Representations of $\mathrm{GL}_n(\mathcal{O})$ in functions on Grassmannians

Question

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. The natural representation of the group $\GL_n(\mathcal{O})$ in the space of locally constant functions on the Grassmannian $\Gr_{i,n}$ of $i$-dimensional linear subspaces in $\mathbb{F}^n$ is known to be multiplicity free, see the post Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?

Is it possible to give an explicit formula for a non-zero function on $\Gr_{i,n}$ in every irreducible $\GL_n(\mathcal{O})$-submodule?

Furthermore let $P$ be the stabilizer of a certain fixed $i$-dimensional subspace, so that $\Gr_{i,n}=\GL_n(\mathcal{O})/P$. Then every irreducible subspace of functions on $\Gr_{i,n}$ contains a unique (up to a proportionality) $P$-invariant vector.

Is it possible to describe this vector explicitly?

Answer 1

The answer is given in this paper, by Uri Onn and myself. It might not be very easy to read (I am sorry for this, but this paper and its companion were the first ones each of us wrote and we were young and inexperience).

You are asking of an explicit description of the $P$-invariant vectors in the space of locally constant functions on $\text{Gr}_{i,n}=\text{GL}_n(\mathcal{O})/P$. Each such locally constant function is actually invariant under a small enough congurence subgroup, so we may replace $\text{GL}_n(\mathcal{O})$ by its corresponding quotients. Below I fix an integer $k$, set $G= \text{GL}_n(\mathcal{O}/p^k)$ and by abuse of notation regard $P$ as its image in $G$. These are finite groups and the function spaces below are finite dimensional.

Let $V$ be the space of functions over $G/P$ and conisder the sub vector space of $P$-invariants, $H=V^P$, which we identify with the space of functions on $P\backslash G/P$. $H$ has a natural basis, given by $\delta$ functions (or characteristic functions of $P$-orbits on $G/P$, considered as elements in $V$). However, $H$ aslo has a natural (Hecke-)algebra structure, and what you are looking for really is the idempotents in this algebra, which form another basis for $H$.

What we give in the abovementioned paper is the transition matrix between these two bases. In fact, we consider a third basis, which we call a cellular basis, and we give the transition matrices between each of the previous ones and this basis. The advantage of doing so is that these matrices are triangular, one is upper triangular and the other is lower triangular.

We give the matrices rather explicitly, using a combintaorial type formulation. The formulas we came with are actually based on many explicit compuation we made using a computer package that deals with q-series. Unfortunately, I do not remeber the deatils of these computations now (maybe Uri Onn does, and you could certainly email him and ask).

In reading the paper, my advice would be to start by considering the case $k=1$ (that is, to work over the residue field). Then $P\backslash G/P$ is a set of size $i+1$ (assuming $i<n/2$, which does not reduce the generality, by an easy duality argument), which you may think as possible dimensions of a subspaces of an $i$-dim vector space. After getting this, try $k=2$ and then you'll get a good feeling of what's going on.

Please let me know if it helps. If not, I can try to provide more explanations.