arXiv subjects
Publications and source records attributed to Hiraku Nakajima.
We study finite dimensional representations of the quantum affine algebra, using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.
Let $G$ be a nontrivial finite subgroup of $\SL_n(\C)$. Suppose that the quotient singularity $\C^n/G$ has a crepant resolution $π\colon X\to \C^n/G$ (i.e. $K_X = \shfO_X$). There is a slightly imprecise conjecture, called the McKay correspondence, stating that there is a relation between the Grothendieck group (or (co)homology group) of $X$ and the representations (or conjugacy classes) of $G$ with a ``certain compatibility'' between the intersection product and the tensor product (see e.g. \cite{Maizuru}). The purpose of this paper is to give more precise formulation of the conjecture when $X$ can be given as a certain variety associated with the Hilbert scheme of points in $\C^n$. We give the proof of this new conjecture for an abelian subgroup $G$ of $\SL_3(\C)$.
The Jack symmetric polynomials $P_λ^{(α)}$ form a class of symmetric polynomials which are indexed by a partition $λ$ and depend rationally on a parameter $α$. They reduced to the Schur polynomials when $α=1$, and to other classical families of symmetric polynomials for several specific parameters. It is well-known that Schur polynomials can be realized as certain elements of homology groups of Grassmann manifolds. The purpose of this paper is to give a similar geometric realization for Jack polynomials. However, spaces which we use are totally different. Our spaces are Hilbert schemes of points on a surface X which is the total space of a line bundle L over the projective line. The parameter $α$ in Jack polynomials relates to our surface X by $α= - $, where C is the zero section, and is the self-intersection number of C.
Various constructions of the affine Lie algebra action on the homology group of moduli spaces of instantons on 4-manifolds are discussed. The analogy between the local-global principle and the role of mass is also explained. The detailed proofs are given in separated papers \cite{Na-algebra,Na-Hilbert}.
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