# Descent equalities and the inductive McKay condition for types B and E

## TL;DR

This paper proves the inductive McKay condition for certain finite simple groups of Lie types B, E6, E6 (twisted), and E7, using descent arguments and Shintani's norm map, advancing the understanding of the McKay conjecture.

## Contribution

The authors establish the inductive McKay condition for specific Lie type groups, extending previous methods with descent arguments and uniform proofs for global requirements.

## Key findings

- Proved the inductive McKay condition for types B, E6, ^2E6, and E7.
- Developed descent arguments using Shintani's norm map.
- Verified local requirements through detailed analysis of normalizers.

## Abstract

We establish the inductive McKay condition introduced by Isaacs-Malle-Navarro \cite{IMN} for finite simple groups of Lie types $\tB_l$ ($l\geq 2$), $\tE_6$, $^2\tE_6$ and $\tE_7$, thus leaving open only the types $\tD$ and $^2\tD$. We bring to the methods previously used by the authors for type $\tC$ \cite{CS17C} some descent arguments using Shintani's norm map. This provides for types different from $ \tA, \tD, {}^2\tD$ a uniform proof of the so-called global requirement of the criterion given by the second author in \cite[2.12]{S12}. The local requirements from that criterion are verified through a detailed study of the normalizers of relevant Levi subgroups and their characters.