This research is on topics in the representation theory of p-adic groups in the context of Langlands program. It concerns both the classical representations on complex vector spaces and p-adic representations. In the area of p-adic representations, the investigator will work with Peter Schneider on problems in the theory of locally analytic and Banach space representations. This is a new research area, with lots of open problems. The theory of p-adic representations is totally different from the classical theory of complex representations. In both theories, we can ask similar questions, but the answers are often completely different. This research will contribute to the development of the theory of locally analytic and Banach space representations. The research in the classical representation theory is motivated by Arthur's conjectures. The investigator will work with Chris Jantzen on problems related to geometric R-groups and Arthur packets. In addition, she will study the effects of the Aubert involution on Arthur parameters and Arthur packets. The local questions considered in the project appear in the consideration of local factors of terms of Arthur's trace formula. The obtained information will play a role in future arithmetic questions related to automorphic forms.
Langlands program is a mathematical philosophy and area of mathematical research. It originates in a series of conjectures formulated by Robert Langlands, describing deep and hidden connections between number theory, representation theory and automorphic forms. James Arthur's advanced research resulted in generalizations of Langlands' conjectures. After decades of mathematical research, it is amazing to see that Langlands' conjectures turn out to be true in so many different situations. The unifying principle of Langlands' conjectures is now investigated in a new area: the theory of p-adic representations. The work of Schneider and Teitelbaum on locally analytic and Banach space representations is one of the cornerstones of p-adic Langlands program. The problems in the representation theory investigated by this project are of importance for Langlands program.
