Monday, January 22, 2024
Time  Items 

All day 

4:00pm 
01/22/2024  4:00pm We consider the family of periodic measures for the full diagonal action on the space of unimodular lattices. This family is important and natural due to its tight relation to class groups in number fields. We show that many natural families of measures on the space of lattices can be approximated using this family (in the weak sense). E.g., in our work we show that for any 0<c\leq 1, the measure cm_{X_n} can be approximated this way, where m_{X_n} denotes the Haar probability measure on X_n. Moreover, we show that non ergodic measures can be approximated. Our proof is based on the effective equidistribution of Hecke neighbors due to Clozel, Oh and Ulmo and on constructions of special number fields. We will discuss the results, alternative ways to attack the problem, and our method of proof. This talk is based on a joint work with Omri Solan. Location:
KT205
01/22/2024  4:30pm Consider a simple algebraic group G of classical type and its Lie algebra g. Let F be an algebraically closed field of characteristic p>> 0, we write g_F for the Fform of g. Let U^{\chi}_{F, \lambda} be the central reduction of U(g_F) with respect to a pair \lambda\in \mathfrak{h}^*/W and \chi\in \mathfrak{g}^{*}_\mathbb{F}^{(1)}. We focus on the case where \lambda is integral regular and \chi is nilpotent. In [BM12], there exists an algebra A^{0}_e (the fiber of the noncommutative Springer resolution over a nilpotent element e\in g) equipped with an isomorphism K_0(U^{\chi}_{F, \lambda} mod)\cong K_0(A^{0}_emod)$. This isomorphism sends classes of simple modules to classes of simple modules. Let Y_e be the set of simple modules of A^{0}_e. Let Q_e be the reductive part of the centralizer of e in G. As Q_e acts on A^{0}_e by algebra automorphisms, the finite set Y_e has the structure of a Q_ecentrally extended set. In this work, we study this centrally extended structure of Y_e when the partition of e has few rows. In particular, we first prove that Y_e can be determined by certain numerical invariants of the Springer fiber B_e. Next, we introduce a variety B_e^{gr} \subset B_e that shares the same set of numerical invariants as B_e. We then reconstruct Y_e from B_e^{gr} using categorical tools. The main result is that the derived category D^b(B_e^{gr}) admits a complete exceptional collection that is compatible with the Q_eaction. The objects in this collection can be regarded as the points of Y_e in a suitable sense. As applications, we obtain an algorithm to compute the multiplicities of orbits in Y_e, which provides some numerical information on cells in affine Weyl group. Location:
KT217
