# Lecture 2 (Monday, August 23)

## A Lightning Introduction to Groups and Representations

:::{.remark}
Throughout, *finite type* means finitely generated over the base field.

:::

:::{.remark}
Which $G$ are important for equivariant cohomology of the flag variety, and equivariant \(\K\dash\)theory.
We'll consider only connected reductive groups, and work over $k \da \CC$.
:::

:::{.definition title="Pertaining to Linear Algebraic Groups"}
\envlist

- A group $G \in \Alg\Grp$ be is a **linear algebraic group** if

  - The coordinate ring $\CC[G]$ is a reduced (so no nonzero nilpotents) $\CC\dash$algebra of finite type.
  - $G$ is a group where multiplication $m:G^{\times 2}\to G$ and inverseion $i:G\to G$ are algebraic morphisms


- A **maximal torus** of $G$ is a torus not properly contained in any other torus of the form $(\CC\units)^{\times n}$.
- A **Cartan** subgroup is the centralizer of a maximal torus.
  Note that maximal torii are the same as Cartans in the connected reductive case.
- $G$ is **unipotent** if every representation has a nonzero fixed vector.
- The **unipotent radical** $R_u(G)\leq G$ is a maximal closed connected normal subgroup of $G$.
- $G$ is **reductive** iff $R_u(G) = \ts{e}$.

:::

:::{.proposition title="?"}
To study $\Rep(G)^{\irr}$ for $G\in \Alg\Grp$ linear, we can assume that $G$ is reductive.
:::

:::{.proof title="?"}
Let $V\in \Rep(G)^{\irr}$, we'll show that the unipotent radical acts trivially.
Then $V$ is the data of

1. $G\to \GL(V)$ for some $V$, a morphism of varieties and algebraic groups
2. There is an action map $G\times V\to V$.

Let $V_0 = \Fix(R_u(G)) \subseteq V$ be the fixed points of $R_u(G)$, by restricting the $G$ action to an $R_u(G)\leq G$ action by a subgroup.
We know $V_0 \neq 0$, and we have for every $g\in G, r\in R_u(G), v\in V_0$.
We'd like to show $V_0 = V$, which means that $R_u(G)$ acts trivially.
So we'll show $r$ fixes every $gv$:
\[
r(gv) = g(g\inv r g)v \in g R_u(G) v = gv
,\]
using that $R_u(G)$ fixes $v$.
So $V_0$ is $G\dash$stable, and since $V_0$ is irreducible and $V$ is irreducible, we get equality.
:::

:::{.remark}
So $R_u(G)$ won't matter for irreducible representations, or in turn for equivariant \(\K\dash\)theory, and we can assume $R_u(G) = \ts{e}$ is trivial.
If $G$ is not reductive, just replace it with $R/R_u(G)$, which is a reductive linear algebraic group when $G$ is a linear algebraic group since $R_u(G) \normal G$.

Next question: how can we relate compact groups to complex reductive groups?
:::

:::{.remark}
Let $K \in \Lie\Grp$ be compact, and set $\CC[K]$ to be the $\CC\dash$span of matrix coefficients of finite dimensional representations of $K$.
For $V$ a finite-dimensional representation of $K$ (just a continuous representation of a compact group), define
\[
\phi: V\dual \tensor_\CC V &\to \CC[K] \\
f\tensor v &\mapsto \qty{k \mapsvia{\phi_{f, v}}  f(kv)}
.\]
:::

:::{.fact}
$\CC[K]$ is a finite type reduced algebra.
Such algebras correspond to an affine variety, i.e. it is the ring of functions on some affine variety.
Thus $\CC[K] = \CC[G]$ for $G\in \Aff\Var\slice{\CC}$ where $K \subseteq G$.
:::

:::{.theorem title="Chevalley"}
\envlist

1. $G$ is a *reductive* algebraic group.
2. Every locally finite continuous representation of $K$ extends uniquely to an algebraic representation of $G$, and every algebraic representation of $G$ restricts to a locally finite representation of $K$.

:::

:::{.remark}
So despite $\CC[G]$ being infinite dimensional, every representation is contained in some finite dimensional piece.
Note that there is an equivalence of categories between algebraic and compact groups, but there are differences: e.g. there are no irreducible infinite dimensional representations of compact groups.

> Side note, see stuff by David Vogan!

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:::{.remark}
The next result reduces representations to Cartans, which are *almost* tori, and is along the lines of what Langlands was originally thinking about.
:::

:::{.theorem title="Cartan-Weyl"}
There is a bijection
\[
\hat{G} \da \correspond{
  \text{Irreducible representations} \\
  \text{of }G
}
&\mapstofrom
\correspond{
  \text{Irreducible dominant representations} \\
  \text{of a Cartan subgroup } H\leq G
}
\]

Moreover,

1. If $G$ is finite, $\ts{e} = B \supseteq = \ts{e}$, so there is no reduction in this case, noting that the centralizer ends up being the whole group.
2. If $G$ is connected reductive, then $T=H$ and there reduce to dominant characters of a torus.
:::

:::{.remark}
See David Vogan's orange book on unitary representations of real reductive groups.
:::

:::{.exercise title="?"}
Try proving this directly!
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:::{.definition title="Dominant characters"}
Define 
\[
X(T) \da\ts{T \mapsvia{f} \CC\units \st f \text{ is algebraic}}
,\]
which is a moduli of irreducible representations of $G$.
Then 
\[
X(T) \supseteq D_\ZZ \da \ts{\chi \in X \st \chi \text{ is dominant for } B}
.\]
Note that this may make more sense after seeing root systems.
:::

:::{.remark}
Given $\lambda \in D_\ZZ$, define a $G\dash$equivariant line bundle on the flag variety as $\mcl(\lambda) \da (G\cross \CC_{-\lambda})/B$, 
where $(-\lambda)t \da \lambda(t)\inv$.
This can be extended to a representation of $B$ by 
\[
B \to B/R_u(B) \cong T \mapsvia{\lambda} \CC\units
.\]
This makes sense thinking of a Borel as upper-triangular matrices, tori as diagonal matrices, and unipotent as strictly upper triangular.
So we can extend representations by making them trivial on a normal subgroup?

\todo[inline]{Check}

We refer to $\lambda$ as the map and $\CC_{\lambda}$ as the vector space in the representation $G\to \GL(V)$.
Note that $B$ acts on the right of $G\times \CC_{-\lambda}$ by 
\[
(g, z)b \da (gb, b\inv z) \da (gb, \lambda(b)\inv z)
.\]
:::

:::{.fact}
$\mcl(\lambda)$ is an algebraic variety.
:::