# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Tatsuya Akasaka and Masaki Kashiwara, Finite-dimensional representations of quantum affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997), no. 5, 839 – 867. · Zbl 0915.17011 · doi:10.2977/prims/1195145020 · doi.org [2] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1 – 15. · Zbl 0482.58013 · doi:10.1112/blms/14.1.1 · doi.org [3] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523 – 615. · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017 · doi.org [4] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch and topological \? theory for singular varieties, Acta Math. 143 (1979), no. 3-4, 155 – 192. · Zbl 0474.14004 · doi:10.1007/BF02392091 · doi.org [5] Jonathan Beck, Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), no. 3, 555 – 568. · Zbl 0807.17013 [6] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5 – 171 (French). · Zbl 0536.14011 [7] A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480 – 497. · Zbl 0275.14007 · doi:10.2307/1970915 · doi.org [8] James B. Carrell and Andrew John Sommese, \?*-actions, Math. Scand. 43 (1978/79), no. 1, 49 – 59. · Zbl 0416.32022 · doi:10.7146/math.scand.a-11762 · doi.org [9] Vyjayanthi Chari and Andrew Pressley, Fundamental representations of Yangians and singularities of \?-matrices, J. Reine Angew. Math. 417 (1991), 87 – 128. · Zbl 0726.17014 [10] Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. · Zbl 0839.17009 [11] Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras and their representations, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 59 – 78. · Zbl 0855.17009 [12] Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras at roots of unity, Represent. Theory 1 (1997), 280 – 328. · Zbl 0891.17013 [13] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. · Zbl 0879.22001 [14] C. De Concini, G. Lusztig, and C. Procesi, Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), no. 1, 15 – 34. · Zbl 0646.14034 [15] V. G. Drinfel$$^{\prime}$$d, A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13 – 17 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 2, 212 – 216. [16] Geir Ellingsrud and Stein Arild Strømme, Towards the Chow ring of the Hilbert scheme of \?², J. Reine Angew. Math. 441 (1993), 33 – 44. · Zbl 0814.14003 [17] E. Frenkel and E. Mukhin, Combinatorics of $$q$$-characters of finite-dimensional representations of quantum affine algebras, preprint, math.QA/9911112. · Zbl 1051.17013 [18] E. Frenkel and N. Reshetikhin, The $$q$$-characters of representations of quantum affine algebras and deformations of $$\mathcal W$$-algebras, preprint, math.QA/9810055. · Zbl 0973.17015 [19] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005 [20] V. Ginzburg, $${\mathfrak{g}}$$-modules, Springer’s representations and bivariant Chern classes, Adv. in Math. 61 (1986), 1-48. · Zbl 0601.22008 [21] Victor Ginzburg and Éric Vasserot, Langlands reciprocity for affine quantum groups of type \?_\?, Internat. Math. Res. Notices 3 (1993), 67 – 85. · Zbl 0785.17014 · doi:10.1155/S1073792893000078 · doi.org [22] Victor Ginzburg, Mikhail Kapranov, and Éric Vasserot, Langlands reciprocity for algebraic surfaces, Math. Res. Lett. 2 (1995), no. 2, 147 – 160. · Zbl 0914.11040 · doi:10.4310/MRL.1995.v2.n2.a4 · doi.org [23] I. Grojnowski, Affinizing quantum algebras: From $$D$$-modules to $$K$$-theory, preprint, 1994. · Zbl 0819.17009 [24] I. Grojnowski, Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), no. 2, 275 – 291. · Zbl 0879.17011 · doi:10.4310/MRL.1996.v3.n2.a12 · doi.org [25] G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Remarks on fermionic formula, preprint, math.QA/9812022. · Zbl 1032.81015 [26] David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153 – 215. · Zbl 0613.22004 · doi:10.1007/BF01389157 · doi.org [27] A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515 – 530. · Zbl 0837.16005 · doi:10.1093/qmath/45.4.515 · doi.org [28] M. Kleber, Finite dimensional representations of quantum affine algebras, preprint, math.QA/9809087. [29] Peter B. Kronheimer and Hiraku Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), no. 2, 263 – 307. · Zbl 0694.53025 · doi:10.1007/BF01444534 · doi.org [30] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), no. 2, 169 – 178. · Zbl 0473.20029 · doi:10.1016/0001-8708(81)90038-4 · doi.org [31] George Lusztig, Equivariant \?-theory and representations of Hecke algebras, Proc. Amer. Math. Soc. 94 (1985), no. 2, 337 – 342. , https://doi.org/10.1090/S0002-9939-1985-0784189-2 David Kazhdan and George Lusztig, Equivariant \?-theory and representations of Hecke algebras. II, Invent. Math. 80 (1985), no. 2, 209 – 231. · Zbl 0613.22003 · doi:10.1007/BF01388604 · doi.org [32] George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599 – 635. · Zbl 0715.22020 [33] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447 – 498. · Zbl 0703.17008 [34] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365 – 421. · Zbl 0738.17011 [35] G. Lusztig, Affine quivers and canonical bases, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 111 – 163. · Zbl 0776.17013 [36] George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. · Zbl 0788.17010 [37] George Lusztig, Cuspidal local systems and graded Hecke algebras. II, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 217 – 275. With errata for Part I [Inst. Hautes Études Sci. Publ. Math. No. 67 (1988), 145 – 202; MR0972345 (90e:22029)]. · Zbl 0841.22013 [38] G. Lusztig, On quiver varieties, Adv. Math. 136 (1998), no. 1, 141 – 182. · Zbl 0915.17008 · doi:10.1006/aima.1998.1729 · doi.org [39] G. Lusztig, Bases in equivariant \?-theory. II, Represent. Theory 3 (1999), 281 – 353. · Zbl 0999.20036 [40] -, Quiver varieties and Weyl group actions, preprint. · Zbl 0958.20036 [41] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. · Zbl 0899.05068 [42] A. Maffei, Quiver varieties of type $$A$$, preprint, math.AG/9812142. [43] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. · Zbl 0797.14004 [44] Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365 – 416. · Zbl 0826.17026 · doi:10.1215/S0012-7094-94-07613-8 · doi.org [45] Hiraku Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515 – 560. · Zbl 0970.17017 · doi:10.1215/S0012-7094-98-09120-7 · doi.org [46] -, Lectures on Hilbert schemes of points on surfaces, Univ. Lect. Ser. 18, AMS, 1999. CMP 2000:02 [47] Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583 – 591. · Zbl 0735.16009 · doi:10.1007/BF01231516 · doi.org [48] Yoshihisa Saito, Quantum toroidal algebras and their vertex representations, Publ. Res. Inst. Math. Sci. 34 (1998), no. 2, 155 – 177. · Zbl 0982.17008 · doi:10.2977/prims/1195144759 · doi.org [49] Y. Saito, K. Takemura, and D. Uglov, Toroidal actions on level 1 modules of \?_\?(\Hat \?\?_\?), Transform. Groups 3 (1998), no. 1, 75 – 102. · Zbl 0915.17012 · doi:10.1007/BF01237841 · doi.org [50] Reyer Sjamaar and Eugene Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), no. 2, 375 – 422. · Zbl 0759.58019 · doi:10.2307/2944350 · doi.org [51] Reyer Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2) 141 (1995), no. 1, 87 – 129. · Zbl 0827.32030 · doi:10.2307/2118628 · doi.org [52] Toshiyuki Tanisaki, Hodge modules, equivariant \?-theory and Hecke algebras, Publ. Res. Inst. Math. Sci. 23 (1987), no. 5, 841 – 879. · Zbl 0655.14004 · doi:10.2977/prims/1195176035 · doi.org [53] R. W. Thomason, Algebraic \?-theory of group scheme actions, Algebraic topology and algebraic \?-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539 – 563. [54] R. W. Thomason, Equivariant algebraic vs. topological \?-homology Atiyah-Segal-style, Duke Math. J. 56 (1988), no. 3, 589 – 636. · Zbl 0655.55002 · doi:10.1215/S0012-7094-88-05624-4 · doi.org [55] R. W. Thomason, Une formule de Lefschetz en \?-théorie équivariante algébrique, Duke Math. J. 68 (1992), no. 3, 447 – 462 (French). · Zbl 0813.19002 · doi:10.1215/S0012-7094-92-06817-7 · doi.org [56] M. Varagnolo and E. Vasserot, Double-loop algebras and the Fock space, Invent. Math. 133 (1998), no. 1, 133 – 159. · Zbl 0904.17014 · doi:10.1007/s002220050242 · doi.org [57] -, On the $$K$$-theory of the cyclic quiver variety, preprint, math.AG/9902091. [58] E. Vasserot, Affine quantum groups and equivariant \?-theory, Transform. Groups 3 (1998), no. 3, 269 – 299. · Zbl 0969.17009 · doi:10.1007/BF01236876 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.