# on geometric Satake and functions

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:

$f_{\lambda}(x)=Tr(Fr_{\bar{x}},IC_{\lambda,\bar{x}})$.

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

$H^{*}(\overline{Gr}^{\lambda},IC_{\lambda})=\bigoplus\limits_{\nu\leq\lambda}H^{2(\rho,\nu)}(S_{\nu}\cap\overline{Gr}^{\lambda},IC_{\lambda})$

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

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