Wenhan Dai

Springer Theory, Representation Theory, and Complex Geometry

This seminar will be about classical Springer theory with an emphasis on concreteness. In the first-half semester, we concentrate on the construction of Springer representations, as well as an alternative construction via perverse sheaves; we will dedicate the second-half semester to more explicit and combinatorial topics, such as parametrizations of nilpotent orbits in classical types using (generalized) tableaux, and Bala-Carter theory to tackle with exceptional cases and the fundamental groups of nilpotent orbits. While time permitting, the explicit generalized Springer Correspondence due to Lusztig follows at last.

• Instructor: Marc Besson.
• Time: Every Tuesday, 1–3pm.
• Venue: Quan 09.

Click here to download my notes for Talk 10.

Syllabus

• (Talk 1) Zhiyuan Zhang: Symplectic Manifolds and Moment Maps, [CG, 1.1 & 1.4].
• (Talk 2) Shurui Liu: Basics of Flag Varieties, [CG, 3.1].
• (Talk 3) Zhanwei Zhang: Nilpotent Cone, [CG, 3.2].
• (Talk 4) Zhaoxing Gu: Steinberg Variety, [CG, 3.3].
• (Talk 5) Xingjian Zhou: Convolution and Borel Moore Homology, [CG, 2.6 & 2.7].
• (Talk 6) Xingjian Zhou: Lagrangian Construction, [CG, 3.4].
• (Talk 7) Xingzhu Fang & Haoda Li: Weyl Group Representations I, [CG, 3.5].
• (Talk 8) Xingzhu Fang & Haoda Li: Weyl Group Representations II, [CG, 3.6].
• (Talk 9) Jianxiang Tan: Jacobson Morozov, [CG, 3.7].
• (Talk 10) Wenhan Dai: Springer Theory via Perverse Sheaves, [Lus].
• (Talk 11) Marc Besson: Kostant Dynkin Classification of Orbits, [CM, Chap 3].
• (Talk 12) Shurui Liu: Partition Type Classification , [CM, Chap 5].
• (Talk 13) Shurui Liu: More Topology on Nilpotent Orbits, [CM, Chap 6].
• (Talk 14) Marc Besson: Induced Nilpotent Orbits, [CM, Chap 7].
• (Talk 15) Marc Besson: Lusztig’s Symbols, [CM, Chap 10].

References

1. [CG] Chriss, N., Ginzburg, V., “Representation Theory and Complex Geometry,” Birkhauser 1997.
2. [CM] Collingwood, D., McGovern, W., “Nilpotent Orbits in Semisimple Lie Algebras,” Van Nostrant Reinhold Mathematics Series 1993.
3. [Lus] Lusztig, G. Green polynomials and singularities of unipotnt classes, Advances in Mathematics 1981.
4. [Yun] Yun, Z. Lectures on Springer Theories and Orbital Integrals, Geometry of Moduli Spaces and Representation Theory. Park City Mathematics series 2015.