Infinite-dimensional $p$-adic groups, semigroups of double cosets, and inner functions on Bruhat–Tits buildings

We construct $p$-adic analogues of operator colligations and their characteristic functions. Consider a $p$-adic group $\mathbf G=\mathrm{GL}(\alpha+k\infty,\mathbb Q_p)$, a subgroup $L=\mathrm O(k\infty,\mathbb Z_p)$ of $\mathbf G$ and a subgroup $\mathbf K=\mathrm O(\infty,\mathbb Z_p)$ which is diagonally embedded in $L$. We show that the space $\Gamma=\mathbf K\setminus\mathbf G/\mathbf K$ of double cosets admits the structure of a semigroup and acts naturally on the space of $\mathbf K$-fixed vectors of any unitary representation of $\mathbf G$. With each double coset we associate a ‘characteristic function’ that sends a certain Bruhat–Tits building to another building (the buildings are finite-dimensional) in such a way that the image of the distinguished boundary lies in the distinguished boundary. The second building admits the structure of a (Nazarov) semigroup, and the product in $\Gamma$ corresponds to the pointwise product of characteristic functions.