Orthogonal subsets of root systems and the orbit method
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M. V. Ignat′ev
Translated by: the author - St. Petersburg Math. J. 22 (2011), 777-794
- DOI: https://doi.org/10.1090/S1061-0022-2011-01167-7
- Published electronically: June 27, 2011
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Abstract:
Let $k$ be the algebraic closure of a finite field, $G$ a Chevalley group over $k$, $U$ the maximal unipotent subgroup of $G$. To each orthogonal subset $D$ of the root system of $G$ and each set $\xi$ of $|D|$ nonzero scalars in $k$ one can assign the coadjoint orbit of $U$. It is proved that the dimension of such an orbit does not depend on $\xi$. An upper bound for this dimension is also given in terms of the Weyl group.References
- Carlos A. M. André, The basic character table of the unitriangular group, J. Algebra 241 (2001), no. 1, 437–471. MR 1839342, DOI 10.1006/jabr.2001.8734
- Carlos A. M. André and Ana Margarida Neto, Super-characters of finite unipotent groups of types $B_n$, $C_n$ and $D_n$, J. Algebra 305 (2006), no. 1, 394–429. MR 2264135, DOI 10.1016/j.jalgebra.2006.04.030
- M. Boyarchenko and V. Drinfeld, A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic, arXiv: math.RT/0609769v1.
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
- M. V. Ignat′ev and A. N. Panov, Coadjoint orbits of the group $\textrm {UT}(7,K)$, Fundam. Prikl. Mat. 13 (2007), no. 5, 127–159 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 156 (2009), no. 2, 292–312. MR 2379743, DOI 10.1007/s10958-008-9267-0
- M. V. Ignat′ev, Subregular characters of the unitriangular group over a finite field, Fundam. Prikl. Mat. 13 (2007), no. 5, 103–125 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 156 (2009), no. 2, 276–291. MR 2379742, DOI 10.1007/s10958-008-9266-1
- M. V. Ignat′ev, Basic subsystems in the root systems $B_n$ and $D_n$ and associated coadjoint orbits, Vestn. Samar. Gos. Univ. Estestvennonauchn. Ser. 3 (2008), 124–148 (Russian, with English and Russian summaries). MR 2473733
- M. V. Ignat′ev, Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups, Mat. Zametki 86 (2009), no. 1, 65–80 (Russian, with Russian summary); English transl., Math. Notes 86 (2009), no. 1-2, 65–80. MR 2588639, DOI 10.1134/S0001434609070074
- D. Kazhdan, Proof of Springer’s hypothesis, Israel J. Math. 28 (1977), no. 4, 272–286. MR 486181, DOI 10.1007/BF02760635
- A. A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics, vol. 64, American Mathematical Society, Providence, RI, 2004. MR 2069175, DOI 10.1090/gsm/064
- A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
- A. A. Kirillov, Variations on the triangular theme, Lie groups and Lie algebras: E. B. Dynkin’s Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 169, Amer. Math. Soc., Providence, RI, 1995, pp. 43–73. MR 1364453, DOI 10.1090/trans2/169/05
- Shantala Mukherjee, Coadjoint orbits for $A^+_{n-1},\ B^+_n$, and $D^+_n$, J. Lie Theory 16 (2006), no. 3, 455–469. MR 2248140
- A. N. Panov, Involutions in $S_n$ and associated coadjoint orbits, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 349 (2007), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 16, 150–173, 244 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 151 (2008), no. 3, 3018–3031. MR 2742857, DOI 10.1007/s10958-008-9016-4
- Bhama Srinivasan, Representations of finite Chevalley groups, Lecture Notes in Mathematics, vol. 764, Springer-Verlag, Berlin-New York, 1979. A survey. MR 551499
- Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
- Robert Steinberg, Abstract homomorphisms of simple algebraic groups (after A. Borel and J. Tits), Séminaire Bourbaki, 25ème année (1972/1973), Lecture Notes in Math., Vol. 383, Springer, Berlin, 1974, pp. Exp. No. 435, pp. 307–326. MR 0414732
Bibliographic Information
- M. V. Ignat′ev
- Affiliation: Department of Algebra and Geometry, Samara State University, Ak. Pavlova 1, Samara 443011, Russia
- Email: mihail.ignatev@gmail.com
- Received by editor(s): April 14, 2010
- Published electronically: June 27, 2011
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 777-794
- MSC (2010): Primary 17B22
- DOI: https://doi.org/10.1090/S1061-0022-2011-01167-7
- MathSciNet review: 2828828