Orthogonal subsets of root systems and the orbit method

Ignat′ev, M.

doi:10.1090/S1061-0022-2011-01167-7

Orthogonal subsets of root systems and the orbit method
HTML articles powered by AMS MathViewer

by 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

PDF | Request permission

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