Weight space corresponding to $\lambda$ what is $-\lambda$?

Question

Let $G$ be an algebraic group and let $T\subset G$ be a torus. Let $\lambda:T\to \Bbb C^\times$ be a character. Let $V$ be a $G$-module, and $V'$ be a nonzero $\lambda$-weight space of $G$ (pick the $\lambda$ at the start so it is a weight).

What does it mean to say "consider the $-\lambda$ weight space"? Isn't the character group $X(T)$ a multiplicative group. Then what is $-\lambda$?

Answer 1

Technically you're right, and it would be justified to talk about $\lambda^{-1}$ instead of $\lambda$. In practice we prefer to use additive notation in $X(T)$ because we really want to use weight diagrams of representations. If you haven't seen them yet, you soon will!

Anyway, if $T$ is some group of diagonal matrices and $$ \lambda:diag(x_1,x_2,\ldots,x_n)\mapsto x_1^{m_1}x_2^{m_2}\cdots x_n^{m_n} $$ is a weight, then we really want to equate the vector of exponents $(m_1,m_2,\ldots,m_n)$ with $\lambda$. In this spirit $-\lambda$ gets equated with the vector of exponents $(-m_1,-m_2,\ldots,-m_n)$.

Further points:

• The set of all possible weights appearing in some f.d. representation naturally form a f.g. free abelian group. We are very much used to handling such beasts additively. Like $\Bbb{Z}^n$.
• We want to turn $X(T)$ into a vector space over $\Bbb{R}$ (you will see this soon) by forming the tensor product $X(T)\otimes_{\Bbb{Z}}\Bbb{R}$. When we do this, conjugation action by $N_G(T)$ (i.e. essentially of the Weyl group) is then done via linear transformationa. It turns out that they are orthogonal transformations. For all that to make sense we want to turn the product of $X(T)$ into an addition.
• In the Lie algebra side, when we in a sense take logarithms of pretty much everything, this group operation becomes additive.

Answer 2

With some more thought, this is my potential self answer.

We use the correspondence $X(T)\cong \Bbb Z^n$, and then we identify $\mu$ with $(\mu_1,\dots,\mu_n)$, where it certainly makes sense to talk about $-\mu$.

In fact, indeed under this correspondence $\lambda,\mu\in X(T)$ then $\lambda\mu$ under the correspondence is $(\lambda_1+\mu_1,\dots,\lambda_n+\mu_n)$.

Example: For $\text{GL}_n$, write the map $\epsilon_i((a_{ij}))=a_{ii}$. Then we can consider the characters, which are given by: $$\{\epsilon_1^{a_1}\dots\epsilon_n^{a_n}\}_{a_i\in \Bbb Z}$$ and this is sent to $(a_1,\dots,a_n)\in \Bbb Z^n$, and $\mu\nu((a_{ij}))=\mu((a_{ij}))\nu((a_{ij}))$ shows us that these indices add, and hence get sent to the sum of the indices in $\Bbb Z^n$.