\) the \(n{\hbox{-}}\)dimensional natural representation as column vectors, \begin{align*} V = \left\{{{\left[ {v_1, \cdots, v_n} \right]} {~\mathrel{\Big|}~}v_j \in k}\right\} .\end{align*}

Then \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*}

Consider the basis vectors \(\mathbf{e}_j\), then \begin{align*} \begin{bmatrix} a_1 & 0 & 0 \\ 0 & \ddots & 0\\ 0 & 0 & a_n \end{bmatrix} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = a_j \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = a_1^0 a_2^0 \cdots a_j^0 \cdots a_n^0 \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} .\end{align*}

Here the weights are of the form \(\varepsilon_j\mathrel{\vcenter{:}}={\left[ {0, 0, \cdots, 1, \cdots, 0} \right]}\) with a \(1\) in the \(j\)th spot, so we have \begin{align*} V = V_{\varepsilon_1} \oplus V_{\varepsilon_2} \oplus \cdots \oplus V_{{\varepsilon_n}} .\end{align*}

For \(V = {\mathbb{C}}\), we have \(t.v = (a_1^0 \cdots a_n^0)v\) and \(V = V_{(0, 0, \cdots, 0)}\).

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 \({\mathfrak{sl}}_3\):

Image

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

3 Wednesday, August 26

3.1 Review

\(A_n\) corresponds to \({\mathfrak{sl}}(n+1, k)\) (mnemonic: \(A_1\) corresponds to \({\mathfrak{sl}}(2)\))

3.2 Root Systems and Weights

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

Image

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

Every root \(\alpha\in I\) can be expressed as either positive integer linear combination (or negative) 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 \(S_3\):

Image

\(W\) acts transitively on bases.

\(X(T) \subseteq {\mathbb{Z}}\Phi\), recalling that \(X(T) = \hom(T, {\mathbb{G}}_m) = {\mathbb{Z}}^n\) for some \(n\). Denote \({\mathbb{Z}}\Phi\) the root lattice and \(X(T)\) the weight lattice.

Let \(G = {\mathfrak{sl}}(2, {\mathbb{C}})\) then \(X(T) = {\mathbb{Z}}\omega\) where \(\omega = 1\), \({\mathbb{Z}}\Phi = {\mathbb{Z}}\left\{{\alpha}\right\}\) Then there is one weight \(\alpha\), and the root lattice \({\mathbb{Z}}\Phi\) is just \(2{\mathbb{Z}}\). However, the weight lattice is \({\mathbb{Z}}\omega = {\mathbb{Z}}\), and these are not equal in general.

There is partial ordering on \(X(T)\) given by \(\lambda \geq \mu \iff \lambda - \mu = \sum_{\alpha\in \Delta} n_\alpha \alpha\) where \(n_\alpha \geq 0\). (We say \(\lambda\) dominates \(\mu\).)

We extend scalars for the weight lattice to obtain \(E \mathrel{\vcenter{:}}= X(T) \otimes_{\mathbb{Z}}{\mathbb{R}}\cong {\mathbb{R}}^n\), a Euclidean space with an inner product \({\left\langle {{\,\cdot\,}},~{{\,\cdot\,}} \right\rangle}\).

For \(\alpha\in \Phi\), define its coroot \(\alpha^\vee\mathrel{\vcenter{:}}={2\alpha \over {\left\langle {\alpha},~{\alpha} \right\rangle}}\). Define the simple coroots as \(\Delta^\vee\mathrel{\vcenter{:}}=\left\{{\alpha_i^\vee}\right\}_{i=1}^n\), which has a dual basis \(\Omega \mathrel{\vcenter{:}}=\left\{{\omega_i}\right\}_{i=1}^n\) the fundamental weights. These satisfy \({\left\langle {\omega_i},~{\alpha_j^\vee} \right\rangle} = \delta_{ij}\).

Important because we can index irreducible representations by fundamental weights.

A weight \(\lambda\in X(T)\) is dominant iff \(\lambda \in {\mathbb{Z}}^{\geq 0} \Omega\), i.e. \(\lambda = \sum n_i \omega_i\) with \(n_i \in {\mathbb{Z}}^{\geq 0}\).

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 \(Z(\lambda) = U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}}^+)} \lambda \twoheadrightarrow L(\lambda)\).

\(L(\lambda)\) as a finite-dimensional \(U({\mathfrak{g}}){\hbox{-}}\)module \(\iff\) \(\lambda\) is dominant, i.e. \(\lambda \in X(T)_+\).

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