Separable representations for automorphism groups of infinite symmetric spaces

Pickrell, Doug

doi:10.1016/0022-1236(90)90078-y

Cited by 37 publications

(19 citation statements)

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“…Borel subset of H either has measure zero or has a measure zero complement. The classification of U(∞)-ergodic probability measures on H has been obtained by Pickrell [15,14]. In this paper, the Olshanski-Vershik approach [13] will be followed, see also [3, §4, §5].…”

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confidence: 99%

Infinite random matrices & ergodic decomposition of finite and infinite Hua–Pickrell measures

Qiu

2017

Advances in Mathematics

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The ergodic decomposition of a family of Hua-Pickrell measures on the space of infinite Hermitian matrices is studied. Firstly, we show that the ergodic components of Hua-Pickrell probability measures have no Gaussian factors, this extends a result of Alexei Borodin and Grigori Olshanski. Secondly, we show that the sequence of asymptotic eigenvalues of Hua-Pickrell random matrices is balanced in certain sense and has a "principal value" coincides with the γ 1 parameter of ergodic components. This allow us to complete the program of Borodin and Olshanski on the description of the ergodic decomposition of Hua-Pickrell probability measures. Finally, we extend the aforesaid results to the case of infinite Hua-Pickrell measues. By using the theory of σ-finite infinite determinantal measures recently introduced by A. I. Bufetov, we are able to identify the ergodic decomposition of Hua-Pickrell infinite measures to some explicit σ-finite determinantal measures on the space of point configurations in R * . The paper resolves a problem of Borodin and Olshanski.

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confidence: 99%

Infinite random matrices & ergodic decomposition of finite and infinite Hua–Pickrell measures

Qiu

2017

Advances in Mathematics

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“…For one-dimensional representations Π ±1 the sequence consists of unit operators. The group GL is one of (G, K)-pairs considered in Olshanski's theory of representations of infinite-dimensional classical groups, see [22], [24], [16].…”

Section: 2mentioning

confidence: 99%

“…In the both cases classification of K-spherical representations of G is known, see[23],[24] 8. For instance, we can consider the field of formal Laurent series over Q, then the group PGL 2 over this field acts in a natural way by automorphisms of the tree T 9.…”

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confidence: 99%

On the group of spheromorphisms of a homogeneous non-locally finite tree

Neretin¹

2020

Izv. Math.

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Consider a tree T, all whose vertices have countable valence; its boundary is the Baire space B ≃ N N ; continued fractions expansions identify the set of irrational numbers R \ Q with B. Removing k edges from T we get a forest consisting of copies of T. A spheromorphism (or hierarchomorphism) of T is an isomorphisms of two such subforests considered as a transformation of T or of B. Denote the group of all spheromorphisms by Hier(T). We a show that the correspondence R \ Q ≃ B sends the Thompson group realized by piecewise PSL2(Z)-transformations to a subgroup of Hier(T). We construct some unitary representations of the group Hier(T), show that the group of automorphisms Aut(T) is spherical in Hier(T), and describe the train (enveloping category) of Hier(T).

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“…On representations of infinite-dimensional classical p-adic groups. Basic representation theory of infinite-dimensional classical groups and infinite symmetric groups was developed in 70-80s, see [21], [22], [13] for symmetric groups and [12], [14], [16], [19], [16], [7] for classical groups. These works had various continuations, see, e.g., [7], [1], [15], [3], and further references in [11].…”

Section: C) In Both Cases (Symplectic and Orthogonal)mentioning

confidence: 99%