On 21/03/2020, I had a quick read through structure theory of linear algebraic groups so that I know certain terminologies, how they are defined.

I learn these from Fiona Murnaghan’s short note about linear algebraic groups and Daniel Bump’s note, starting from section 4.

Also, I am stuck at a certain argument when reading Bushnell, Henniart’s book Local Langlands for GL(2) and spend some time with it but with no progress.

I will try to write something if I have time.

OK, here is roughly what I learnt on this day:

Structure of $\text{GL}_n(F)$

Center of $GL_n$ is denoted $Z$.

In matrix manners, the diagonal torus $T$ is subgroup of all diagonal matrices, the standard Borel subgroup $B$ consists of all upper triangular matrices. The unipotent radical $U$ consists of all upper triangular unipotent matrices. Note $T$ normalizes $U$ and $B$ is semidirect product $TU$. Let $W$, the \vocab{Weyl group} of $G$, is the group $N_G(T)/T$ where $N_G(T)$ normalizer of $T$. For $G=GL_n(F)$, $W$ is subgroup of all monomial matrices (one nonzero entry in each row and each column) so $W$ can be identified with $S_n$.

In different view, by picking $n$-dimensional vector space $V$ over $F$, $GL_n(F)$ can be identified with $GL(V)$. Define a flag $W_{\bullet}$ in $V$ to be strictly increasing sequence of subspaces $W_0\subset W_1\subset \cdots \subset W_m=V$. Subgroup of $GL(V)$ that stabilizes flag $W_{.}$, i.e. with the property that $gW_i=W_i$ for all $i$ is called parabolic subgroup of $G$ associated to flag $W_{\bullet}$.

If ${v_1,\ldots, v_n}$ basis of $V$ then stabilizer of flag $\{(v_1)\subset (v_1,v_2)\subset \cdots \subset (v_1,\ldots, v_n)\}$ is called Borel subgroup. In the case of $GL(V)$, stabilizer of any two such (maximal) flags are conjugate under $GL(V)$.

If $W_{\bullet}={W_0\subset W_1\subset \cdots \subset W_m}$ then inside the associated parabolic subgroup $P$, there exists normal subgroup $N$ consisting of elements who operating trivially on $W_{i+1}/W_i$ for $0\le i \le m-1$, which is called unipotent radical of $P$. There is a semidirect product decomposition $P=MN$ with $M=\prod_{i=0}^{m-1} GL(W_{i+1}/W_i)$. The decomposition $P=MN$ is called Levi decomposition of $P$ with $M$ called Levi subgroup of $P$.

Let $K=GL_n(\mathfrak{o})$ denote subgroups of elements in $G$ in $\mathfrak{o}$ and whose determinant is unit in $\mathfrak{o}$. This is a maximal open compact subgroup of $GL_n(F)$, as shown in following exercise.

Exercise. Let $V$ be $n$-dimensional vector space over $F$. Let $L$ be lattice on $V$, i.e. $\mathfrak{o}$-submodule rank $n$. Show that stabilizer of $L$ is open compact subgroup of $G=GL(V)$. If $C$ is any open compact subgroup then there is lattice $L$ such that $C$ lies in stabilizer of $L$. Hence, up to conjugacy, $K$ is the unique maximal open compact subgroup of $GL_n(F)$.

For every integer $m\ge 1$, the map $\mathfrak{o} \to \mathfrak{o}/\mathfrak{p}^m$ induces map $K\to GL_n(\mathfrak{o}/\mathfrak{p}^m)$. The kernel $K_m$ of this map is called principal congruence subgroup of level $m$. We also define $K_0$ to be $K$. For all $m\ge 1$, we have $K_m = 1_n+\mathfrak{p}^m M_n(\mathfrak{o})$. $K_m$ are open compact subgroups of $G$ and gives basis of neighborhood at the identity.

Bruhat decomposition gives $G=\bigsqcup_{w\in W}BwB=\bigsqcup_{w\in W}BwU$. Proof use row reduction (on the left) and column reduction (on the right) by elementary operations.

Cartan decomposition gives, for $A={\text{diag}(\Pi^{m_1},\ldots,\Pi^{m_n}): m_i\in \mathbb{Z}_{\ge 0}, m_1\le \cdots \le m_n}$ then $G=\bigsqcup_{a\in A} KaK$.

Iwasawa decomposition gives $G=KB$.

Iwahori factorization gives for $m\ge 1$ then $K_m=(K_m\cap U^-)(K_m\cap T) (K_m\cap U)$ where $U^-$ subgroup of all lower triangular unipotent matrices.

In language of algebraic groups

(Very roughly, so that I can read other things)

An algebraic group is an algebraic variety $G$ defined over some field $F$ with morphims $m:G\times G\to G$ and $\text{inv}: G\to G$ becomes multiplication and inverse map on $G(E)$ making $G(E)$ of $E$-rational points (i.e. $G(E)=G\cap E^n$) into a group when $E$ is commutative $F$-algebra. Affine algebraic group is when $G$ is affine variety, example is multiplicative group $G_m$, with $G(E)=E^{\times}$.

A torus is group $T$ that is isomorphic to direct product of $G_m^k$ for some $k$. If the isomorphism is defined over $F$ we say $T$ split (over $F$).

For affine algebraic group $G$ over $F$. By representation we mean morphism $\rho:G\to GL_n$ for some $n$ such that $\rho:G(E)\to GL_n(E)$ group hom for any commutative $F$-algebra $E$.

By Jordan decomposition, $g\in G$ then exists $g_s,g_u\in G$ such that $g=g_sg_u=g_ug_s$, $g_s$ semisimple, $g_u$ unipotent. Hence $g\in G$ is called semisimple if $g=g_s$ and unipotent if $g=g_u$.

$G$ is unipotent if all its elements are unipotnent. Group $G$ has maximal normal unipotent subgroup $U$, called unipotent radical. If $U$ is trivial then $G$ is reductive. If it is reductive and has no nontrivial normal tori then $G$ is semisimple. For example group of $\left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \right\}$ is not reductive since $\left\{ \begin{pmatrix} 1 & x\\ 0 & 1 \end{pmatrix} \right\}$ is normal unipotent subgroup. Group $SL_n$ is semisimple. Group $GL_n$ is reductive but not semisimple.

Maximal torus $T$ is subgroup as large as possible such that $T$ product of multiplicative groups. If $T$ splits over $F$ then $G$ is $F$-split. All maximal tori in $G(F)$ are conjugate if $F$ algebraically closed.

If $G$ is $F$-split reductive group and $T$ is $F$-split maximal torus, $N$ normalizer of $T$ then $N/T$ is Weyl group $W$.

See also this short note.