Let $G(F)/G(O)$ the affine grassmanian with $F=k((t))$ where $k$ is a finite field.

For $\lambda$ a dominant cocharacter, we have by Cartan decomposition the schubert strata $\overline{Gr^{\lambda}}$. Let $IC_{\lambda}$ be the intersection complex on $\overline{Gr}^{\lambda}$.

We can associate to $IC_{\lambda}$ the function $f_{\lambda}$ defined by:


Mirkovic and Vilonen defined for a cocharacter $\nu\in X_{*}(T)$ a strata $S_{\nu}$ such that


where $2\rho$ is the sum of all positive roots.

My question is what is the value of $f_{\lambda}(x)$ for $x\in S_{\nu}\cap\overline{Gr}^{\lambda}$?


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