## 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

### Keywords:

building; Hall-Littlewood polynomial; $$C^*$$-algebra

### Citations:

Zbl 0958.51015; Zbl 0267.14010; Zbl 0808.51014
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