On extensions of characters of affine pro-$p$ Iwahori--Hecke algebra

TL;DR

This paper computes extensions between characters of the affine pro-$p$ Iwahori--Hecke algebra for reductive groups over non-Archimedean local fields, revealing connections to blocks and L-packets in rank one cases.

Contribution

It provides explicit calculations of extensions of characters of the affine pro-$p$ Iwahori--Hecke algebra, linking algebraic structures to representation theory concepts like blocks and L-packets.

Findings

Computed extensions between characters of the algebra

Established relations between blocks and L-packets in rank one

Enhanced understanding of the algebra's representation theory

Abstract

Let $K$ be a non-discrete non-Archimedean local field with residue characteristic $p$. Let $G$ be the group of $K$ rational points of a algebraic connected reductive group defined over $K$. In this article we compute the extensions between characters of affine pro-$p$ Iwahori--Hecke algebra $\mathcal{H}^{\text{aff}}$ over an algebraically closed field $R$ of characteristic $p$. In rank one case we deduce the relation between the blocks and $L$-packets.