Adjoint groups, regular unipotent elements and discrete series characters
Author:
G. I. Lehrer
Journal:
Trans. Amer. Math. Soc. 214 (1975), 249-260
MSC:
Primary 20C15; Secondary 20G40
DOI:
https://doi.org/10.1090/S0002-9947-1975-0384915-9
MathSciNet review:
0384915
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: It is shown that if G is a finite Chevalley group or twisted type over a field of characteristic p and U is a maximal p-subgroup of G then any nonlinear irreducible character of U vanishes on regular elements. For groups of adjoint type the linear content of the restriction to U of a discrete series character J of G is calculated and it is deduced that J takes the value 0 or on regular elements of U
.
- [1] A. Borel and T. A. Springer, Rationality properties of linear algebraic groups. II, Tohoku Math. J. (2) 20 (1968), 443–497. MR 244259, https://doi.org/10.2748/tmj/1178243073
- [2] Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150 (French). MR 207712
- [3] Roger W. Carter, Simple groups of Lie type, John Wiley & Sons, London-New York-Sydney, 1972. Pure and Applied Mathematics, Vol. 28. MR 0407163
- [4] C. Chevalley, Séminaire sur la classification des groupes de Lie algébriques, 2 vols., Secrétariat mathématique, Paris, 1958. MR 21 #5696.
- [5] I. M. Gel′fand and M. I. Graev, Construction of irreducible representations of simple algebraic groups over a finite field, Dokl. Akad. Nauk SSSR 147 (1962), 529–532 (Russian). MR 0148765
- [6] Robert B. Howlett, On the degrees of Steinberg characters of Chevalley groups, Math. Z. 135 (1973/74), 125–135. MR 360781, https://doi.org/10.1007/BF01189349
- [7] G. I. Lehrer, Discrete series and regular unipotent elements, J. London Math. Soc. (2) 6 (1973), 732–736. MR 318288, https://doi.org/10.1112/jlms/s2-6.4.732
- [8] T. A. Springer, Some arithmetical results on semi-simple Lie algebras, Inst. Hautes Études Sci. Publ. Math. 30 (1966), 115–141. MR 206171
- [9] T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192
- [10] Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
- [11] Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49–80. MR 180554
- [12] -, Lectures on Chevalley groups, Mimeographed notes, Yale University, 1967.
Retrieve articles in Transactions of the American Mathematical Society with MSC: 20C15, 20G40
Retrieve articles in all journals with MSC: 20C15, 20G40
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1975-0384915-9
Article copyright:
© Copyright 1975
American Mathematical Society