Reductive group terminology questions

Question

$\DeclareMathOperator{\Hom}{Hom}$ $\newcommand{\g}{\mathfrak{g}}$

I am a beginner in the subject of reductive groups and I am hoping someone might be able to walk me through some basic terminology. This is not homework so any and all remarks, even those not directly related to what I ask, are helpful. In some sense I am asking many questions, so if moderators wish that I ask separate questions, I will of course do so instead. But I view all of these as trivially tied together and I think it will be easy for someone knowledgeable to answer them all rapid-fire.

Let $k$ be an algebraically closed field of arbitrary characteristic. Let $G$ be a connected reductive group over $k$. Let $B=TU$ be a fixed Borel subgroup of $G$, where $T$ is a torus and $U$ is the unipotent radical of $B$. Let $X^*(T):=\Hom(T, GL_1)$ be the group of characters or the weight lattice and $X_*(T):=\Hom(GL_1, T)$ be the group of cocharacters or the coweight lattice. Now let $\g$ be the Lie algebra of $G$. The adjoint representation of $G$ is given by conjugation on $\g$. A root is a nontrivial weight of $G$ that occurs in the action of $T$ on $\g$. The choice of $B$ determines a set of positive roots. A positive root that is not the sum of two positive roots is called a simple root. If everything I have written so far makes sense, then I am pretty good up until here.

My questions are the following:

1. What is the meaning of a coroot? I have been told that there is a pairing between the weight and coweight lattices, and those coweights corresponding to the roots are called coroots. What precisely is this pairing and what is the meaning of "corresponding to?" And what would the coroots for, say, $GL_n$ look like (with the standard choices of $B$ and $T$)?

2. Similar to #2, what is a positive coroot and a simple coroot? I have some good guesses but I want to be sure.

3. What is a dominant weight and a dominant coweight? Why would we care about such weight/coweights?

4. Are there only finitely many dominant weights/coweights?

5. Finally, what are the fundamental weights and fundamental coweights? It seems that these are necessarily dominant, but I am not sure what their precise definition is. Are there only finitely many of these?

Any and all hints or remarks that you feel may clarify the situation are of course welcome.

Answer 1

1 . The pairings you are referring to come from the natural bilinear map

$$X^{\ast}(T) \times X_{\ast}(T) \rightarrow \mathbb Z$$
defined as follows: if $\chi$ is a character of $T$, and $\eta$ is a cocharacter of $T$, then $\chi \circ \eta$ is a morphism of linear algebraic groups $\operatorname{GL}_1 \rightarrow \operatorname{GL}_1$. Consequently, there exists an integer $n$ such that $\chi \circ \eta(x) = x^n$. We then set $\langle \chi, \eta \rangle = n$. It's a good exercise to work what this is saying for the special case where $T$ is the group of size $n$ diagonal invertible matrices.

Definition of coroot: for each root $\alpha$ of $T$ in $G$, let $T_{\alpha}$ be the connected component of the kernel of $\alpha$. There is a unique cocharacter $\alpha^{\vee} \in X_{\ast}(T)$ such that (i) $\langle \alpha, \alpha^{\vee} \rangle = 2$ and (ii) $T$ is generated by $T_{\alpha}$ and $\operatorname{Im} T_{\alpha}$. Each $\alpha^{\vee}$ is the coroot attached to the root $\alpha$.

Actually, it is better to work with a pairing of vector spaces than of abelian groups. The abelian groups $X^{\ast}(T)$ and $X_{\ast}(T)$ are both free of the same rank, so the vector spaces $V^{\ast} = X^{\ast}(T) \otimes_{\mathbb Z} \mathbb Q$ and $V_{\ast} = X_{\ast}(T) \otimes_{\mathbb Z} \mathbb Q$ are of the same dimension, and the pairing above induces a nondegenerate pairing

$$V^{\ast} \times V_{\ast} \rightarrow \mathbb Q$$

$$(v,w) \mapsto \langle v, w\rangle.$$

It is useful to think of $X^{\ast}(T)$ being a lattice inside $V^{\ast}$, and likewise for $X_{\ast}(T)$ inside $V_{\ast}$.

It is not difficult to work out the coroots for $\operatorname{GL}_n$. They are "identical" to the roots. If you have trouble with this, ask a new question.

2 . They are what you expect. Positive (resp. simple) coroots are those corresponding to positive (resp. simple) roots. The set of simple roots and the set of simple coroots are each linearly independent sets in $X^{\ast}(T)$ (resp. $X_{\ast}(T)$).

3 - 5: The fundamental weights are dual to the simple coroots. That is, if $\alpha_1, ... , \alpha_n$ are the simple roots, then the fundamental weights $\omega_1, ... , \omega_n$ are the unique elements of $V^{\ast}$ such that $\langle \omega_i, \alpha_j^{\vee} \rangle = \delta_{ij}$.

Similarly, the fundamental coweights $\omega_1^{\vee}, ... , \omega_n^{\vee} \in V_{\ast}$ are dual to the simple roots: $\langle \alpha_i, \omega_j^{\vee} \rangle = \delta_{ij}$.

Dominant weights (resp. coweights) are the elements in $V^{\ast}$ (resp. $V_{\ast}$) of the form $a_1 \omega_1 + \cdots + a_n \omega_n$ (resp. $a_1 \omega_1^{\vee} + \cdots + a_n \omega_n^{\vee}$), where $a_i$ are nonnegative integers. Of course, there are infinitely many of these whenever $T \subsetneq G$.

There are several reasons to care about dominant/fundamental weights and coweights. An interesting result is that the sum of the fundamental weights $\omega_1 + \cdots + \omega_n$ is equal to half the sum of the positive roots. These give you the modulus character on the nonunimodular topological group $B(\mathbb C)$ (in the Euclidean, not Zariski, topology).

Also, the positive chamber $\{ c_1 \omega + \cdots + c_n \omega : \mathbb R \ni c_i \geq 0\}$ shows up in Arthur's modified trace formula. I don't know too much about this myself, still learning.