# What does the weights of Lie group mean?

Let $\Delta=\{\alpha_1,\alpha_2\}$ be the simple root system

of the exceptional Lie group $G_2$ with $\alpha_1$ is short and $\alpha_2$ is long, so $\lambda_1=2\alpha_1+\alpha_2,\lambda_2=3\alpha_1+2\alpha_2$ are the fundamental dominant weights. Let $T$ be the maximal torus of $G_2,$ then $H^*(BT;Z)$ is a polynomial algebra on two generators. Can we see $\alpha_1$ and $\alpha_2$ as the generators? What about $\lambda_1$ and $\lambda_2$?

-

Now back to your question. The elements α1 and α2 form a basis of BT, where T now refers to the maximal torus of G2. So you get an isomorphism H*(BT;ℤ) $\xrightarrow{\sim}$ ℤ[α1, α2]. But λ1 and λ2 also form a basis of BT. So you get another isomoprhism H*(BT;ℤ) $\xrightarrow{\sim}$ ℤ[λ1, λ2].