**Speaker:** Aron Heleodoro (University of Hong Kong)

**Abstract:** Let $F$ be a discretely valued local field with an algebraically closed residue field $k$, e.g.\ $F=\bC((t)), \overline{\bF}((t))$, or $W(\overline{\bF})$. Let $G$ be a connected reductive algebraic group over $F$. One can associate to $G$ the loop group $LG$ which is a (perfect) ind-scheme over $k$. I will define $D(\frac{LG}{LG})$ a category of constructible \’etale sheaves on the quotient stack $\frac{LG}{LG}$ and explain that this category has a semi-orthogonal decomposition coming from a geometric stratification with finitely presented strata associated to its Newton points. I will also explain how this sheaf theory allows us to compute the categorical trace of the affine Hecke category as a subcategory of $D(\frac{LG}{LG})$. In the first part, I will review the theory in the finite case, namely for $G$ defined over $k$ and considering $D(\frac{G}{G})$ and the finite Hecke category.

This is joint work with Xuhua He and Xinwen Zhu.

**Time: **16:00~19:00, Oct. 28th, 2024

**Location:** Room 1410, SIMIS, Block A, No. 657 Songhu Road, Yangpu District, Shanghai