Mirabolic Howe Duality

Abstract

We introduce the space

{\mathcal{M}\mathcal{V}}_{\mathcal{n}\mathcal{|}\mathcal{m}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{MV}_{n|m}$$\end{document}

as the space of G-invariant functions on the variety of triples of two partial flag varieties and a vector space. Subsequently, we define the action of the mirabolic quantum Schur algebra via convolution multiplication. Finally, we present the geometric approach of the mirabolic Howe duality.

Keywords

Howe duality / mirabolic quantum groups / flag variety / 17B37

References

[1] Bao H, Kujawa J, Li Y, Wang W. Geometric Schur duality of classical type. Transform. Groups, 2018, 23(2): 329-389

[2] Baumann PThe q-Weyl group of a q-Schur algebra, 1999

[3] Beilinson AA, Lusztig G, MacPherson R. A geometric setting for the quantum deformation of GL_n. Duke Math. J., 1990, 61(2): 655-677

[4] Brauer R. On algebras which are connected with the semisimple continuous groups. Ann. of Math. (2), 1937, 38(4): 857-872

[5] Brown WP. An algebra related to the orthogonal group. Michigan Math. J., 1955, 3: 1-22

[6] Chari V, Pressley A. Quantum affine algebras and affine Hecke algebras. Pacific J. Math., 1996, 174(2): 295-326

[7] Chen C., Fan Z., Geometric Approach to I-Quantum Group of Affine Type D. 2024, arXiv:2403.04461

[8] Chen Q, Fan Z, Wang Q. Affine flag varieties of type D. J. Algebra, 2025, 674: 257-275

[9] Deng B, Du J, Fu QA Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory, 2012CambridgeCambridge University Press 401

[10] Dipper R, Doty S, Hu J. Brauer algebras, symplectic Schur algebras and Schur–Weyl duality. Trans. Amer. Math. Soc., 2008, 360(1): 189-213

[11] Doty S. New versions of Schur–Weyl duality. Finite Groups 2003, 2004BerlinWalter de Gruyter59-71

[12] Drinfel’d VG. Hopf algebras and the quantum Yang–Baxter equation. Dokl. Akad. Nauk SSSR, 1985, 283(5): 1060-1064

[13] Ehrig M, Stroppel C. Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality. Adv. Math., 2018, 331: 58-142

[14] Fan Z., Lai C.-J., Li Y., Luo L., Wang W., Affine flag varieties and quantum symmetric pairs. Mem. Amer. Math. Soc., 2020, 265(1285): v+123 pp.

[15] Fan Z, Lai C-J, Li Y, Luo L, Wang W, Watanabe H. Quantum Schur duality of affine type C with three parameters. Math. Res. Lett., 2020, 27(1): 79-114

[16] Fan Z., Lai C.-J., Li Y., Luo L., Wang W., Affine Hecke algebras and quantum symmetric pairs. Mem. Amer. Math. Soc., 2023, 281(1386): ix+92 pp.

[17] Fan Z, Li Y. Geometric Schur duality of classical type, II. Trans. Amer. Math. Soc. Ser. B, 2015, 2: 51-92

[18] Fan Z, Li Y. Affine flag varieties and quantum symmetric pairs, II, Multiplication formula. J. Pure Appl. Algebra, 2019, 223(10): 4311-4347

[19] Fan Z., Zhang Z., Ma H., Geometric approach to mirabolic Schur–Weyl duality of type A. J. Lond. Math. Soc. (2), 2025, 112(5): Paper No. e70343, 53 pp.

[20] Flicker YZ. Affine Schur duality. J. Lie Theory, 2021, 31(3): 681-718

[21] Fu Q. Integral affine Schur–Weyl reciprocity. Adv. Math., 2013, 243: 1-21

[22] Futorny V, Křižka L, Zhang J. Quantum Howe duality and invariant polynomials. J. Algebra, 2019, 530: 326-367

[23] Gavrilik AM, Klimyk AU. q-deformed orthogonal and pseudo-orthogonal algebras and their representations. Lett. Math. Phys., 1991, 21(3): 215-220

[24] Godelle E. Generic Hecke algebra for Renner monoids. J. Algebra, 2011, 343: 224-247

[25] Grojnowski I, Lusztig G. On bases of irreducible representations of quantum GL_n. Kazhdan–Lusztig Theory and Related Topics (Chicago, IL, 1989), 1992Providence, RIAmer. Math. Soc.167-174 139

[26] Jimbo M. A q-analogue of U(gl(N+1))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U({\frak{gl}}(N+1))$$\end{document}, Hecke algebra, and the Yang–Baxter equation. Lett. Math. Phys., 1986, 11(3): 247-252

[27] Lai KF, Zhang RB. Multiplicity free actions of quantum groups and generalized Howe duality. Lett. Math. Phys., 2003, 64(3): 255-272

[28] Luo L., Xu Z., Geometric Howe dualities of finite type. Adv. Math., 2022, 410: part B, Paper No. 108751, 38 pp.

[29] Magyar P, Weyman J, Zelevinsky A. Multiple flag varieties of finite type. Adv. Math., 1999, 141(1): 97-118

[30] Noumi M, Umeda T, Wakayama M. A quantum dual pair (sl2,on)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$({\frak{sl}}_{2},{\frak{o}}_{n})$$\end{document} and the associated Capelli identity. Lett. Math. Phys., 1995, 34(1): 1-8

[31] Noumi M, Umeda T, Wakayama M. Dual pairs, spherical harmonics and a Capelli identity in quantum group theory. Compositio Math., 1996, 104(3): 227-277

[32] Pouchin G. A geometric Schur–Weyl duality for quotients of affine Hecke algebras. J. Algebra, 2009, 321(1): 230-247

[33] Quesne C. Complementarity of su_q(3) and u_q(2) and q-boson realization of the su_q(3) irreducible representations. J. Phys. A, 1992, 25(22): 5977-5998

[34] Rosso D. The mirabolic Hecke algebra. J. Algebra, 2014, 405: 179-212

[35] Rosso D. Mirabolic quantum sl2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\frak{sl}}_{2}$$\end{document}. Transform. Groups, 2018, 23(1): 217-255

[36] Sartori A, Tubbenhauer D. Webs and q-Howe dualities in types BCD. Trans. Amer. Math. Soc., 2019, 371(10): 7387-7431

[37] Solomon L. The affine group, I. Bruhat decomposition. J. Algebra, 1972, 20: 512-539

[38] Varagnolo M, Vasserot E. On the decomposition matrices of the quantized Schur algebra. Duke Math. J., 1999, 100(2): 267-297

[39] Wang W. Lagrangian construction of the (gl_n, gl_m)-duality. Commun. Contemp. Math., 2001, 3(2): 201-214

[40] Zhang RB. Howe duality and the quantum general linear group. Proc. Amer. Math. Soc., 2003, 131(9): 2681-2692

RIGHTS & PERMISSIONS

Peking University