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

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