Endoscopic codes for unitary groups over the reals

Abstract

OF THE DISSERTATION Endoscopic Codes For Unitary Groups Over The Reals By DMITRY RUBANOVICH Dissertation Director: Diana Shelstad Transfer factors, originally defined by Langlands and Shelstad for the transfer of orbital integrals, play a central role in the theory of endoscopy. Spectral transfer factors, for the dual transfer of traces, have been defined for real groups by Shelstad. The theory shows that for discrete series representations of unitary groups the spectral transfer factors determine a bijection between the representations in a packet and certain binary words. The binary word thus associated to a representation may be called its endoscopic code. Such a code is difficult to calculate from the definition by transfer factors. Low dimensional examples suggest that there is an alternative approach, directly in terms of the Harish-Chandra data of the representation, which provides fast calculation of spectral transfer factors. This thesis presents a new direct construction of the endoscopic code of a discrete series representation of any unitary group directly from its HarishChandra data and, conversely, identifies a discrete series representation from any particular given endoscopic code. An explicit algorithm is given and implemented in Mathematica .