On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic
Tian, Hongfei
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https://hdl.handle.net/2142/99314
Description
Title
On the center of the ring of invariant differential operators on semisimple groups over fields of positive characteristic
Author(s)
Tian, Hongfei
Issue Date
2017-10-30
Director of Research (if dissertation) or Advisor (if thesis)
Haboush, William J.
Doctoral Committee Chair(s)
Bergvelt, Maarten J.
Committee Member(s)
Yong, Alexander
Nevins, Thomas A.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Representation theory
Positive characteristic
Invariant differential operators
Semisimple center
Abstract
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 $G$.
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