This learning seminar will focus on the representation theory of reductive Lie groups, exploring possible connections between two different ways of describing those representations:
One one hand, we have the orbit method initiated by Kirillov [Ki]. The idea of the orbit method is that one should be able to attach representations of a group G to G-orbits on the dual Lie algebra g*. Those coadjoint orbits have the structure of symplectic manifolds, hence the orbit method falls within the broader idea of quantization, turning a symplectic manifold into a Hilbert space, and the Poisson algebra of functions on the symplectic manifold into an algebra of operators.
On the other hand, we have the Langlands classification, describing irreducible representations in terms of another dual, the "Langlands dual group" of G. This classification was refined in the book [ABV] by Adams, Barbasch, and Vogan, using geometry and perverse sheaves on the space of Langlands parameters.
While the dual Lie algebra and the Langlands dual group are different beasts, they are not completely unrelated: For example, (semisimple) conjugacy classes in the complexification of the former coincide with conjugacy classes in the Lie algebra of the latter. And, one can sense instances of "quantization" all over the Langlands program, whose importance is yet to be revealed.
References (more to be added):
[ABV] Jeffrey Adams, Dan Barbasch, and David A. Vogan, Jr., The Langlands classification and irreducible characters for real reductive groups. Progress in Mathematics, 104. Birkhäuser Boston, Inc., Boston, MA, 1992. xii+318 pp.
[Ki] A.A. Kirillov, Lectures on the orbit method. Graduate Studies in Mathematics, 64, Providence, RI: American Mathematical Society, 2004.
[Ven] Akshay Venkatesh, Geometric quantization and representation theory. Notes by Tony Feng and Niccolo Ronchetti from a 2017 course at Stanford.
[LMM] Ivan Losev, Lucas Mason-Brown, and Dmytro Matvieievskyi, Unipotent Ideals and Harish-Chandra Bimodules. arXiv:2108.03453.
This seminar will start with two talks by (our neighbor) Jeff Adams (University of Maryland), to motivate our inquiries. Then, we will spend several weeks covering the material of [ABV], before turning our attention to quantization and the orbit method.
The seminar will meet Thursdays, 3–5pm, in Krieger 413 (the math lounge). We will also try to accommodate a hybrid format, so, if you are interested but not in the area, contact Yiannis for the links.