AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Affine flag varieties and quantum symmetric pairs
About this Title
Zhaobing Fan, School of science, Harbin Engineering University, Harbin, China 150001, Chun-Ju Lai, Department of Mathematics, University of Virginia, Charlottesville, VA 22904, Yiqiang Li, Department of Mathematics, University at Buffalo, SUNY, Buffalo, NY 14260, Li Luo, Department of Mathematics, East China Normal University, Shanghai, China 200241 and Weiqiang Wang, Department of Mathematics, East China Normal University, Shanghai, China 200241
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 265, Number 1285
ISBNs: 978-1-4704-4175-3 (print); 978-1-4704-6138-6 (online)
DOI: https://doi.org/10.1090/memo/1285
Published electronically: April 1, 2020
Keywords: Affine flag variety,
affine quantum symmetric pair,
canonical basis.
MSC: Primary 17B37, 20G25, 14F43.
Table of Contents
Chapters
- 1. Introduction
1. Affine flag varieties, Schur algebras, and Lusztig algebras
- 2. Constructions in affine type $A$
- 3. Lattice presentation of affine flag varieties of type $C$
- 4. Multiplication formulas for Chevalley generators
- 5. Coideal algebra type structures of Schur algebras and Lusztig algebras
2. Lusztig algebras and coideal subalgebras of $\mathbf {U}(\widehat {\mathfrak {sl}}_n)$
- 6. Realization of the idempotented coideal subalgebra $\dot {\mathbf {U}}^{\mathfrak {c}}_n$ of $\mathbf {U}(\widehat {\mathfrak {sl}}_n)$
- 7. A second coideal subalgebra of quantum affine $\mathfrak {sl}_\mathfrak {n}$
- 8. More variants of coideal subalgebras of quantum affine $\mathfrak {sl}_n$
3. Schur algebras and coideal subalgebras of $\mathbf {U}(\widehat {\mathfrak {gl}}_n)$
- 9. The stabilization algebra $\dot {\mathbf K}^{\mathfrak {c}}_n$ arising from Schur algebras
- 10. Stabilization algebras arising from other Schur algebras
- A. Constructions in finite type $C$
Abstract
The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$. In this paper we study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type $C$. We show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine $\mathfrak {sl}$ and $\mathfrak {gl}$ types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine $\mathfrak {sl}$ type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotented quantum affine $\mathfrak {gl}$ and its canonical basis.\frenchspacing
- 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). MR 751966
- A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
- Huanchen Bao, Kazhdan-Lusztig theory of super type D and quantum symmetric pairs, Represent. Theory 21 (2017), 247–276. MR 3696376, DOI 10.1090/ert/505
- Huanchen Bao, Jonathan Kujawa, Yiqiang Li, and Weiqiang Wang, Geometric Schur duality of classical type, Transform. Groups 23 (2018), no. 2, 329–389. MR 3805209, DOI 10.1007/s00031-017-9447-4
- H. Bao, Y. Li, and W. Wang, A geometric setting for the coideal algebra $\dot {\mathbf {U}}^\imath$ and compatibility of canonical bases, Appendix to [H. Bao, J. Kujawa, Y Li, and W. Wang, Geometric Schur duality of classical type. Transform. Groups 23 (2018), no. 2, 329–389], Transform. Groups 23 (2018), no. 2, 329–389, DOI: 10.1007/s00031-017-9447-4, MR3805209.
- Tom Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), no. 3, 209–216. MR 1996415, DOI 10.1007/s00031-003-0606-4
- Martina Balagović and Stefan Kolb, Universal K-matrix for quantum symmetric pairs, J. Reine Angew. Math. 747 (2019), 299–353. MR 3905136, DOI 10.1515/crelle-2016-0012
- H. Bao and W. Wang, A new approach to Kazhdan-Lusztig theory of type $B$ via quantum symmetric pairs, Astérisque 402, Société mathématique de France.
- Huanchen Bao and Weiqiang Wang, Canonical bases arising from quantum symmetric pairs, Invent. Math. 213 (2018), no. 3, 1099–1177. MR 3842062, DOI 10.1007/s00222-018-0801-5
- Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
- Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J. Math. 174 (1996), no. 2, 295–326. MR 1405590
- Richard Dipper and Stephen Donkin, Quantum $\textrm {GL}_n$, Proc. London Math. Soc. (3) 63 (1991), no. 1, 165–211. MR 1105721, DOI 10.1112/plms/s3-63.1.165
- Bangming Deng and Jie Du, Monomial bases for quantum affine $\mathfrak {sl}_n$, Adv. Math. 191 (2005), no. 2, 276–304. MR 2103214, DOI 10.1016/j.aim.2004.03.008
- Bangming Deng, Jie Du, and Qiang Fu, A double Hall algebra approach to affine quantum Schur-Weyl theory, London Mathematical Society Lecture Note Series, vol. 401, Cambridge University Press, Cambridge, 2012. MR 3113018
- Bangming Deng, Jie Du, Brian Parshall, and Jianpan Wang, Finite dimensional algebras and quantum groups, Mathematical Surveys and Monographs, vol. 150, American Mathematical Society, Providence, RI, 2008. MR 2457938
- Jie Du and Qiang Fu, Quantum affine $\mathfrak {gl}_n$ via Hecke algebras, Adv. Math. 282 (2015), 23–46. MR 3374521, DOI 10.1016/j.aim.2015.06.007
- Jie Du and Qiang Fu, The integral quantum loop algebra of $\mathfrak {gl}_n$, Int. Math. Res. Not. IMRN 20 (2019), 6179–6215. MR 4031235, DOI 10.1093/imrn/rnx300
- Richard Dipper and Gordon James, The $q$-Schur algebra, Proc. London Math. Soc. (3) 59 (1989), no. 1, 23–50. MR 997250, DOI 10.1112/plms/s3-59.1.23
- Richard Dipper and Gordon James, $q$-tensor space and $q$-Weyl modules, Trans. Amer. Math. Soc. 327 (1991), no. 1, 251–282. MR 1012527, DOI 10.1090/S0002-9947-1991-1012527-1
- V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
- Jie Du, Kazhdan-Lusztig bases and isomorphism theorems for $q$-Schur algebras, Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 121–140. MR 1197832, DOI 10.1090/conm/139/1197832
- Michael Ehrig and Catharina Stroppel, Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality, Adv. Math. 331 (2018), 58–142. MR 3804673, DOI 10.1016/j.aim.2018.01.013
- Z. Fan, C. Lai, Y. Li, L. Luo and W. Wang, Affine Hecke algebras and quantum symmetric pairs, Mem. Amer. Math. Soc. (to appear), arXiv:1609.06199v2.
- Zhaobing Fan and Yiqiang Li, Geometric Schur duality of classical type, II, Trans. Amer. Math. Soc. Ser. B 2 (2015), 51–92. MR 3402700, DOI 10.1090/S2330-0000-2015-00008-9
- Z. Fan and Y. Li, Positivity of canonical basis under comultiplication, Int. Math. Res. Notices (to appear), DOI: 10.1093/imrn/rnz047.
- Zhaobing Fan and Yiqiang Li, Affine flag varieties and quantum symmetric pairs, II. Multiplication formula, J. Pure Appl. Algebra 223 (2019), no. 10, 4311–4347. MR 3958094, DOI 10.1016/j.jpaa.2019.01.011
- R. M. Green, Hyperoctahedral Schur algebras, J. Algebra 192 (1997), no. 1, 418–438. MR 1449968, DOI 10.1006/jabr.1996.6935
- R. M. Green, The affine $q$-Schur algebra, J. Algebra 215 (1999), no. 2, 379–411. MR 1686197, DOI 10.1006/jabr.1998.7753
- I. Grojnowski and G. Lusztig, On bases of irreducible representations of quantum $\textrm {GL}_n$, Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 167–174. MR 1197834, DOI 10.1090/conm/139/1197834
- Victor Ginzburg and Éric Vasserot, Langlands reciprocity for affine quantum groups of type $A_n$, Internat. Math. Res. Notices 3 (1993), 67–85. MR 1208827, DOI 10.1155/S1073792893000078
- Victor Ginzburg, Nicolai Reshetikhin, and Éric Vasserot, Quantum groups and flag varieties, Mathematical aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992) Contemp. Math., vol. 175, Amer. Math. Soc., Providence, RI, 1994, pp. 101–130. MR 1302015, DOI 10.1090/conm/175/01840
- Roger Howe, Affine-like Hecke algebras and $p$-adic representation theory, Iwahori-Hecke algebras and their representation theory (Martina-Franca, 1999) Lecture Notes in Math., vol. 1804, Springer, Berlin, 2002, pp. 27–69. MR 1979924, DOI 10.1007/978-3-540-36205-0_{2}
- Nagayoshi Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215–236 (1964). MR 165016
- N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak {p}}$-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. MR 185016
- Michio Jimbo, A $q$-analogue of $U({\mathfrak {g}}{\mathfrak {l}}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252. MR 841713, DOI 10.1007/BF00400222
- M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
- Masaki Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383–413. MR 1262212, DOI 10.1215/S0012-7094-94-07317-1
- Stefan Kolb, Quantum symmetric Kac-Moody pairs, Adv. Math. 267 (2014), 395–469. MR 3269184, DOI 10.1016/j.aim.2014.08.010
- David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
- Gail Letzter, Symmetric pairs for quantized enveloping algebras, J. Algebra 220 (1999), no. 2, 729–767. MR 1717368, DOI 10.1006/jabr.1999.8015
- Gail Letzter, Coideal subalgebras and quantum symmetric pairs, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 117–165. MR 1913438
- Chun-Ju Lai and Li Luo, An elementary construction of monomial bases of modified quantum affine $\mathfrak {gl}_n$, J. Lond. Math. Soc. (2) 96 (2017), no. 1, 15–27. MR 3687937, DOI 10.1112/jlms.12049
- G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
- George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
- G. Lusztig, Cells in affine Weyl groups and tensor categories, Adv. Math. 129 (1997), no. 1, 85–98. MR 1458414, DOI 10.1006/aima.1997.1645
- G. Lusztig, Aperiodicity in quantum affine $\mathfrak {g}\mathfrak {l}_n$, Asian J. Math. 3 (1999), no. 1, 147–177. Sir Michael Atiyah: a great mathematician of the twentieth century. MR 1701926, DOI 10.4310/AJM.1999.v3.n1.a7
- George Lusztig, Transfer maps for quantum affine $\mathfrak {s}\mathfrak {l}_n$, Representations and quantizations (Shanghai, 1998) China High. Educ. Press, Beijing, 2000, pp. 341–356. MR 1802182
- G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442
- Y. Li and W. Wang, Positivity vs negativity of canonical basis, Proceedings for Lusztig’s 70th birthday conference, Bulletin of Institute of Mathematics Academia Sinica (New Series), 13 (2018), No. 2, 143–198, DOI: 10.21915/BIMAS.2018201.
- Kevin McGerty, On the geometric realization of the inner product and canonical basis for quantum affine $\mathfrak {s}\mathfrak {l}_n$, Algebra Number Theory 6 (2012), no. 6, 1097–1131. MR 2968635, DOI 10.2140/ant.2012.6.1097
- Masatoshi Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), no. 1, 16–77. MR 1413836, DOI 10.1006/aima.1996.0066
- A. L. Onishchik and È. B. Vinberg, Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites. MR 1064110
- G. Pouchin, A geometric Schur-Weyl duality for quotients of affine Hecke algebras, J. Algebra 321 (2009), no. 1, 230–247. MR 2469358, DOI 10.1016/j.jalgebra.2008.09.018
- Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. MR 1062796, DOI 10.1007/BF01231516
- Daniel S. Sage, The geometry of fixed point varieties on affine flag manifolds, Trans. Amer. Math. Soc. 352 (2000), no. 5, 2087–2119. MR 1491876, DOI 10.1090/S0002-9947-99-02295-3
- Olivier Schiffmann, Lectures on Hall algebras, Geometric methods in representation theory. II, Sémin. Congr., vol. 24, Soc. Math. France, Paris, 2012, pp. 1–141 (English, with English and French summaries). MR 3202707
- O. Schiffmann and E. Vasserot, Geometric construction of the global base of the quantum modified algebra of $\widehat {\mathfrak {gl}}_n$, Transform. Groups 5 (2000), no. 4, 351–360. MR 1800532, DOI 10.1007/BF01234797
- Michela Varagnolo and Eric Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), no. 2, 267–297. MR 1722955, DOI 10.1215/S0012-7094-99-10010-X