For a simple finite dimensional Lie algebra \(\widehat{g}\) of type ADE, let \(g\) be the corresponding (untwisted) affine Lie algebra and \(U_q(\widehat g)\) its quantum affine algebra. In this paper, the author studies finite dimensional representations of \(U_q(\widehat g)\) using geometry of quiver varieties. His purpose is to solve the following conjecture affirmatively, that is, an equivariant \(K\)-homology group of the quiver variety gives the quantum affine algebra \(U_q(\widehat g)\), and to derive results whose analogues are known for \(H_q\).
In §1, the author recalls a new realization of \(U_q(\widehat g)\), called Drinfeld realization and introduces the quantum loop algebra \(U_q(Lg)\) as a subquotient of \(U_q(\widehat g)\), which will be studied rather than \(U_q(\widehat g)\). The basic results are recalled on finite dimensional representations of \(U_\varepsilon(Lg)\). And, several useful concepts are introduced.
In §2, the author introduces two types of quiver varieties \({\mathcal M}(w)\) and \({\mathcal M}_0(\infty, w)\) as analogues of \(T^*{\mathcal B}\) and the nilpotent cone \(\mathcal N\) respectively. Their elementary properties are given.
In §3–§8, the author prepares some results on quiver varieties and \(K\)-theory which will be used in later sections.
In §9–§11, the author considers an analogue of the Steinberg variety \[ Z(w) = {\mathcal M}(w)\times _{{\mathcal M}_0(\infty,w)}{\mathcal M}(w) \] and its equivariant \(K\)-homology \(K^{G_w\times \mathbb{C}^*}(Z(w))\). An algebra homomorphism is constructed from \(U_q(Lg)\) to \(K^{G_w\times \mathbb{C}^*}(Z(W)) \otimes_{\mathbb{Z}[q,q^{-1}]}\mathbb{Q}(q)\).
In §12, the author shows that the above homomorphism induces a homomorphism from \(U^\mathbb{Z}_q(Lg)\) to \(K^{G_w\times \mathbb{C}^*}(Z(w))/\text{torsion}\).
In §13, the author introduces a standard module \(M_{x,a}\). Thanks to a result in §7, it is proved to be isomorphic to \(H_*({\mathcal M}(w)^A_x,\mathbb{C})\) via the Chern character homomorphism. Also, it is shown that \(M_{x,a}\) is a finite dimensional \(l\)-highest weight module. It is conjectured that \(M_{x,a}\) is a tensor product of \(l\)-fundamental representations in some order, which is proved when the parameter is generic in §14.1.
In §14, it is verified that the standard modules \(M_{x,a}\) and \(M_{y,a}\) are isomorphic if and only if \(x\) and \(y\) are contained in the same stratum. Furthermore, the author shows that the index set \(\{\rho\}\) of the stratum coincides with the set \({\mathcal P} =\{P\}\) of \(l\)-dominant \(l\)-weights of \(M_{0,a}\), the standard module corresponding to the central fiber \(\pi^{-1}(0)\). And, the multiplicity formula \([M(P) : L(Q)] =\dim H^*(i^!_x IC({\mathcal M}^{\text{reg}}_0(\rho_Q)))\) is proved. The result here is simpler than the case of the affine Hecke algebra: nonconstant local systems never appear.
Let \(\text{Res }M(P)\) be the restriction of \(M(P)\) to a \(U_\varepsilon(g)\)-module. In §15, the author shows the multiplicity formula \([\text{Res }M(P) : L(w - v)] \dim H^*(i_x^! IC({\mathcal M}^{\text{reg}}_0(v, w)))\). This result is compatible with the conjecture that \(M(P)\) is a tensor product of \(l\)-fundamental representations since the restiction of an \(l\)-fundamental representation is simple for type \(A\), and Kostka polynomials give tensor product decompositions.
Two examples are given where \({\mathcal M}^{\text{reg}}_0(v, w)\) can be described explicitly.
As mentioned in the Introduction of this paper, \(U_q(\widehat{g})\) has another realization, called the Drinfeld new realization, which can be applied to any symmetrizable Kac-Moody algebra \(g\), not necessarily a finite dimensional one. This generalization also fits the result in this paper, since quiver varieties can be defined for arbitrary finite graphs. If finite dimensional representations are replaced by \(l\)-integrable representations, parts of the result in this paper can be generalized to a Kac-Moody algebra \(g\), at least when it is symmetric.
If equivariant \(K\)-homology is replaced by equivariant homology, one should get the Yangian \(Y(g)\) instead of \(U_q(\widehat{g})\). The conjecture is motivated again by the analogy of quiver varieties with \(T^*\mathcal B\). As an application, the affirmative solution of the conjecture implies that the representation theory of \(U_q(\widehat g)\) and that of the Yangian are the same.