Computation of multiplicity of irreducible representation in some representation via geometric Satake correspondence

Question

Let $G$ be a reductive algebraic group over $\mathbb{C}$. Let $\operatorname{Gr}_{G}$ be the corresponding affine Grassmannian ($\operatorname{Gr}_{G}(\mathbb{C})=G(\mathbb{C}((z)))/G(\mathbb{C}[[z]])$). Let $\operatorname{Perv}_{G(\mathbb{C}[[z]])}\operatorname{Gr}_{G}$ be the category of $G(\mathbb{C}[[z]])$-equivariant perverse sheaves on $\operatorname{Gr}_{G}$. (which via Satake correspondence is equivalent to the category $\operatorname{Rep}\check{G}$). Suppose I have a perverse sheaf $\mathcal{P}$ from $\operatorname{Perv}_{G(\mathbb{C}[[z]])}\operatorname{Gr}_{G}$. Then it corresponds to some representation of $\check{G}$. Actually it is just isomorphic to the sum of sheaves $\operatorname{IC}^{\lambda}$ ($\operatorname{IC}$-extensions of constant shifted sheaves on $G(\mathbb{C}[[z]])$ - orbits). Suppose I have fixed some dominant coweight $\lambda$. Then $\operatorname{IC}^{\lambda}$ appears in $\mathcal{P}$ with some multiplicity.

The question is: how to compute this multiplicity geometrically? (should I something like look at the stalk of $\mathcal{H}^{-\operatorname{dim}(\operatorname{Gr}^{\lambda})}(\mathcal{P})$ in the point $z^{\lambda}$?) Thanks!

Answer 1

Yes, the multiplicity space is given by $\mathcal H_{z^\lambda}^{- dim Gr^\lambda}(\mathcal P) $. See, for example, the argument given in the proof of Proposition 3.1 of https://arxiv.org/pdf/math/0304176.pdf.