Infinite-dimensional \(p\)-adic groups, semigroups of double cosets, and inner functions on Bruhat-Tits buildings. (English) Zbl 1327.22020

Neretin, Yu A.

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.

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

22E50	Representations of Lie and linear algebraic groups over local fields
51E24	Buildings and the geometry of diagrams

Keywords:

Bruhat-Tits building; characteristic function; infinite matrices