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DSpace SAF Submission Ingestion Package generated from Vireo submission #11707 on 2018-03-13 at 10:08:13 Haboush, William J. Bergvelt, Maarten J. Yong, Alexander Nevins, Thomas A. Tian, Hongfei 2018-03-13T15:45:10Z 2018-03-13T15:45:10Z 2017-10-30 2017-12 In this thesis we prove the existence of Jordan Decomposition in $D_{G/k}$, the ring of invariant differential operators on a semisimple algebraic group over a field of positive characteristic, and its corollaries. In particular, we define the semisimple center of $D_{G/k}$, denoted by $Z_s(D_{G/k})$, as the set of semisimple elements of its center. Then we show that if $G$ is connected, the semisimple center $Z_s(D_{G/k})$ contains $Z_s(D_{G/k}^{(\nu)})$ for any positive interger $\nu$, where $Z_s(D_{G/k}^{(\nu)})$ is the ring of invariant differential operators on a Frobenius kernel derived from $G$. Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-03-13 without embargo terms The student, Hongfei Tian, accepted the attached license on 2017-10-26 at 14:48. The student, Hongfei Tian, submitted this Dissertation for approval on 2017-10-26 at 14:49. This Dissertation was approved for publication on 2017-10-30 at 14:06. Made available in DSpace on 2018-03-13T15:45:10Z (GMT). No. of bitstreams: 3 TIAN-DISSERTATION-2017.pdf: 353673 bytes, checksum: 654c1827f749c284e8038970119f5d4d (MD5) LICENSE.txt: 4209 bytes, checksum: 1b71d6987cf524877ecdafae0da9fc15 (MD5) PROQUEST_LICENSE.txt: 4555 bytes, checksum: 4f51e6376179b66e0a9b8383e9dfa5ac (MD5) Previous issue date: 2017-10-30 application/pdf http://hdl.handle.net/2142/99314 en Copyright 2017 Hongfei Tian Representation theory Positive characteristic Invariant differential operators Semisimple center On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.