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Infinite-dimensional $$p$$-adic groups, semigroups of double cosets, and inner functions on Bruhat-Tits buildings. (English) Zbl 1327.22020
Izv. Math. 79, No. 3, 512-553 (2015); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 79, No. 3, 87-130 (2015).
Let $$G=\mathrm{GL}(\infty ,\mathbb Q_p)$$ be the group of all invertible infinite matrices $$g$$ over $$\mathbb Q_p$$, such that $$g-I$$ has only finitely many nonzero entries. Identifying a kind of an orthogonal subgroup $$K$$ in $$G$$, the author studies the space $$\Gamma =K\setminus G/K$$ of double cosets. It is shown that $$\Gamma$$ admits the structure of a semigroup and acts naturally on the space of $$K$$-fixed vectors of any unitary representation of $$G$$. In this framework, the author finds $$p$$-adic analogs of operator colligations and their characteristic functions. With each double coset, a characteristic function acting between certain Bruhat-Tits buildings is associated. The product in $$\Gamma$$ corresponds to the pointwise product of characteristic functions defined using the semigroup structure on some buildings constructed by M. Nazarov [J. Funct. Anal. 128, No. 2, 384–438 (1995; Zbl 0833.22023)]. Open problems are formulated.

##### MSC:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields 5.1e+25 Buildings and the geometry of diagrams
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