Birne Binegar
Okalahoma State, USA
The Atlas Project: Theoretical Consequences of a Phenomenological Approach to the Representation Theory of Reductive Lie Groups

Lecture 1: A Combinatorial Parameterization of Nilpotent Orbits In this talk, I'll quickly review the Dynkin, partition, and Bala-Carter classifications of the nilpotents orbits of a complex semisimple Lie algebra $\frak{g}$ and show how the latter leads to a parameterization of complex nilpotent orbits in terms of certain pairs $\left[ \Gamma,\gamma\right]$ where $\Gamma$ is a certain subset of the simple roots of $\frak{g}$ and $\gamma$ is a certain subset of $\Gamma$. Simple recipes for passing back and forth between the combinatorial parameters and the conventional parameters are described.

Lecture 2: Twisted Induction and Duality The Springer correspondence and Lusztig's cell decomposition of $\widehat{W}$, the set of equivalence classes of the irreducible representations, are reviewed. Together these results induce a partitioning of the set of nilpotent orbits into cells of orbits. Reinterpreting a formula of due to Barbasch and Vogan, we introduce a notion of twisted induction and duality that replicates the Lusztig cell structure of $G\backslash\mathcal{N}$ induced by the Springer correspondence and leads to a simple, intrinsic formulation of the Spaltenstein duality map on $G\backslash \mathcal{N}$.

Lecture 3: Introduction to the Atlas Point of View In this lecture I'll introduce the setting of the Atlas project and describe how the Atlas software classifies and organizes the admissible representations of linear algebraic groups.

Lecture 4. Exploring the Atlas In this lecture I'll provide a hands-on demonstration of how to use the Atlas software to explore the phenomenology of the admissible dual.

Lecture 5. Cells of Harish-Chandra Modules The notion of cells of Harish-Chandra modules is introduced. I'll describe how the Atlas software not only computes the cells but also the entire $W$-graph structure of the admissible dual.

Lecture 6. Cells and Orbits In this lecture, I'll describe how results of Spaltenstein, Vogan and the Atlas software allow one to explicitly attach to each irreducible admissible representation a particular nilpotent orbit.

Lecture 7. Cells and Primitive Ideals In this lecture, I'll describe how results from Atlas computatins enable one to organize the admissible dual into subsets sharing the same annihilator in $U(\frak{g})$.