On Upper Bound Conjecture on Wavefront Set and Generalized Shahidi Conjecture

In the first part of the thesis, we study the upper bound of wavefront sets of irreducible admissible representations of connected reductive groups defined over non-Archimedean local fields of characteristic zero. We formulate a new conjecture on the upper bound and show that it can be reduced to that of anti-discrete series representations, namely, those whose Aubert-Zelevinsky duals are discrete series. Then, under certain assumptions, we show that this conjecture is equivalent to the Jiang conjecture on the upper bound of wavefront sets of representations in local Arthur packets and also equivalent to an analogous conjecture on the upper bound of wavefront sets of representations in ABV packets.

In the second part of the thesis, we propose the generalized Shahidi conjecture: for each Aubert-Zelevinsky dual of an $L$-packet, there exists at least one member whose wavefront set achieves the conjectural upper bound. This generalizes the well-known Shahidi conjecture that tempered $L$-packet has a generic member. For classical groups, we prove that assuming the upper bound conjecture on wavefront set, the generalized Shahidi conjecture for general $L$-parameter can be reduced to that of almost discrete $L$-parameters.