Let $G=SL\_n$, it is proven that $R:=H\_\*(Gr\_G)\cong \mathbb{C}[\sigma\_1,...,\sigma\_{n-1}]$ where $\sigma\_i$ are of degree $2i$ as a polynomial ring generated by $n-1$ variables and the ring structure comes from the fact the $Gr\_G$ is homotopic to $\Omega K$ ($K$ is the maximal compact subgroup of $G$) and $\Omega K$ has a group structure. It is known that $R$ can be viewed as a subring of symmetric function $Sym$ by mapping each $\sigma\_i$ to $h\_i$ ($i$-th complete homogeneous symmetric function of degree $i$ ). 



> 
> My question is: Why should we view $R$ as a subring in $Sym$ in such
>  way? What is the geometric reason behind it?
> 
> 
> 



how to view homology of affine Grassmannian as a subring of symmetric function
You should look at the [book](https://arxiv.org/pdf/1301.3569.pdf) of Lam, Lapointe, Morse, Schilling, Shimozono and Zabrocki. More specifically, under k-Schur functions and how/why they constitute a basis of $H\_\*(Gr\_{SL\_k})$. They mainly work in K-theory, but one of the main results in *loc. cit* originally proved by [Lam](https://arxiv.org/pdf/math/0603125.pdf) is that there is a natural Hopf algebra isomorphism between the Hopf subalgebra of k-Schur functions $\Lambda\_{(k)}:=\mathbb{Q}[h\_1,\ldots,h\_{k-1}]\subset \Lambda$ and $H\_\*(Gr\_{SL\_k})$. See the references therein for the Hopf algebra structures.


Geometrically the k-Schurs are just the Schubert basis in the (K-)homology of the affine Grassmannian. Multiplication by $h\_i$ is more interesting in this basis, it's given by the k-Pieri rule. 


