A projection formula for the ind-Grassmannian

Let $X = \bigcup_k X_k$ be the ind-Grassmannian of codimension $n$ subspaces of an infinite-dimensional torus representation. If $\cE$ is a bundle on $X$, we expect that $\sum_j (-1)^j \Lambda^j(\cE)$ represents the $K$-theoretic fundamental class $[\cO_Y]$ of a subvariety $Y \subset X$ dual to $\cE^*$. It is desirable to lift a $K$-theoretic "projection formula" from the finite-dimensional subvarieties $X_k$, but such a statement requires switching the order of the limits in $j$ and $k$. We find conditions in which this may be done, and consider examples in which $Y$ is the Hilbert scheme of points in the plane, the Hilbert scheme of an irreducible curve singularity, and the affine Grassmannian of $SL(2,\C)$. In the last example, the projection formula becomes an instance of the Weyl-Ka\c{c} character formula, which has long been recognized as the result of formally extending Borel-Weil theory and localization to $Y$ \cite{S}. See also \cite{C3} for a proof of the MacDonald inner product formula of type $A_n$ along these lines.