The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups

Abstract We investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition, we obtain a parametrization of the isomorphism classes of all root data. By working at the level of root data, we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms, such embeddings were constructed by Benjamin Martin. In an unpublished manuscript, Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. Using our investigations into root data we give new proofs of Asai's results and generalize them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.

[1]  Jay Taylor.  Action of automorphisms on irreducible characters of symplectic groups , 2016, Journal of Algebra.

[2]  M. Geck,et al.  Reductive Groups and Steinberg Maps , 2016, The Character Theory of Finite Groups of Lie Type.

[3]  Jay Taylor.  GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS , 2014, Nagoya Mathematical Journal.

[4]  J. Michel.  The development version of the CHEVIE package of GAP3 , 2013, 1310.7905.

[5]  Noor Kafil-Hussain.  Polo , 2007, BMJ : British Medical Journal.

[6]  F. Shahidi,et al.  Generic Transfer for General Spin Groups , 2004, math/0411035.

[7]  G. Lusztig,et al.  Representations of reductive groups over finite fields , 2004 .

[8]  A. Aubert,et al.  SUPPORTS UNIPOTENTS DE FAISCEAUX CARACTÈRES , 2003, Journal of the Institute of Mathematics of Jussieu.

[9]  Hisatoshi Ikai.  Spin Groups Over a Commutative Ring and the Associated Root Data , 2003 .

[10]  B. Martin.  Étale slices for representation varieties in characteristic p , 1999 .

[11]  M. Geck,et al.  On the existence of a unipotent support for the irreducible characters of a finite group of Lie type , 1999 .

[12]  R. Steinberg.  The Isomorphism and Isogeny Theorems for Reductive Algebraic Groups , 1999 .

[13]  D. Voce,et al.  Tame automorphisms of finitely generated abelian groups , 1998, Proceedings of the Edinburgh Mathematical Society.

[14]  Götz Pfeiffer,et al.  CHEVIE — A system for computing and processing generic character tables , 1996, Applicable Algebra in Engineering, Communication and Computing.

[15]  George Lusztic.  A unipotent support for irreducible representations , 1992 .

[16]  Jean Michel,et al.  Representations of Finite Groups of Lie Type , 1991 .

[17]  George Lusztig,et al.  Characters of reductive groups over a finite field , 1984 .

[18]  Charles W. Curtis,et al.  Representations of finite groups of Lie type , 1979 .

[19]  G. Lusztig.  On the finiteness of the number of unipotent classes , 1976 .

[20]  J. Humphreys,et al.  Linear Algebraic Groups , 1975 .

[21]  R. Tennant.  Algebra , 1941, Nature.

[22]  O. Gabber,et al.  Pseudo-reductive Groups , 2010 .

[23]  C. Bonnafé.  Sur les caractères des groupes réductifs finis à centre non connexe : applications aux groupes spéciaux linéaires et unitaires , 2018, Astérisque.

[24]  F. Murnaghan,et al.  LINEAR ALGEBRAIC GROUPS , 2005 .

[25]  Ronald L. Graham,et al.  The graph of generating sets of an abelian group , 1999 .

[26]  M. Geck.  On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes , 1996, Documenta Mathematica.

[27]  A. Borel.  Linear Algebraic Groups , 1991 .

[28]  George Lusztig,et al.  On the representations of reductive groups with disconnected cen-tre , 1988 .

[29]  J. Jantzen.  Representations of algebraic groups , 1987 .

[30]  M. Demazure.  Structure des schémas en groupes réductifs , 1970 .