Isomorphisms of Root Data

Title of dissertation: CLASSICAL INVARIANTS OF PRINCIPAL SERIES AND ISOMORPHISMS OF ROOT DATA Robert Alexander McLean II Doctor of Philosophy, 2016 Dissertation directed by: Professor Jeffrey Adams Department of Mathematics We develop some new techniques to calculate the Schur indicator for selfdual irreducible Langlands quotients of the principal series representations. Using these techniques we derive some new formulas for the Schur indicator and the realquaternionic indicator. We make progress towards developing an algorithm to decide whether or not two root data are isomorphic. When the derived group has cyclic center, we solve the isomorphism problem completely. An immediate consequence is a clean and precise classification theorem for connected complex reductive groups whose derived groups have cyclic center. CLASSICAL INVARIANTS OF PRINCIPAL SERIES AND ISOMORPHISMS OF ROOT DATA by Robert Alexander McLean II Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2016 Advisory Committee: Professor Jeffrey Adams, Chair/Advisor Professor Thomas Haines Professor Xuhua He Professor Jonathan Rosenberg Professor Rabindra Mohapatra, Dean’s Representative c © Copyright by Robert Alexander McLean II 2016