Special Unipotent Arthur Packets for Real Reductive Groups

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Let $\GR$ be a real reductive group. In this thesis we study the unitary representations of $\GR$. In particular, we study the special Arthur unipotent parameters and the associated packets of irreducible representations of $\GR$. It is conjectured that these unipotent representations form the building blocks for all unitary representations of $\GR$.


To understand unipotent representations, we will need to compute the following invariants of irreducible representations of $\GR$: complex associated variety and the theta associated variety. Even though these invariants are theoretically understood, there are no known (at least to this author) results/algorithms to compute them explicitly.


The primary results of this thesis provide algorithms to compute these invariants explicitly in many cases. We then use these invariants to compute information about unipotent Arthur packets, and in favorable cases, their entire contents explicitly. In unfavorable cases, we show how to extract more information from our results by using the stable sum formula.


We have implemented these algorithms into the Atlas of Lie Groups software, available at www.liegroups.org. We also provide some tables of data compiled using the output from Atlas.