On explicit computation of Curtis homomorphisms for $GL(n,\mathbb{F}_q)$

TL;DR

This paper provides explicit calculations of Curtis homomorphisms for $GL(n,_q)$, linking them to Gelfand-Tsetlin diagrams and Whittaker functions, enhancing understanding of their structure and computation.

Contribution

It introduces explicit formulas for Curtis homomorphisms in $GL(n,_q)$ and connects them to Gelfand-Tsetlin diagrams using Whittaker functions and Gauss-Bruhat decomposition.

Findings

Explicit formulas for Curtis homomorphisms in $GL(n,_q)$

Connection between homomorphisms and Gelfand-Tsetlin diagrams

Method for computing Whittaker functions via Gauss-Bruhat decomposition

Abstract

In this note we give explicit computations of certain types of Curtis homomorphisms and interpret them in terms of Gelfand-Tsetlin diagrams. Namely, this interpretation follows from Gelfand-Tsetlin formulas for the $GL(n,\mathbb{F}_q)$-Whittaker functions associated with principal series representations via exploiting the method in arXiv:0705.2886 for $GL(n,\mathbb{$})$-Whittaker functions. The explicit computations of Whittaker functions are based on computing the Gauss-Bruhat decomposition for certain elements in $GL(n,\mathbb{F}_q)$.