Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups. (English. Russian original) Zbl 1182.20041
Math. Notes 86, No. 1, 65-80 (2009); translation from Mat. Zametki 86, No. 1, 65-80 (2009).
Summary: We consider a specific class of coadjoint orbits of maximal unipotent subgroups in classical groups over a finite field; namely, orbits associated with orthogonal subsets in root systems. We derive a formula for the dimension of these orbits in terms of the Weyl group and construct polarizations for canonical forms on the orbits. As a consequence, we describe all possible dimensions of irreducible representations of such unipotent groups.
MSC:
20G05 | Representation theory for linear algebraic groups |
17B22 | Root systems |
20G40 | Linear algebraic groups over finite fields |
Keywords:
root systems; coadjoint orbits; dimensions of orbits; unipotent groups; Weyl groups; irreducible representations; irreducible complex characters; polarizations of linear forms; classical groups over finite fieldsReferences:
[1] | A. A. Kirillov, Lectures on the Orbit Method, in University Series (Nauchnaya Kniga, Novosibirsk, 2002), Vol. 10 [in Russian]. |
[2] | D. Kazhdan, ”Proof of Springer’s conjecture,” Israel J. Math. 28(4), 272–286 (1977). · Zbl 0391.22006 · doi:10.1007/BF02760635 |
[3] | A. N. Panov, ”Involutions in S n and associated coadjoint orbits,” in Problems of Theory of Algebras and Groups Representations. 16, Zap. Nauchn. Sem. POMI (POMI, St. Petersburg, 2007), Vol. 349, pp. 150–173 [in Russian]; [J.Math. Sci. 151 (3), 3018–3031 (2008)]. |
[4] | M. V. Ignat’ev, ”Basic subsystems in the root systems B n and D n and associated coadjoint orbits,” Vestnik SamGU, Estestvennonauchn. Ser., No. 3, 124–148 (2008). |
[5] | B. Srinivasan, Representations of Finite Chevalley Groups: A Survey, in Lecture Notes in Math. (Springer-Verlag, Berlin-New York, 1974), Vol. 764. · Zbl 0434.20022 |
[6] | C. A. M. André and A. M. Neto, ”Super-characters of finite unipotent groups of types B n, C n and D n,” J. Algebra 305(1), 394–429 (2006). · Zbl 1104.20041 · doi:10.1016/j.jalgebra.2006.04.030 |
[7] | S. Mukherjee, Coadjoint Orbits for A n-1 + , B n + and D n + , arXiv: math. RT/0501332 [in Russian]. |
[8] | N. Bourbaki, Groupes et algebres de Lie. Groupes de Coxeter et systemes de Tits. Groupes engendres par des reflexions. Systemes de racines (Paris, Hermann, 1968; Mir, Moscow, 1972). |
[9] | C. A. M. André, ”Basic sums of coadjoint orbits of the unitriangular group,” J. Algebra 176(3), 959–1000 (1995). · Zbl 0837.20050 · doi:10.1006/jabr.1995.1280 |
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.