Irreducible Representations of Finite Groups of Lie Type: On the Irreducible Restriction Problem and Some Local-Global Conjectures by Amanda A. Schaeffer Fry A Dissertation Submitted to the Faculty of the Department of Mathematics In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy In the Graduate College The University of Arizona 2 0 1 3 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the disser- tation prepared by Amanda A. Schaeffer Fry entitled Irreducible Representations of Finite Groups of Lie Type: On the Irreducible Restriction Problem and Some Local-Global Conjectures and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Date: April 3, 2013 Pham Huu Tiep Date: April 3, 2013 Klaus Lux Date: April 3, 2013 David Savitt Date: April 3, 2013 Dinesh Thakur Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Date: April 3, 2013 Pham Huu Tiep 3 Statement by Author This dissertation has been submitted in partial fulfillment of re- quirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to bor- rowers under rules of the Library. Brief quotations from this dissertation are allowable without spe- cial permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or repro- duction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Signed: Amanda A. Schaeffer Fry 4 Acknowledgments First and foremost, I would like to thank my advisor, Professor P.H. Tiep, for all of his support, guidance, and helpful discussions. Most of what I have learned in this project has been learned through him; his patience has been endless and his knowledge has been irreplaceable. His guidance has not only helped prepare me for my future endeavors, but has helped to make them possible. I would also like to thank Professor K. Lux for numerous helpful discussions, for introducing me to computations with GAP, and for getting me interested in algebra, and more specifically group theory, in the first place. Thank you as well to my additional committee members, Professors Dinesh Thakur and David Savitt, for their help and support both in the dissertation process and in my earlier years at the University of Arizona, as well as to Professor Bryden Cais for his participation and support on my comprehensive exam committee. My doctoral study was made possible in part by a graduate fellowship from the National Physical Science Consortium. I am forever grateful for the opportunities this fellowship has given me, including the internship at the National Security Agency during summer 2010. Thank you to my supervisor at the Agency, Dr. Art Drisko, and my other Agency colleagues and fellow NPSCers. I am also thankful for research assistantships through Professor Tiep's NSF grants DMS-0901241 and DMS-1201374 in summers 2011 and 2012, as well as teaching assistantships in spring 2011, summer 2011, and spring 2012. Thank you to all of my students for giving me new insight into mathematics and for some of my favorite memories of my graduate school years. Thank you to my parents, brother, and sister-in-law for all of their support throughout the years. I am also very grateful for my husband, Brendan Fry, for his love and support, and (perhaps most importantly) for reminding me to have some fun. (By the way, congratulations to Brendan on his own defense, which is just a few hours after mine!) I also want to thank Professors Klaus Lux, Doug Ulmer, and Tom Kennedy for pushing my application through early when it became clear that Brendan and I would be staying at UA for graduate school. Finally, thank you to my office-mates, current and past, for making math building room 513 an entertaining and supportive place to work. 5 Table of Contents List of Tables .................................. 8 Abstract ..................................... 9 Chapter 1. Introduction .......................... 10 1.1. The Aschbacher-Scott Program and the Irreducible Restriction Problem 13 1.2. Local-Global Conjectures . 19 Chapter 2. Preliminaries .......................... 23 2.1. Groups of Lie Type . 23 2.1.1. Exceptional Groups of Lie Type . 24 2.2. The Finite Classical Groups . 25 2.2.1. The Classical Groups as Matrix Groups . 25 2.2.2. The Classical Groups as Groups of Lie Type . 29 2.3. Representations . 31 2.3.1. Characters and Blocks of Finite Groups . 32 2.3.2. Representations of Finite Classical Groups . 34 2.3.3. Some Deligne-Lusztig Theory . 35 2.4. Centralizers of Semisimple Elements of Unitary and Symplectic Groups 37 2.4.1. Centralizers of Semisimple Elements of GUn(q)......... 38 2.4.2. Centralizers of Semisimple Elements of Sp2n(q), q even . 40 Chapter 3. Bounds for Character Degrees of Unitary Groups . 46 3.1. The Largest Degree of a Unipotent Character in Finite Unitary Groups 47 3.2. Proof of Theorem 3.0.3 . 52 a Chapter 4. The Blocks and Brauer Characters of Sp6(2 ).... 67 4.1. On Bonnaf´e-Rouquier'sMorita Equivalence . 67 4.2. Low-Dimensional Representations of Sp6(q).............. 70 4.2.1. Weil Characters of Sp2n(q).................... 72 4.2.2. The Proof of Theorem 1.1.1 . 74 a 4.3. Unipotent Blocks of Sp6(2 )....................... 77 a 4.4. Non-Unipotent Blocks of Sp6(2 ).................... 83 a 4.4.1. Non-Unipotent Block Distributions for Irr(Sp6(2 )) . 83 4.4.2. `j(q2 +1) ............................. 84 4.4.3. 3 6= `j(q2 + q +1)......................... 84 4.4.4. 3 6= `j(q2 − q +1)......................... 85 4.4.5. `j(q − 1).............................. 85 6 Table of Contents|Continued 4.4.6. `j(q +1).............................. 88 a 4.4.7. Non-Unipotent Brauer Characters for Sp6(2 )......... 91 a Chapter 5. Cross-Characteristic Representations of Sp6(2 ) and Their Restrictions to Proper Subgroups .............. 95 5.1. Some Preliminary Observations . 95 5.1.1. Other Notes on Sp6(q), q even . 97 5.2. A Basic Reduction . 97 5.3. Restrictions of Irreducible Characters of Sp6(q) to G2(q)....... 103 5.3.1. Fusion of Conjugacy Classes in G2(q) into Sp6(q)....... 104 5.3.2. The Complex Case . 112 5.3.3. The Modular Case . 114 5.3.4. Descent to Subgroups of G2(q).................. 117 ± 5.4. Restrictions of Irreducible Characters of Sp6(q) to the Subgroups O6 (q) 117 5.5. Restrictions of Irreducible Characters to Maximal Parabolic Subgroups . 125 5.5.1. Descent to Subgroups of P3 ................... 137 5.6. The case q =2 .............................. 141 Chapter 6. Restrictions of Complex Representations of Finite Unitary Groups of Low Rank to Certain Subgroups ....... 145 6.1. A Condition On (n; q).......................... 147 6.2. Reducing the Problem . 149 6.3. The Case n =5............................... 152 6.3.1. GAP Computation for q =2................... 157 6.4. On the Remaining Cases 4 ≤ n < 10 . 160 a Chapter 7. Sp6(2 ) is \Good" for the McKay, Alperin Weight, and Related Local-Global Conjectures ............... 162 7.1. Preliminaries and Notation . 162 a a 7.2. Radical Subgroups of Sp6(2 ) and Sp4(2 )............... 164 7.3. Characters of NG(Q)........................... 175 7.3.1. Characters of Some Relevant Subgroups . 176 7.3.2. Q = Q1 .............................. 178 7.3.3. Q = Q2 .............................. 179 7.3.4. Q = Q3 .............................. 180 7.3.5. Q = Q1;1 .............................. 181 7.3.6. Q = Q2;1 .............................. 183 7.3.7. Q = Q1;1;1 ............................. 184 7.3.8. Q = P ............................... 185 7 Table of Contents|Continued 7.3.9. Q = R ............................... 187 7.3.10. Q = Q(3) .............................. 188 7.3.11. Q = Q(2) .............................. 189 7.4. The Maps . 190 7.4.1. The maps ΩQ ........................... 190 7.4.2. The maps ∗Q ........................... 196 a 7.4.3. The Maps for Sp4(2 )...................... 201 a a 7.5. Sp6(2 ) and Sp4(2 ) are Good for the Conjectures . 203 References .................................... 219 8 List of Tables Table 2.1. Identifications of Finite Chevalley Groups with Finite Classical Groups . 30 ∗ ∗ Table 4.1. Semisimple Classes of G = Sp6(q) with Small [G : CG∗ (s)]20 .. 70 Table 4.2. Weil Characters of Sp2n(q) [27, Table 1] . 73 Table 4.3. `−Brauer Characters in Unipotent Blocks of G = Sp6(q), `j(q−1), ` 6=3 ..................................... 78 Table 4.4. `−Brauer Characters in Unipotent Blocks of G = Sp6(q), ` = 3j(q − 1) ................................... 79 Table 4.5. `−Brauer Characters in Unipotent Blocks of G = Sp6(q), `j(q + 1) 79 2 Table 4.6. `−Brauer Characters in Unipotent Blocks of G = Sp6(q), `j(q − q + 1), ` 6=3 ................................. 80 2 Table 4.7. `−Brauer Characters in Unipotent Blocks of G = Sp6(q), `j(q + q + 1), ` 6=3 ................................. 81 2 Table 4.8. `−Brauer Characters in Unipotent Blocks of G = Sp6(q), `j(q + 1) 82 a Table 4.9. `−Brauer Characters in Non-Unipotent Blocks of G = Sp6(2 ), ` 6=2 ..................................... 93 a Table 4.10. `−Brauer Characters in Non-Unipotent Blocks of G = Sp6(2 ), ` 6= 2, Continued .