Representations and corresponding operators induced by Hecke algebras. (English) Zbl 1332.05152
Summary: In this paper, by establishing free-probabilistic models on the Hecke algebras \(\mathcal H(G_p)\), we construct canonical free probability spaces \((\mathcal H(G_p),\psi_p)\), where \(G_p=\mathrm{GL}_2(\mathbb Q_p)\), for primes \(p\). Dependent upon such free-probabilistic structures, we study corresponding representations of \(\mathcal H(G_p)\), and consider spectral properties of operators realized under representations.
MSC:
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 05E10 | Combinatorial aspects of representation theory |
| 11G15 | Complex multiplication and moduli of abelian varieties |
| 11R04 | Algebraic numbers; rings of algebraic integers |
| 11R09 | Polynomials (irreducibility, etc.) |
| 11R47 | Other analytic theory |
| 11R56 | Adèle rings and groups |
| 46L10 | General theory of von Neumann algebras |
| 46L40 | Automorphisms of selfadjoint operator algebras |
| 46L53 | Noncommutative probability and statistics |
| 46L54 | Free probability and free operator algebras |
| 47L15 | Operator algebras with symbol structure |
| 47L30 | Abstract operator algebras on Hilbert spaces |
| 47L55 | Representations of (nonselfadjoint) operator algebras |
Keywords:
free probability; free moments; free cumulants; Hecke algebra; normal Hecke subalgebra; free probability spaces; representations; operators; Hilbert spacesReferences:
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