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Quiver varieties and finite dimensional representations of quantum affine algebras. (English) Zbl 0981.17016
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.
Reviewer: Li Fang (Hangzhou)

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
14L30 Group actions on varieties or schemes (quotients)
16G20 Representations of quivers and partially ordered sets
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
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