# A sharp upper bound for the size of Lusztig series

## TL;DR

This paper establishes a simple formula to bound the maximum size of Lusztig series in finite groups of Lie type, providing explicit bounds for classical groups based on their Lie rank and characteristic.

## Contribution

It introduces a new upper bound formula for Lusztig series sizes and explicitly determines the maximum for large q in classical groups.

## Key findings

- Derived a simple upper bound formula for Lusztig series sizes.
- Explicitly calculated maximum sizes for classical groups when q is large.
- Connected the bounds to Lie rank and defining characteristic.

## Abstract

The paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group $G$ of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group $G$ in terms of its Lie rank and defining characteristic. When $G$ is specified as $G(q)$ and $q$ is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of $G$.