×

On the Steinberg module of Chevalley groups. (English) Zbl 1091.20031

Let \(G\) be a simple Chevalley group with root system \(\Phi\). If \(K\) is a field, then the Steinberg module \(S_K\) for \(G(K)\) is the top homology of the Tits building \(T_K\). It contains important cohomological information. By the Solomon-Tits theorem \(S\) is a cyclic module for the group ring \(K[G(K)]\). Now suppose \(\mathcal D\) is a Euclidean domain with field of fractions \(K\). It is then of interest to know that \(S_K\) is also cyclic as a \(K[G(\mathcal D)]\)-module. This has been shown by Ash and Rudolph for \(G=PSL_n\) and by Gunnells for the symplectic case. See A. Ash and L. Rudolph [Invent. Math. 55, 241-250 (1979; Zbl 0426.10023)] and P. E. Gunnells [Duke Math. J. 102, No. 2, 329-350 (2000; Zbl 0988.11023)].
The author now gives new proofs, valid for all \(A\), \(B\), \(C\), \(D\) types and also for types \(E_6\), \(E_7\). What these types have in common is that there is a maximal parabolic subgroup with Abelian unipotent radical. First one needs a completely new presentation of \(S_K\) as a \(K[G(K)]\)-module. This presentation involves only \(SL_2\) relations and works for any \(\Phi\). To get to the main result requires extensive ‘word processing’. A representative in \(K[G(K)]\) of an element of \(S_K\) is modified repeatedly until it lies in \(K[G(\mathcal D)]\). Although this word processing is guided by the Euclidean norm amongst other things, the method probably extends to cases where \(\mathcal D\) is only a UFD.

MSC:

20G10 Cohomology theory for linear algebraic groups
20G30 Linear algebraic groups over global fields and their integers
11F75 Cohomology of arithmetic groups
20G05 Representation theory for linear algebraic groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ash, A.: A note on minimal modular symbols. Proc. Amer. Math. Soc. 96(3), 394-396 (1986) · Zbl 0589.20033 · doi:10.1090/S0002-9939-1986-0822426-7
[2] Ash, A., Gunnells, P.E., McConnell, M.: Cohomology of congruence subgroups of SL4(?). J. Number Theory 94(1), 181-212 (2002) · Zbl 1006.11026 · doi:10.1006/jnth.2001.2730
[3] Ash, A., McConnell, M.: Experimental indications of three-dimensional Galois representations from the cohomology of SL(3,Z). Experiment. Math. 1(3), 209-223 (1992) · Zbl 0780.11029
[4] Ash, A., Pinch, R., Taylor, R.: An extension of Q attached to a nonselfdual automorphic form on GL(3). Math. Ann. 291(4), 753-766 (1991) · Zbl 0737.11015 · doi:10.1007/BF01445238
[5] Ash, A., Rudolph, L.: The modular symbol and continued fractions in higher dimensions. Invent. Math. 55(3), 241-250 (1979) · Zbl 0426.10023 · doi:10.1007/BF01406842
[6] R. Bieri and B. Eckmann, Groups with homological duality generalizing Poincare duality. Invent. Math. 20, 103-124 (1973) · Zbl 0274.20066
[7] Borel, A., Serre, J-P.: Corners and arithmetic groups. Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault. Comment. Math. Helv. 48, 436-491 (1973) · Zbl 0274.22011
[8] Brown, K.S.: Cohomology of groups. Graduate Texts in Mathematics. Springer- Verlag, New York-Berlin, 87, x+306, (1982)
[9] Bruhat, F., Tits, J.: Groupes reductifs sur un corps local. Inst. Hautes Etudes Sci. Publ. Math. No. 41 , 5-251 (1972) · Zbl 0254.14017 · doi:10.1007/BF02715544
[10] Carter, R.W.: Simple groups of Lie type. Pure and Applied Mathematics. John Wiley & Sons, London-New York-Sydney, 28, viii+331, (1972) · Zbl 0248.20015
[11] Chevalley, C.: Sur certains groupes simples. Tohoku Math. J. 7(2), 14-66 (1955) · Zbl 0066.01503 · doi:10.2748/tmj/1178245104
[12] Séminaire C. Chevalley, 1956-1958. Classification des groupes de Lie algébriques, Secrétariat Math., 11 rue Pierre Curie, Paris, (1958)
[13] Clark, D.A., Murty, R.M.: The Euclidean algorithm for Galois extensions of Q. J. Reine Angew. Math. 459, 151-162 (1995) · Zbl 0814.11049
[14] Gunnells, P.E.: Modular symbols for Q-rank one groups and Voronoi reduction. J. Number Theory 75(2), 198-219 (1999) · Zbl 0977.11023 · doi:10.1006/jnth.1998.2347
[15] Gunnells, P.E.: Symplectic modular symbols. Duke Math. J. 102(2), 329-350 (2000) · Zbl 0988.11023 · doi:10.1215/S0012-7094-00-10226-8
[16] Gunnells, P.E.: Computing Hecke eigenvalues below the cohomological dimension. Experiment. Math. 9(3), 351-367 (2000) · Zbl 1037.11037
[17] Gunnells, P.E.: Finiteness of minimal modular symbols for SLn. J. Number Theory 82(1), 134-139 (2000) · Zbl 0987.11040 · doi:10.1006/jnth.1999.2481
[18] Gunnells, P.E., McConnell, M.: Hecke operators and -groups associated to self-adjoint homogeneous cones. J. Number Theory 100(1), 46-71 (2003) · Zbl 1088.11039 · doi:10.1016/S0022-314X(02)00041-0
[19] Iwahori, N., Matsumoto, H.: On some Bruhat decomposition and the structure of the Hecke rings of -adic Chevalley groups. Inst. Hautes Etudes Sci. Publ. Math. No. 25, 5-48 (1965) · Zbl 0228.20015 · doi:10.1007/BF02684396
[20] Humphreys, J.E.: Introduction to Lie algebras and representation theory, Springer, New York, (1978) · Zbl 0447.17001
[21] Kostant, B.: Groups over Z, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 90-98 (1966)
[22] Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989), Edited by J.-P. Labesse and J. Schwermer. Lecture Notes in Math., 1447, Springer, Berlin, (1990)
[23] Lee, R., Szczarba, R.H.: On the homology of congruence subgroups and K3(Z). Proc. Nat. Acad. Sci. U.S.A. 72, 651-653 (1975) · Zbl 0305.18008 · doi:10.1073/pnas.72.2.651
[24] Lee, R., Szczarba, R.H.: On the homology and cohomology of congruence subgroups. Invent. Math. 33 (1), 15-53 (1976) · Zbl 0332.18015 · doi:10.1007/BF01425503
[25] Lee, R., Szczarba, R.H.: The group K3(Z) is cyclic of order forty-eight. Ann. of Math. (2) 104(1), 31-60 (1976) · Zbl 0341.18008
[26] Lee, R., Szczarba, R.H.: On the torsion in K4(Z) and K5(Z). Duke Math. J. 45(1), 101-129 (1978) · Zbl 0385.18009 · doi:10.1215/S0012-7094-78-04508-8
[27] Manin, Ju.I.: Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR Ser. Mat. 36, 19-66 (1972) · Zbl 0243.14008
[28] Ono, T.: Sur les groupes de Chevalley. J. Math. Soc. Japan 10, 307-313 (1958) · Zbl 0085.01705 · doi:10.2969/jmsj/01030307
[29] Reeder, M.: Modular symbols and the Steinberg representation. In [22] 287-302 · Zbl 0722.20030
[30] Reeder, M.: The Steinberg module and the cohomology of arithmetic groups. J. Algebra 141(2), 287-315 (1991) · Zbl 0824.20040 · doi:10.1016/0021-8693(91)90233-X
[31] Richardson, R., Rohrle, G. Steinberg, R.: Parabolic subgroups with abelian unipotent radical. Invent. Math. 110(3), 649-671 (1992) · Zbl 0786.20029 · doi:10.1007/BF01231348
[32] Solomon, L.: The Steinberg character of a finite group with BN-pair. 1969 Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968) Benjamin, New York, 213-221
[33] Schwermer, J.: Cohomology of arithmetic groups, automorphic forms and L-functions. In [22], 1-29
[34] Steinberg, Robert, Prime power representations of finite linear groups. Canad. J. Math. 8, 580-591 (1956) · Zbl 0073.01502
[35] Steinberg, R.: Lectures on Chevalley groups, Yale University, New Haven, Conn., (1968) · Zbl 1196.22001
[36] Weinberger, P.J.: On Euclidean rings of algebraic integers. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc., Providence, R. I., 321-332 (1973) · Zbl 0245.76023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.