Boundary unitary representations—right-angled hyperbolic buildings

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

Boundary unitary representations—right-angled hyperbolic buildings

Uri Bader ^1, and Jan Dymara ^2,

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

Abstract
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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.

Keywords: Hecke algebra., Unitary representation, hyperbolic building.

Mathematics Subject Classification: Primary: 22D10; Secondary: 20C08, 20E4.

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), 49-69. doi: 10.3934/jmd.2011.5.49.
[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), 731-741. doi: 10.1007/s00209-002-0442-6.
[3]	N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Masson, Paris, 1981.
[4]	M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9.
[5]	S. Buyalo and V. Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/036.
[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), 339-358. doi: 10.4310/MRL.2013.v20.n2.a10.
[7]	M. Caspers, Absence of Cartan subalgebras for Hecke von Neumann algebras, preprint, arXiv:1601.00593.
[8]	M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. Reine Angew. Math., 420 (1991), 85-98.
[9]	M. Davis, The Geometry and Topology of Coxeter Groups, London Mathematical Society Monographs Series, 32, Princeton University Press, Princeton, NJ, 2008.
[10]	J. Dymara and D. Osajda, Boundaries of right-angled hyperbolic buildings, Fund. Math., 197 (2007), 123-165. doi: 10.4064/fm197-0-6.
[11]	J. Dymara, Thin buildings, Geom. Topol., 10 (2006), 667-694. doi: 10.2140/gt.2006.10.667.
[12]	A. Figà-Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups, Lecture Notes in Pure and Applied Mathematics, 8, Marcel Dekker, Inc., New York, 1983.
[13]	A. Figà-Talamanca and T. Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Mem. Amer. Math. Soc., 110 (1994), xii+68. doi: 10.1090/memo/0531.
[14]	Ł. Garncarek, Analogs of principal series representations for Thompson's groups $F$ and $T$, Indiana Univ. Math. J., 61 (2012), 619-626. doi: 10.1512/iumj.2012.61.4572.
[15]	Ł. Garncarek, Boundary representations of hyperbolic groups, preprint, arXiv:1404.0903.
[16]	Ł. Garncarek, Factoriality of Hecke-von Neumann algebras of right-angled Coxeter groups, J. Funct. Anal., 270 (2016), 1202-1219. doi: 10.1016/j.jfa.2015.11.014.
[17]	R. Howe and A. Moy, Hecke algebra isomorphisms for $GL(n)$ over a $p$-adic field, J. Algebra, 131 (1990), 388-424. doi: 10.1016/0021-8693(90)90182-N.
[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), 215-236.
[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), 5-48.
[20]	D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165-184. doi: 10.1007/BF01390031.
[21]	B. Rémy and M. Ronan, Topological groups of Kac-Moody type, right-angled twinnings and their lattices, Comment. Math. Helv., 81 (2006), 191-219. doi: 10.4171/CMH/49.

show all references

References:

[1]	U. Bader and R. Muchnik, Boundary unitary representations-irreducibility and rigidity, J. Mod. Dyn., 5 (2011), 49-69. doi: 10.3934/jmd.2011.5.49.
[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), 731-741. doi: 10.1007/s00209-002-0442-6.
[3]	N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Masson, Paris, 1981.
[4]	M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9.
[5]	S. Buyalo and V. Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/036.
[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), 339-358. doi: 10.4310/MRL.2013.v20.n2.a10.
[7]	M. Caspers, Absence of Cartan subalgebras for Hecke von Neumann algebras, preprint, arXiv:1601.00593.
[8]	M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. Reine Angew. Math., 420 (1991), 85-98.
[9]	M. Davis, The Geometry and Topology of Coxeter Groups, London Mathematical Society Monographs Series, 32, Princeton University Press, Princeton, NJ, 2008.
[10]	J. Dymara and D. Osajda, Boundaries of right-angled hyperbolic buildings, Fund. Math., 197 (2007), 123-165. doi: 10.4064/fm197-0-6.
[11]	J. Dymara, Thin buildings, Geom. Topol., 10 (2006), 667-694. doi: 10.2140/gt.2006.10.667.
[12]	A. Figà-Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups, Lecture Notes in Pure and Applied Mathematics, 8, Marcel Dekker, Inc., New York, 1983.
[13]	A. Figà-Talamanca and T. Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Mem. Amer. Math. Soc., 110 (1994), xii+68. doi: 10.1090/memo/0531.
[14]	Ł. Garncarek, Analogs of principal series representations for Thompson's groups $F$ and $T$, Indiana Univ. Math. J., 61 (2012), 619-626. doi: 10.1512/iumj.2012.61.4572.
[15]	Ł. Garncarek, Boundary representations of hyperbolic groups, preprint, arXiv:1404.0903.
[16]	Ł. Garncarek, Factoriality of Hecke-von Neumann algebras of right-angled Coxeter groups, J. Funct. Anal., 270 (2016), 1202-1219. doi: 10.1016/j.jfa.2015.11.014.
[17]	R. Howe and A. Moy, Hecke algebra isomorphisms for $GL(n)$ over a $p$-adic field, J. Algebra, 131 (1990), 388-424. doi: 10.1016/0021-8693(90)90182-N.
[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), 215-236.
[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), 5-48.
[20]	D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165-184. doi: 10.1007/BF01390031.
[21]	B. Rémy and M. Ronan, Topological groups of Kac-Moody type, right-angled twinnings and their lattices, Comment. Math. Helv., 81 (2006), 191-219. doi: 10.4171/CMH/49.

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