Geometric Satake, Springer correndence, and small representations

開催日時
2015/01/27 火 16:30 - 18:00
場所
3号館110講演室
講演者
Anthony Henderson
講演者所属
シドニー大学
概要

Let $G$ be a connected reductive group and $W$ its Weyl group. Consider the functor $\Phi$ from representations of $G$ to representations of $W$ defined by taking the zero weight space. This functor contains important information, but is hard to describe in general. Note that when $G=GL_n$, the restriction of $\Phi$ to the subcategory of representations whose weights $(a_1,\cdots,a_n)$ satisfy $a_1+\cdots+a_n=0$ and $a_i\geq -1$ is essentially the famous Schur functor. In particular, this restriction is of the form $\mathrm{Hom}_{GL_n}(E,-)$ where $E$ is a tilting module that carries a commuting $S_n$-action.

For general $G$, the analogous subcategory to consider is that of small representations, and the restriction of $\Phi$ to this subcategory was studied by Broer and Reeder in the complex case. However, there is no representation analogous to $E$ in other types. In joint work with Pramod Achar (Louisiana State University) and Simon Riche (Universit\'e Blaise Pascal - Clermont-Ferrand II), we describe the restriction of $\Phi$ geometrically, in terms of the perverse sheaves on the affine Grassmannian of the complex dual group $G^\vee$ that correspond to small representations under geometric Satake; this makes sense for any ground field. As we show, the correct substitute for $E$ is the Springer sheaf on the nilpotent cone of $G^\vee$, with its $W$-action that gives rise to the Springer correspondence.