Let $G=GL\_n(\mathbb{F}\_q)$ be the (finite) group of all linear invertible transformations of the vector space $(\mathbb{F}\_q)^n$ over the finite field $\mathbb{F}\_q$.


$G$ acts naturally on the Grassmannian $Gr\_{k,n}$ of $k$-dimensional linear subspaces of $(\mathbb{F}\_q)^n$.


**What is the length of the corresponding representation of $G$ in the space of complex valued functions on $Gr\_{k,n}$?**



Length of representation of $GL_n(\mathbb{F}_q)$ in functions on Grassmannian
I believe that this is answered in Proposition 5.1 of [this](https://arxiv.org/abs/1811.08675) paper, which says that it is $\text{min}(k,n-k) + 1$.


