Mod ℓ Weil representations and Deligne–Lusztig inductions for unitary groups

Imai, Naoki; Tsushima, Takahiro

doi:10.1090/ert/685

Mod $\ell$ Weil representations and Deligne–Lusztig inductions for unitary groups
HTML articles powered by AMS MathViewer

by Naoki Imai and Takahiro Tsushima;

Represent. Theory 29 (2025), 35-59

DOI: https://doi.org/10.1090/ert/685

Published electronically: February 11, 2025

HTML | PDF

Abstract:

We study the mod $\ell$ Weil representation of a finite unitary group and related Deligne–Lusztig inductions. In particular, we study their decomposition as representations of a symplectic group, and give a construction of a mod $\ell$ Howe correspondence for $(\mathrm {Sp}_{2n},\mathrm {O}_2^-)$ including the case where $p=2$.

References

Anne-Marie Aubert, Séries de Harish-Chandra de modules et correspondance de Howe modulaire, J. Algebra 165 (1994), no. 3, 576–601 (French, with French summary). MR 1275922, DOI 10.1006/jabr.1994.1132
Michel Broué and Jean Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. Reine Angew. Math. 395 (1989), 56–67 (French). MR 983059, DOI 10.1515/crll.1989.395.56
Cédric Bonnafé, Representations of $\textrm {SL}_2(\Bbb F_q)$, Algebra and Applications, vol. 13, Springer-Verlag London, Ltd., London, 2011. MR 2732651, DOI 10.1007/978-0-85729-157-8
Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180, DOI 10.1007/978-1-4612-4404-2
François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841, DOI 10.1017/CBO9781139172417
Paul Fong and Bhama Srinivasan, The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982), no. 1, 109–153. MR 671655, DOI 10.1007/BF01389188
Paul Gérardin, Weil representations associated to finite fields, J. Algebra 46 (1977), no. 1, 54–101. MR 460477, DOI 10.1016/0021-8693(77)90394-5
Robert M. Guralnick, Kay Magaard, Jan Saxl, and Pham Huu Tiep, Cross characteristic representations of symplectic and unitary groups, J. Algebra 257 (2002), no. 2, 291–347. MR 1947325, DOI 10.1016/S0021-8693(02)00527-6
Robert M. Guralnick and Pham Huu Tiep, Cross characteristic representations of even characteristic symplectic groups, Trans. Amer. Math. Soc. 356 (2004), no. 12, 4969–5023. MR 2084408, DOI 10.1090/S0002-9947-04-03477-4
Ryoshi Hotta and Kiyoshi Matsui, On a lemma of Tate-Thompson, Hiroshima Math. J. 8 (1978), no. 2, 255–268. MR 486178
Gerhard Hiss and Gunter Malle, Low-dimensional representations of special unitary groups, J. Algebra 236 (2001), no. 2, 745–767. MR 1813499, DOI 10.1006/jabr.2000.8513
Roger E. Howe, On the character of Weil’s representation, Trans. Amer. Math. Soc. 177 (1973), 287–298. MR 316633, DOI 10.1090/S0002-9947-1973-0316633-5
Luc Illusie, Théorie de Brauer et caractéristique d’Euler-Poincaré (d’après P. Deligne), The Euler-Poincaré characteristic (French), Astérisque, vol. 82, Soc. Math. France, Paris, 1981, pp. 161–172 (French). MR 629127
Naoki Imai and Takahiro Tsushima, Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences, Eur. J. Math. 9 (2023), no. 2, Paper No. 31, 34. MR 4581884, DOI 10.1007/s40879-023-00620-5
Naoki Imai and Takahiro Tsushima, Shintani lifts for Weil representations of unitary groups over finite fields, Math. Res. Lett. 31 (2024), no. 5, 1471–1491. MR 4837562, DOI 10.4310/mrl.241211044338
Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341, DOI 10.1017/CBO9780511629235
Martin W. Liebeck, E. A. O’Brien, Aner Shalev, and Pham Huu Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939–1008. MR 2654085, DOI 10.4171/JEMS/220
G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), no. 2, 125–175. MR 463275, DOI 10.1007/BF01390002
Masahiko Saito, Représentations unitaires des groupes symplectiques, J. Math. Soc. Japan 24 (1972), 232–251 (French). MR 299728, DOI 10.2969/jmsj/02420232
Ken-ichi Shinoda, The characters of Weil representations associated to finite fields, J. Algebra 66 (1980), no. 1, 251–280. MR 591256, DOI 10.1016/0021-8693(80)90123-4
Pham Huu Tiep and Alexander E. Zalesskii, Some characterizations of the Weil representations of the symplectic and unitary groups, J. Algebra 192 (1997), no. 1, 130–165. MR 1449955, DOI 10.1006/jabr.1996.6943
André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI 10.1007/BF02391012
Donald L. White, Decomposition numbers of $\textrm {Sp}(4,q)$ for primes dividing $q\pm 1$, J. Algebra 132 (1990), no. 2, 488–500. MR 1061493, DOI 10.1016/0021-8693(90)90143-C

AMS Website Down

Representation Theory

Mod $\ell$ Weil representations and Deligne–Lusztig inductions for unitary groups
HTML articles powered by AMS MathViewer

Abstract: