On abstract homomorphisms of some special unitary groups

Rapinchuk, Igor; Ruiter, Joshua

doi:10.1090/proc/15991

On abstract homomorphisms of some special unitary groups
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by Igor A. Rapinchuk and Joshua Ruiter PDF

Proc. Amer. Math. Soc. 150 (2022), 4241-4258 Request permission

Abstract:

We analyze the abstract representations of the groups of rational points of even-dimensional quasi-split special unitary groups associated with quadratic field extensions. We show that, under certain assumptions, such representations have a standard description, as predicted by a conjecture of Borel and Tits [Ann. of Math. (2) 97 (1973), pp. 499–571]. Our method extends the approach introduced by the first author in [Proc. Lond. Math. Soc. (3) 102 (2011), pp. 951–983] to study abstract representations of Chevalley groups and is based on the construction and analysis of a certain algebraic ring associated to a given abstract representation.

References

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Student economy edition, Addison-Wesley Series in Mathematics, Westview Press, Boulder, CO, 2016. For the 1969 original see [ MR0242802]. MR 3525784
A. Borel, Linear algebraic groups. Second enlarged edition, GTM 126, Springer, 1997.
Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150 (French). MR 207712
Armand Borel and Jacques Tits, Compléments à l’article: “Groupes réductifs”, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 253–276 (French). MR 315007
Armand Borel and Jacques Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571 (French). MR 316587, DOI 10.2307/1970833
Mitya Boyarchenko and Igor A. Rapinchuk, On abstract representations of the groups of rational points of algebraic groups in positive characteristic, Arch. Math. (Basel) 107 (2016), no. 6, 569–580. MR 3571148, DOI 10.1007/s00013-016-0971-6
Brian Conrad, Ofer Gabber, and Gopal Prasad, Pseudo-reductive groups, 2nd ed., New Mathematical Monographs, vol. 26, Cambridge University Press, Cambridge, 2015. MR 3362817, DOI 10.1017/CBO9781316092439
Vinay V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math. 100 (1978), no. 2, 303–386. MR 489962, DOI 10.2307/2373853
V. A. Petrov and A. K. Stavrova, Elementary subgroups in isotropic reductive groups, Algebra i Analiz 20 (2008), no. 4, 160–188 (Russian); English transl., St. Petersburg Math. J. 20 (2009), no. 4, 625–644. MR 2473747, DOI 10.1090/S1061-0022-09-01064-4
Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, vol. 9, Springer-Verlag, New York-Berlin, 1982. MR 674652
Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
Igor A. Rapinchuk, On linear representations of Chevalley groups over commutative rings, Proc. Lond. Math. Soc. (3) 102 (2011), no. 5, 951–983. MR 2795729, DOI 10.1112/plms/pdq043
Igor A. Rapinchuk, On abstract representations of the groups of rational points of algebraic groups and their deformations, Algebra Number Theory 7 (2013), no. 7, 1685–1723. MR 3117504, DOI 10.2140/ant.2013.7.1685
Igor A. Rapinchuk, Abstract homomorphisms of algebraic groups and applications, Handbook of group actions. Vol. II, Adv. Lect. Math. (ALM), vol. 32, Int. Press, Somerville, MA, 2015, pp. 397–447. MR 3382035
Igor A. Rapinchuk, On abstract homomorphisms of Chevalley groups over the coordinate rings of affine curves, Transform. Groups 24 (2019), no. 4, 1241–1259. MR 4038092, DOI 10.1007/s00031-018-9494-5
Gary M. Seitz, Abstract homomorphisms of algebraic groups, J. London Math. Soc. (2) 56 (1997), no. 1, 104–124. MR 1462829, DOI 10.1112/S0024610797005176
A. Stavrova, Homotopy invariance of non-stable $K_1$-functors, J. K-Theory 13 (2014), no. 2, 199–248. MR 3189425, DOI 10.1017/is013006012jkt232
A. Stavrova, On the congruence kernel of isotropic groups over rings, Trans. Amer. Math. Soc. 373 (2020), no. 7, 4585–4626. MR 4127856, DOI 10.1090/tran/8091
Michael R. Stein, Surjective stability in dimension $0$ for $K_{2}$ and related functors, Trans. Amer. Math. Soc. 178 (1973), 165–191. MR 327925, DOI 10.1090/S0002-9947-1973-0327925-8
Robert Steinberg, Lectures on Chevalley groups, University Lecture Series, vol. 66, American Mathematical Society, Providence, RI, 2016. Notes prepared by John Faulkner and Robert Wilson; Revised and corrected edition of the 1968 original [ MR0466335]; With a foreword by Robert R. Snapp. MR 3616493, DOI 10.1090/ulect/066