# Mod p Hecke algebras and dual equivariant cohomology I: the case of $GL_2$

## TL;DR

This paper explores the relationship between mod p Hecke algebras and equivariant cohomology for the group GL_2 over a p-adic field, revealing new geometric realizations of supersingular modules.

## Contribution

It demonstrates that supersingular irreducible modules of the pro-p Iwahori-Hecke algebra for GL_2 can be realized via equivariant cohomology of the flag variety of the mod p Langlands dual group.

## Key findings

- Supersingular modules of dimension 2 are realized through equivariant cohomology.
- Provides a geometric interpretation of certain Hecke algebra modules.
- Establishes a link between algebraic and geometric structures in the mod p setting.

## Abstract

Let $F$ be a p-adic local field and $G=GL_2(F)$. Let $\mathcal{H}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of $\mathbb{F}_p$. We show that the supersingular irreducible $\mathcal{H}^{(1)}$-modules of dimension 2 can be realized through the equivariant cohomology of the flag variety of the mod p Langlands dual group of $G$.