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Spherical harmonic analysis on buildings of type \(\tilde A_n\). (English) Zbl 1008.51019

In [Random walks and discrete potential theory (Cortona, 1997), Cambridge Univ. Press, Cambridge, 104-138 (1999; Zbl 0958.51015)] the author showed that the algebra \({\mathcal A}\) generated by the averaging operators \(A_r\) for \(r\in\{1,\ldots,n\}\) on a locally finite thick building \({\mathfrak X}\) of type \(\tilde A_n\) with vertex set \({\mathfrak X}^0\) is commutative and that \({\mathcal A}\) is spanned by the more general averaging operators \(A_k\) for \(k\in{\mathbb N}^n\).
In the paper under review the author classifies all multiplicative functionals on \(\mathcal{A}\). Using formulas for the structure constants of \({\mathcal A}\) with respect to the \(A_k\) found in the previous paper two expressions for these functionals are developed, one in terms of Hall-Littlewood polynomials associated with partitions and the other one as an integral formula on the boundary of \({\mathfrak X}\). Restricting the averaging operators to the Hilbert space \(l^2({\mathfrak X}^0)\) of square summable functions on \({\mathfrak X}^0\), the algebra \({\mathcal A}\) can be regarded as a subalgebra of \({\mathcal L}(l^2({\mathfrak X}^0))\). Its norm closure \(\bar{\mathcal A}\) is a commutative \(C^*\)-algebra and the author calculates the spectrum of \(\bar{\mathcal A}\). Then \(\bar{\mathcal A}\) is isometrically isomorphic to the algebra of continuous symmetric functions on \(Z_n\), the space of \((n+1)\)-tuples of complex numbers of modulus 1 with product 1. This generalises results of P. Cartier [Sem. Bourbaki 1971/72, No. 407, Lect. Notes Math. 317, 123-140 (1973; Zbl 0267.14010)] for \(n=1\) and the author and W. Mlotkowski [J. Aust. Math. Soc 56, 345-383 (1994; Zbl 0808.51014)] for \(n=2\).

MSC:

51E24 Buildings and the geometry of diagrams
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
20E42 Groups with a \(BN\)-pair; buildings
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