A projection formula for the ind-Grassmannian

TL;DR

This paper develops a projection formula for the ind-Grassmannian, linking $K$-theoretic classes of subvarieties to classical formulas like Weyl-Kac, with applications to Hilbert schemes and affine Grassmannians.

Contribution

It establishes conditions under which a $K$-theoretic projection formula can be extended from finite-dimensional subvarieties to the ind-Grassmannian, connecting to classical representation theory results.

Findings

Conditions for switching limits in the projection formula are identified.

Examples include Hilbert schemes of points and affine Grassmannian of SL(2,C).

The formula recovers the Weyl-Kac character formula in a specific case.

Abstract

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.