Motivic nature of character values of depth-zero representations

Motivic proof of a character formula for SL(2)

We give a motivic proof of a character formula for depth zero supercuspidal representations of $p$-adic SL(2). We begin by finding the virtual Chow motives for the character values of all depth zero supercuspidal representations of $p$-adic SL(2), at topologically unipotent elements. Then we find the virtual Chow motives for the values of the Fourier transform of all regular elliptic orbital integrals with depth zero in their Cartan subalgebra, at topologically nilpotent elements. Finally, we prove a character formula for depth zero supercuspidal representations by showing that the formula cor…

Homology of depth-graded motivic Lie algebras and koszulity

The Broadhurst-Kreimer (BK) conjecture describes the Hilbert series of a bigraded Lie algebra A related to the multizeta values. Brown proposed a conjectural description of the homology of this Lie algebra (homological conjecture (HC)), and showed it implies the BK conjecture. We show that a part of HC is equivalent to a presentation of A, and that the remaining part of HC is equivalent to a weaker statement. Finally, we prove that granted the first part of HC, the remaining part of HC is equivalent to either of the following equivalent statements: (a) the vanishing of the third homology group…

2-Calabi-Yau 范畴中的动机簇乘法公式

Motivic cluster multiplication formulas in 2-Calabi-Yau categories

本文利用虚拟 Poincaré 多项式定义动机簇特征标，证明 2-Calabi-Yau 三角范畴中加权量子簇特征标乘法公式的动机版本及加细形式，并在无圈箭图簇范畴中与已有结果对应。

Motivic random variables and representation stability I: Configuration spaces

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain *motivic random variables*, and gives explicit universal formulas for the limits in terms of the exponents of a motivic Euler product for the Kapranov zeta function. The result can be thought of as a weak but explicit version of representation stability for the cohomology of ordered configuration spaces. In t…

通过母题积分研究p-循环McKay对应

The $p$-cyclic McKay correspondence via motivic integration

本文利用推广到商栈的母题积分研究特征p下p阶循环群表示的McKay对应，显式计算了商簇的弦不变量，并证明crepant解消的拓扑欧拉特征数为p，同时将其与Artin-Schreier扩张的计数建立联系。