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 $\lambda\_2$?



What does the weights of Lie group mean?
Let *T* be an arbitrary compact torus.
The second cohomology group of *BT* (with arbitrary coefficients, call that ring *k*) generates the full cohomology freely as an algebra. In other words, if you pick a *k*-basis *x*1, *x*2,... of *H*2(*BT*), then you get an isomorphism of *H*\*(*BT*) with *k*[*x*1, *x*2,...].


Now let's specialise to the case *k*=ℤ. In that case, the second cohomology group of *BT* is canonically isomorphic to the group of characters of *T*, i.e., to the group of homomorphisms from *T* → *S*1. Given a character χ :*T* → *S*1, the corresponding element of *H*2(*BT*) is represented by (the first Chern class of) the complex line bundle *ET* ×T ℂχ → *BT* = *ET*/*T*.


Now back to your question. The elements α1 and α2 form a basis of *BT*, where *T* now refers to the maximal torus of *G*2. 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].




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Let me also answer the question in the title of your question:


By "weight of a Lie group", one means a homomorphism from its maximal torus to *S*1.


