Let $F$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Let $GL\_n(\mathcal{O})$ denote the group of invertible $n\times n$ matrices with entries in $\mathcal{O}$ with the inverse having entries in $\mathcal{O}$.


Consider the natural (unitary) representation of $GL\_2(\mathcal{O})$ in functions on $F\mathbb{P}^1$. It is multiplicity free by the answers here [Is the representation of $GL\_n(\mathcal{O})$ in functions on Grassmannian multiplicity free?](https://mathoverflow.net/questions/350230/is-the-representation-of-gl-n-mathcalo-in-functions-on-grassmannian-multip)


**Is there an explicit description of the irreducible components of the above representation? For example exhibiting of a non-zero function in each irreducible component would be of interest.**



Representation of $GL_2(\mathcal{O})$ in space of functions on projective line
Let $F \mathbb{P}^{n - 1}$ denote $(n - 1)$-dimensional $F$-projective space, which we may identify with $\mathrm{P}\_{(n - 1,1)}(\mathcal{O}) \backslash \mathrm{GL}\_n(\mathcal{O})$, where
$$\mathrm{P}\_{(n - 1,1)}(\mathcal{O}) :=\left\{\begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \in \mathrm{GL}\_n(\mathcal{O}) : a \in \mathrm{GL}\_{n - 1}(\mathcal{O}), \ b \in \mathrm{Mat}\_{(n - 1) \times 1}(\mathcal{O}), \ d \in \mathcal{O}^{\times}\right\}.$$


Let $S^{n - 1}$ denote the $(n - 1)$-dimensional $F$-sphere, which we may identify with $\widetilde{\mathrm{P}}\_{(n - 1,1)}(\mathcal{O}) \backslash \mathrm{GL}\_n(\mathcal{O})$, where
$$\widetilde{\mathrm{P}}\_{(n - 1,1)}(\mathcal{O}) :=\left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \in \mathrm{GL}\_n(\mathcal{O}) : a \in \mathrm{GL}\_{n - 1}(\mathcal{O}), \ b \in \mathrm{Mat}\_{(n - 1) \times 1}(\mathcal{O})\right\}.$$
More explicitly, we can write
$$S^{n - 1} := \left\{(x\_1,\ldots,x\_n) \in F^n : \max\{|x\_1|,\ldots,|x\_n|\} = 1\right\}.$$


Let $C^{\infty}(F\mathbb{P}^{n - 1})$ denote the space of smooth functions on $(n - 1)$-dimensional $F$-projective space. This is the same as the subspace of $C^{\infty}(S^{n - 1})$, the space of smooth function on the $(n - 1)$-dimensional $F$-sphere, consisting of functions that are additionally $\mathrm{Z}(\mathcal{O})$-invariant, where $\mathrm{Z}(\mathcal{O})$ denotes the centre of $\mathrm{GL}\_n(\mathcal{O})$.


The irreducible smooth representations of $\mathrm{GL}\_n(\mathcal{O})$ that appear in the decomposition of $C^{\infty}(F\mathbb{P}^{n - 1})$ are the same as the irreducible smooth representations *with trivial central character* that appear in the decomposition of $C^{\infty}(S^{n - 1})$. These are given explicitly as follows.


First, we define the congruence subgroups
$$K\_0(\mathfrak{p}^m) := \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{GL}\_n(\mathcal{O}) : c \in \mathrm{Mat}\_{1 \times (n - 1)}(\mathfrak{p}^m)\right\},$$
$$K(\mathfrak{p}^m) := \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{GL}\_n(\mathcal{O}) : k - 1\_n \in \mathrm{Mat}\_{n \times n}(\mathfrak{p}^m)\right\}$$
of $\mathrm{GL}\_n(\mathcal{O})$. We consider the induced representation $\mathrm{Ind}\_{K\_0(\mathfrak{p}^m)}^{\mathrm{GL}\_n(\mathcal{O})} 1$. This can be identified with the subspace of $C^{\infty}(S^{n - 1}) \ni f$ consisting of functions satisfying $f(axk) = f(x)$ for all $a \in \mathcal{O}^{\times}$ and $k \in K(\mathfrak{p}^m)$. This is not irreducible unless $m = 0$, since it contains $\mathrm{Ind}\_{K\_0(\mathfrak{p}^{m - 1})}^{\mathrm{GL}\_n(\mathcal{O})} 1$. So we define $\rho\_0$ to be the trivial representation and for $m \geq 1$,
$$\rho\_m := \mathrm{Ind}\_{K\_0(\mathfrak{p}^m)}^{\mathrm{GL}\_n(\mathcal{O})} 1 \ominus \mathrm{Ind}\_{K\_0(\mathfrak{p}^{m - 1})}^{\mathrm{GL}\_n(\mathcal{O})} 1.$$


One can show that this is irreducible and determine its dimension; for $n = 2$, we have that $\dim \rho\_0 = 1$, $\dim \rho\_1 = q$, and $\dim \rho\_m = q^{m - 2} (q^2 - 1)$ for $m \geq 2$.


You also asked about specific nonzero functions in each irreducible component. I prove this in my paper in Proposition 3.4: when viewed as an irreducible component of $C^{\infty}(S^{n - 1})$, an explicit function $P\_m$ in $\rho\_m$ is given as follows:
$$P\_0(x\_1,\ldots,x\_n) = 1,$$
$$P\_1(x\_1,\ldots,x\_n) = \begin{cases}
1 & \text{if $\max\{|x\_1|,\ldots,|x\_{n - 1}|\} \leq q^{-1}$,} \\
-\frac{1}{q} & \text{if $\max\{|x\_1|,\ldots,|x\_{n - 1}|\} = 1$,}
\end{cases}$$
and for $m \geq 2$,
$$P\_m(x\_1,\ldots,x\_n) = \begin{cases}
1 & \text{if $\max\{|x\_1|,\ldots,|x\_{n - 1}|\} \leq q^{-m}$,} \\
-\frac{1}{q - 1} & \text{if $\max\{|x\_1|,\ldots,|x\_{n - 1}|\} = q^{-m + 1}$,} \\
0 & \text{otherwise.}
\end{cases}$$


