Abstract
We study the eigenvalues of the vertex set Hecke algebra of an affine building, and prove, by a direct approach, the Weyl group invariance of any eigenvalue associated to a character. Moreover, we construct the Satake isomorphism of the Hecke algebra and we prove, by this isomorphism, that every eigenvalue arises from a character.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Betori, W., Pagliacci, M.: Harmonic analysis for groups acting on trees. Boll. Unione Mat. Ital., B (6) 3, 333–349 (1984)
Bourbaki, N.: Lie Groups and Lie Algebras. Chapters 4–6. Elements of Mathematics. Springer, Berlin (2002)
Brown, K.: Buildings. Springer, New York (1989)
Figà-Talamanca, A., Picardello, M.A.: Harmonic Analysis on Free Groups. Lectures Notes in Pure and Applied Math., vol. 87. Dekker, New York (1983)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, vol. 9. Springer, New York (1978)
Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
Macdonald, I.G.: Spherical Functions on a Group of p-Adic Type. Publications of the Ramanujan Institute, vol. 2. Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras (1971)
Mantero, A.M., Zappa, A.: Eigenfunctions of the Laplace operators for a building of type \(\widetilde{A_{2}}\). J. Geom. Anal. 10, 339–363 (2000)
Mantero, A.M., Zappa, A.: Eigenfunctions of the Laplace operators for a building of type \(\widetilde{B_{2}}\). Boll. Unione Mat. Ital., B (8) 5, 163–195 (2002)
Mantero, A.M., Zappa, A.: Eigenfunctions of the Laplace operators for a building of type \(\widetilde{G_{2}}\). Boll. Unione Mat. Ital. (9) 2, 483–508 (2009)
Mantero, A.M., Zappa, A.: Macdonald formula for spherical functions on affine buildings. Ann. Fac. Sci. Toulouse (4) 20, 669–758 (2011)
Parkinson, J.: Buildings and Hecke algebras. J. Algebra 297, 1–49 (2006)
Parkinson, J.: Spherical harmonic analysis on affine buildings. Math. Z. 253, 571–606 (2006)
Ronan, M.: Lectures on Buildings. Perspect. in Mathematics. Academic Press, London (1989)
Tits, J.: Reductive groups over local fields. Automorphic forms, representations and L-functions. In: Proc. Sympos. Pure Math., vol. XXXIII, pp. 29–69. Am. Math. Soc., Providence (1979)
Acknowledgements
We would like to acknowledge several useful suggestions of T. Steger during the preparation of this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Italia
About this chapter
Cite this chapter
Mantero, A.M., Zappa, A. (2013). Eigenvalues of the Vertex Set Hecke Algebra of an Affine Building. In: Picardello, M. (eds) Trends in Harmonic Analysis. Springer INdAM Series, vol 3. Springer, Milano. https://doi.org/10.1007/978-88-470-2853-1_12
Download citation
DOI: https://doi.org/10.1007/978-88-470-2853-1_12
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2852-4
Online ISBN: 978-88-470-2853-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)