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2016, 10: 413-437. doi: 10.3934/jmd.2016.10.413

Boundary unitary representations—right-angled hyperbolic buildings

1. 

Department of Mathematics, The Weizmann Institute of Science, Rehovot, 7610001, Israel

2. 

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Received  June 2015 Revised  July 2016 Published  September 2016

We study the unitary boundary representation of a strongly transitive group acting on a right-angled hyperbolic building. We show its irreducibility. We do so by associating to such a representation a representation of a certain Hecke algebra, which is a deformation of the classical representation of a hyperbolic reflection group. We show that the associated Hecke algebra representation is irreducible.
Citation: Uri Bader, Jan Dymara. Boundary unitary representations—right-angled hyperbolic buildings. Journal of Modern Dynamics, 2016, 10: 413-437. doi: 10.3934/jmd.2016.10.413
References:
[1]

U. Bader and R. Muchnik, Boundary unitary representations-irreducibility and rigidity,, J. Mod. Dyn., 5 (2011), 49.  doi: 10.3934/jmd.2011.5.49.  Google Scholar

[2]

M. Bekka and M. Cowling, Some irreducible unitary representations of $G(K)$ for a simple algebraic group $G$ over an algebraic number field $K$,, Math. Z., 241 (2002), 731.  doi: 10.1007/s00209-002-0442-6.  Google Scholar

[3]

N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie,, Chapitres 4, (1981).   Google Scholar

[4]

M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature,, Grundlehren der Mathematischen Wissenschaften, (1999).  doi: 10.1007/978-3-662-12494-9.  Google Scholar

[5]

S. Buyalo and V. Schroeder, Elements of asymptotic geometry,, EMS Monographs in Mathematics, (2007).  doi: 10.4171/036.  Google Scholar

[6]

I. Capdeboscq and A. Thomas, Cocompact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups,, Math. Res. Lett., 20 (2013), 339.  doi: 10.4310/MRL.2013.v20.n2.a10.  Google Scholar

[7]

M. Caspers, Absence of Cartan subalgebras for Hecke von Neumann algebras,, preprint, ().   Google Scholar

[8]

M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices,, J. Reine Angew. Math., 420 (1991), 85.   Google Scholar

[9]

M. Davis, The Geometry and Topology of Coxeter Groups,, London Mathematical Society Monographs Series, (2008).   Google Scholar

[10]

J. Dymara and D. Osajda, Boundaries of right-angled hyperbolic buildings,, Fund. Math., 197 (2007), 123.  doi: 10.4064/fm197-0-6.  Google Scholar

[11]

J. Dymara, Thin buildings,, Geom. Topol., 10 (2006), 667.  doi: 10.2140/gt.2006.10.667.  Google Scholar

[12]

A. Figà-Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups,, Lecture Notes in Pure and Applied Mathematics, (1983).   Google Scholar

[13]

A. Figà-Talamanca and T. Steger, Harmonic analysis for anisotropic random walks on homogeneous trees,, Mem. Amer. Math. Soc., 110 (1994).  doi: 10.1090/memo/0531.  Google Scholar

[14]

Ł. Garncarek, Analogs of principal series representations for Thompson's groups $F$ and $T$,, Indiana Univ. Math. J., 61 (2012), 619.  doi: 10.1512/iumj.2012.61.4572.  Google Scholar

[15]

Ł. Garncarek, Boundary representations of hyperbolic groups,, preprint, ().   Google Scholar

[16]

Ł. Garncarek, Factoriality of Hecke-von Neumann algebras of right-angled Coxeter groups,, J. Funct. Anal., 270 (2016), 1202.  doi: 10.1016/j.jfa.2015.11.014.  Google Scholar

[17]

R. Howe and A. Moy, Hecke algebra isomorphisms for $GL(n)$ over a $p$-adic field,, J. Algebra, 131 (1990), 388.  doi: 10.1016/0021-8693(90)90182-N.  Google Scholar

[18]

N. Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field,, J. Fac. Sci. Univ. Tokyo Sect. I, 10 (1964), 215.   Google Scholar

[19]

N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups,, Inst. Hautes Études Sci. Publ. Math., 25 (1965), 5.   Google Scholar

[20]

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras,, Invent. Math., 53 (1979), 165.  doi: 10.1007/BF01390031.  Google Scholar

[21]

B. Rémy and M. Ronan, Topological groups of Kac-Moody type, right-angled twinnings and their lattices,, Comment. Math. Helv., 81 (2006), 191.  doi: 10.4171/CMH/49.  Google Scholar

show all references

References:
[1]

U. Bader and R. Muchnik, Boundary unitary representations-irreducibility and rigidity,, J. Mod. Dyn., 5 (2011), 49.  doi: 10.3934/jmd.2011.5.49.  Google Scholar

[2]

M. Bekka and M. Cowling, Some irreducible unitary representations of $G(K)$ for a simple algebraic group $G$ over an algebraic number field $K$,, Math. Z., 241 (2002), 731.  doi: 10.1007/s00209-002-0442-6.  Google Scholar

[3]

N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie,, Chapitres 4, (1981).   Google Scholar

[4]

M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature,, Grundlehren der Mathematischen Wissenschaften, (1999).  doi: 10.1007/978-3-662-12494-9.  Google Scholar

[5]

S. Buyalo and V. Schroeder, Elements of asymptotic geometry,, EMS Monographs in Mathematics, (2007).  doi: 10.4171/036.  Google Scholar

[6]

I. Capdeboscq and A. Thomas, Cocompact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups,, Math. Res. Lett., 20 (2013), 339.  doi: 10.4310/MRL.2013.v20.n2.a10.  Google Scholar

[7]

M. Caspers, Absence of Cartan subalgebras for Hecke von Neumann algebras,, preprint, ().   Google Scholar

[8]

M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices,, J. Reine Angew. Math., 420 (1991), 85.   Google Scholar

[9]

M. Davis, The Geometry and Topology of Coxeter Groups,, London Mathematical Society Monographs Series, (2008).   Google Scholar

[10]

J. Dymara and D. Osajda, Boundaries of right-angled hyperbolic buildings,, Fund. Math., 197 (2007), 123.  doi: 10.4064/fm197-0-6.  Google Scholar

[11]

J. Dymara, Thin buildings,, Geom. Topol., 10 (2006), 667.  doi: 10.2140/gt.2006.10.667.  Google Scholar

[12]

A. Figà-Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups,, Lecture Notes in Pure and Applied Mathematics, (1983).   Google Scholar

[13]

A. Figà-Talamanca and T. Steger, Harmonic analysis for anisotropic random walks on homogeneous trees,, Mem. Amer. Math. Soc., 110 (1994).  doi: 10.1090/memo/0531.  Google Scholar

[14]

Ł. Garncarek, Analogs of principal series representations for Thompson's groups $F$ and $T$,, Indiana Univ. Math. J., 61 (2012), 619.  doi: 10.1512/iumj.2012.61.4572.  Google Scholar

[15]

Ł. Garncarek, Boundary representations of hyperbolic groups,, preprint, ().   Google Scholar

[16]

Ł. Garncarek, Factoriality of Hecke-von Neumann algebras of right-angled Coxeter groups,, J. Funct. Anal., 270 (2016), 1202.  doi: 10.1016/j.jfa.2015.11.014.  Google Scholar

[17]

R. Howe and A. Moy, Hecke algebra isomorphisms for $GL(n)$ over a $p$-adic field,, J. Algebra, 131 (1990), 388.  doi: 10.1016/0021-8693(90)90182-N.  Google Scholar

[18]

N. Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field,, J. Fac. Sci. Univ. Tokyo Sect. I, 10 (1964), 215.   Google Scholar

[19]

N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups,, Inst. Hautes Études Sci. Publ. Math., 25 (1965), 5.   Google Scholar

[20]

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras,, Invent. Math., 53 (1979), 165.  doi: 10.1007/BF01390031.  Google Scholar

[21]

B. Rémy and M. Ronan, Topological groups of Kac-Moody type, right-angled twinnings and their lattices,, Comment. Math. Helv., 81 (2006), 191.  doi: 10.4171/CMH/49.  Google Scholar

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