## The Steinberg-Lusztig tensor product theorem, Casselman-Shalika, and LLT polynomials

HTML articles powered by AMS MathViewer

- by Martina Lanini and Arun Ram
- Represent. Theory
**23**(2019), 188-204 - DOI: https://doi.org/10.1090/ert/524
- Published electronically: April 2, 2019
- PDF | Request permission

## Abstract:

In this paper we establish a Steinberg-Lusztig tensor product theorem for abstract Fock space. This is a generalization of the type A result of Leclerc-Thibon and a Grothendieck group version of the Steinberg-Lusztig tensor product theorem for representations of quantum groups at roots of unity. Although the statement can be phrased in terms of parabolic affine Kazhdan-Lusztig polynomials and thus has geometric content, our proof is combinatorial, using the theory of crystals (Littelmann paths). We derive the Casselman-Shalika formula as a consequence of the Steinberg-Lusztig tensor product theorem for abstract Fock space.## References

- Ben Brubaker, Daniel Bump, and Solomon Friedberg,
*Matrix coefficients and Iwahori-Hecke algebra modules*, Adv. Math.**299**(2016), 247â€“271. MR**3519469**, DOI 10.1016/j.aim.2016.05.012 - W. Casselman and J. Shalika,
*The unramified principal series of $p$-adic groups. II. The Whittaker function*, Compositio Math.**41**(1980), no.Â 2, 207â€“231. MR**581582** - Vyjayanthi Chari and Andrew Pressley,
*A guide to quantum groups*, Cambridge University Press, Cambridge, 1994. MR**1300632** - E. Frenkel, D. Gaitsgory, and K. Vilonen,
*Whittaker patterns in the geometry of moduli spaces of bundles on curves*, Ann. of Math. (2)**153**(2001), no.Â 3, 699â€“748. MR**1836286**, DOI 10.2307/2661366 - I. Grojnowski and M. Haiman,
*Affine Hecke algebras and positivity of LLT and Macdonald polynomials*, 2007, available from http://math.berkeley.edu/$\sim$haiman - JĂ©rĂ©mie Guilhot,
*Admissible subsets and Littelmann paths in affine Kazhdan-Lusztig theory*, Transform. Groups**23**(2018), no.Â 4, 915â€“938. MR**3869423**, DOI 10.1007/s00031-018-9495-4 - Thomas J. Haines, Robert E. Kottwitz, and Amritanshu Prasad,
*Iwahori-Hecke algebras*, J. Ramanujan Math. Soc.**25**(2010), no.Â 2, 113â€“145. MR**2642451** - Gordon James and Andrew Mathas,
*A $q$-analogue of the Jantzen-Schaper theorem*, Proc. London Math. Soc. (3)**74**(1997), no.Â 2, 241â€“274. MR**1425323**, DOI 10.1112/S0024611597000099 - Victor G. Kac,
*Infinite-dimensional Lie algebras*, 3rd ed., Cambridge University Press, Cambridge, 1990. MR**1104219**, DOI 10.1017/CBO9780511626234 - M. Kashiwara, T. Miwa, and E. Stern,
*Decomposition of $q$-deformed Fock spaces*, Selecta Math. (N.S.)**1**(1995), no.Â 4, 787â€“805. MR**1383585**, DOI 10.1007/BF01587910 - Masaki Kashiwara and Toshiyuki Tanisaki,
*Kazhdan-Lusztig conjecture for affine Lie algebras with negative level*, Duke Math. J.**77**(1995), no.Â 1, 21â€“62. MR**1317626**, DOI 10.1215/S0012-7094-95-07702-3 - D. Kazhdan and G. Lusztig,
*Tensor structures arising from affine Lie algebras. IV*, J. Amer. Math. Soc.**7**(1994), no.Â 2, 383â€“453. MR**1239507**, DOI 10.1090/S0894-0347-1994-1239507-1 - Friedrich Knop,
*On the Kazhdan-Lusztig basis of a spherical Hecke algebra*, Represent. Theory**9**(2005), 417â€“425. MR**2142817**, DOI 10.1090/S1088-4165-05-00237-2 - M. Lanini, A. Ram, and P. Sobaje,
*Fock space model for decomposition numbers for quantum groups at roots of unity*, to appear in Kyoto Math. J., arXiv:1612.03120. - Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon,
*Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties*, J. Math. Phys.**38**(1997), no.Â 2, 1041â€“1068. MR**1434225**, DOI 10.1063/1.531807 - Bernard Leclerc and Jean-Yves Thibon,
*Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials*, Combinatorial methods in representation theory (Kyoto, 1998) Adv. Stud. Pure Math., vol. 28, Kinokuniya, Tokyo, 2000, pp.Â 155â€“220. MR**1864481**, DOI 10.2969/aspm/02810155 - CĂ©dric Lecouvey,
*Parabolic Kazhdan-Lusztig polynomials, plethysm and generalized Hall-Littlewood functions for classical types*, European J. Combin.**30**(2009), no.Â 1, 157â€“191. MR**2460224**, DOI 10.1016/j.ejc.2008.02.007 - Peter Littelmann,
*Paths and root operators in representation theory*, Ann. of Math. (2)**142**(1995), no.Â 3, 499â€“525. MR**1356780**, DOI 10.2307/2118553 - G. Lusztig,
*Modular representations and quantum groups*, Classical groups and related topics (Beijing, 1987) Contemp. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1989, pp.Â 59â€“77. MR**982278**, DOI 10.1090/conm/082/982278 - George Lusztig,
*Monodromic systems on affine flag manifolds*, Proc. Roy. Soc. London Ser. A**445**(1994), no.Â 1923, 231â€“246. MR**1276910**, DOI 10.1098/rspa.1994.0058 - George Lusztig,
*Errata: â€śMonodromic systems on affine flag manifoldsâ€ť [Proc. Roy. Soc. London Ser. A 445 (1994), no. 1923, 231â€“246; MR1276910]*, Proc. Roy. Soc. London Ser. A**450**(1995), no.Â 1940, 731â€“732. MR**2105507** - Kendra Nelsen and Arun Ram,
*Kostka-Foulkes polynomials and Macdonald spherical functions*, Surveys in combinatorics, 2003 (Bangor), London Math. Soc. Lecture Note Ser., vol. 307, Cambridge Univ. Press, Cambridge, 2003, pp.Â 325â€“370. MR**2011741** - B. C. NgĂ´ and P. Polo,
*RĂ©solutions de Demazure affines et formule de Casselman-Shalika gĂ©omĂ©trique*, J. Algebraic Geom.**10**(2001), no.Â 3, 515â€“547 (French, with English summary). MR**1832331** - Arun Ram,
*Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux*, Pure Appl. Math. Q.**2**(2006), no.Â 4, Special Issue: In honor of Robert D. MacPherson., 963â€“1013. MR**2282411**, DOI 10.4310/PAMQ.2006.v2.n4.a4 - Peng Shan,
*Graded decomposition matrices of $v$-Schur algebras via Jantzen filtration*, Represent. Theory**16**(2012), 212â€“269. MR**2915315**, DOI 10.1090/S1088-4165-2012-00416-2

## Bibliographic Information

**Martina Lanini**- Affiliation: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3FD United Kingdom
- MR Author ID: 990628
- Email: lanini@mat.uniroma2.it
**Arun Ram**- Affiliation: Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010 Australia
- MR Author ID: 316170
- Email: aram@unimelb.edu.au
- Received by editor(s): April 15, 2018
- Received by editor(s) in revised form: January 31, 2019
- Published electronically: April 2, 2019
- Additional Notes: This research was partially supported by grants DP1201001942 and DP130100674.

The first author was partially supported by Australian Research Council grant DP150103525. The first author also acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. - © Copyright 2019 American Mathematical Society
- Journal: Represent. Theory
**23**(2019), 188-204 - MSC (2010): Primary 17B37; Secondary 20C20
- DOI: https://doi.org/10.1090/ert/524
- MathSciNet review: 3933899

Dedicated: Dedicated to Friedrich Knop and Peter Littelmann on the occasion of their 60th birthdays