An exotic Springer correspondence for $F_4$

TL;DR

This paper explores the exotic nilcone of the algebraic group F_4, establishing a finite orbit classification, constructing an exotic Springer correspondence, and linking these geometric structures to representations of affine Hecke algebras.

Contribution

It introduces a new exotic Springer correspondence for F_4's nilcone, revealing finite orbit structure and affine pavings, and connects these to simple modules of affine Hecke algebras.

Findings

Finite number of orbits on the exotic nilcone

Existence of affine pavings for exotic Springer fibers

Geometric classification of certain affine Hecke algebra modules

Abstract

We investigate the structure of the `exotic nilcone' of $F_4$ which is defined by exploiting certain characteristic two phenomena. We show that there are finitely many orbits on this nilcone and construct an associated Springer correspondence. Further to that, we show that all corresponding `exotic Springer fibers' admit an affine paving. We also deduce from this a geometric classification of certain simple modules for the affine Hecke algebra with unequal parameters of type $F_4$.