Unimodularity and atomic Plancherel measure

AbstractLet Γ be a rank-one arithmetic subgroup of a semisimple Lie group G. For fixed K-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of G, whose discrete part encodes the dimensions of the spaces of square-integrable Γ-automorphic forms. It is shown that this distribution converges to the Plancherel measure of G when Γ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices Γ follows from results of DeGeorge-Wallach and Delorme.

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Uniqueness of the Plancherel measure as an invariant measure over the dual objects of compact groups

Proceedings of the Japan Academy ◽

10.3792/pja/1195519965 ◽

1971 ◽

Vol 47 (4) ◽

pp. 346-347

Author(s):

Nobuhiko Tatsuuma

Keyword(s):

Invariant Measure ◽

Compact Groups ◽

Plancherel Measure

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Tensor Square of the Minimal Representation of O(p, q)

Canadian Mathematical Bulletin ◽

10.4153/cmb-2007-005-x ◽

2007 ◽

Vol 50 (1) ◽

pp. 48-55 ◽

Cited By ~ 2

Author(s):

Alexander Dvorsky

Keyword(s):

Tensor Product ◽

Hyperbolic Space ◽

Irreducible Representations ◽

Plancherel Measure ◽

Minimal Representation ◽

Direct Integral ◽

Tensor Square ◽

Real Hyperbolic Space

AbstractIn this paper, we study the tensor product π = σmin ⊗ σmin of the minimal representation σmin of O(p, q) with itself, and decompose π into a direct integral of irreducible representations. The decomposition is given in terms of the Plancherel measure on a certain real hyperbolic space.

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Gaussian fluctuations of Young diagrams under the Plancherel measure

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2006.1808 ◽

2007 ◽

Vol 463 (2080) ◽

pp. 1069-1080 ◽

Cited By ~ 6

Author(s):

Leonid V Bogachev ◽

Zhonggen Su

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Gaussian Process ◽

Central Limit ◽

Point Processes ◽

Plancherel Measure ◽

Young Diagrams ◽

Finite Dimensional ◽

The Central Limit Theorem ◽

Local Correlations

We obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the ‘spectrum’ of partitions λ ⊢ n ∈ (under the Plancherel measure), thus settling a long-standing problem posed by Logan & Shepp. Namely, under normalization growing like , the corresponding random process in the bulk is shown to converge, in the sense of finite-dimensional distributions, to a Gaussian process with independent values, while local correlations in the vicinity of each point, measured on various power scales, possess certain self-similarity. The proofs are based on the Poissonization techniques and use Costin–Lebowitz–Soshnikov's central limit theorem for determinantal random point processes. Our results admit a striking reformulation after the rotation of Young diagrams by 45°, whereby the normalization no longer depends on the location in the spectrum. In addition, we explain heuristically the link with an earlier result by Kerov on the convergence to a generalized Gaussian process.

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