On $p$-adic colligations and 'rational maps' of Bruhat-Tits trees

TL;DR

This paper explores $p$-adic colligations and rational maps of Bruhat-Tits trees, introducing a product of conjugacy classes and characteristic functions that map trees to buildings, with implications for operator colligations.

Contribution

It defines a new product of $p$-adic conjugacy classes and constructs characteristic functions as maps from Bruhat-Tits trees to buildings, extending the theory of operator colligations.

Findings

Defined a product of $p$-adic conjugacy classes.

Constructed characteristic functions mapping trees to buildings.

Analyzed categorical quotients for operator colligations.

Abstract

Consider matrices of order $k+N$ over $p$-adic field determined up to conjugations by elements of $GL$ over $p$-adic integers. We define a product of such conjugacy classes and construct the analog of characteristic functions (transfer functions), they are maps from Bruhat-Tits trees to Bruhat-Tits buildings. We also examine categorical quotient for usual operator colligations.