Program
Introduction to the affine springer fibers
Student No.:50
Time:Mon/Wed 13:00-14:50, 2014-07-02 ~ 2014-07-21
Instructor:Zongbin Chen  [Tsinghua University]
Place:Conference Room 3, Floor 2, Jin Chun Yuan West Building
Starting Date:2014-7-2
Ending Date:2014-7-21

 

Description:

 

The first part of this course will be devoted to the proof of Goresky, Kottwitz andMacpherson [GKM1] of the fundamental lemma in the unramified case, which is a combinatorialidentity between certain orbital integrals. Their strategy is the following: Firstof all, they found a geometric interpretation of the orbital integrals as the number of rationalpoints on the affine Springer fibers, and the fundamental lemma is reformulated asan identity between homologies of the relevant affine Springer fibers. Secondly, assumingthe cohomological purity of the affine Springer fibers, they calculated their T-equivariantcohomology (and so the ordinary cohomology) by thelocalization theorem. The result isin terms of the fixed points and the 1-dimensional ̃T-orbits, this gives an easy comparisonbetween the cohomology of the affine Springer fibers and so the fundamental lemma.

 

The second part will be concerned with the work of Chaudouard-Laumon [CL] and thelecturer [C] on the truncated affine Springer fibers. The aim is to explain a reduction of thepurity conjecture on the affine Springer fibers to that of a certain truncated affine Springerfibers, and a rationality phenomena about the Poincaré polynomials of these truncatedaffine Springer fibers.

 

The course should be accessible to students with some basic knowledge on the algebraicgeometry (the first 3 chapters of Hartshorne’s book will be largely enough, for example).And we will restrict ourselves to the unitary group setting, so no prior knowledge on thelinear algebraic groups is required.

 

Reference:

 
B] R. Bezrukavnikov, The dimension of the fixed point set on affine flag manifolds, Math. Res.

Lett. 3 (1996), 185–189.

 
[BL] A. Beauville, Y. Laszlo, Conformal blocks and generalized theta functions. Comm. Math. Phys.

164 (1994), no. 2, 385–419.

 
[C] Z. Chen, On the fundamental domain of affine Springer fibers,

http://arxiv.org/abs/1303.4630

 
[CL] P-H. Chaudouard, G. Laumon, Sur l’homologie des fibres de Springer affines tronquées, Duke

Math. J. 145 (2008), no. 3, 443–535.

 
[GKM1] M. Goresky, R. Kottwitz, R. Macpherson, Homology of affine Springer fibers in the unramified

case, Duke Math. J. 121 (2004), no. 3, 509-561.

 
[GKM2] M. Goresky, R. Kottwitz, R. Macpherson, Purity of equivalued affine Springer fibers, Represent.

Theory 10 (2006), 130-146.

 
[GKM3] M. Goresky, R. Kottwitz, R. Macpherson, Regular points in affine Springer fibers, Michigan

Math. J. 53 (2005), no. 1, 97-107.

 

[GKM4] M. Goresky, R. Kottwitz, R. Macpherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131, 25-83 (1998).

 

[KL] D. Kazhdan, G. Lusztig, Fixed point varieties on affine flag manifolds, Israel. J. Math. 62(1988), 129-168.

 

EPFL SB Mathgeom/Geom, MA B1 447, Lausanne, CH-1015, Switzerland

E-mail address: zongbin.chen@epfl.ch