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Buildings and Hecke algebras. (English) Zbl 1095.20003
Starting with \(G=\text{PGL}(n+1,F)\), where \(F\) is a local field, consider the group \(K=\text{GLP}(n+1,{\mathcal O})\) over the valuation ring of \(F\).
It is known that the space of bi-\(K\)-invariant, compactly supported functions on \(G\) is a commutative algebra [I. G. Macdonald, Publ. Ramanujan Inst. 2, 79 p. (1971; Zbl 0302.43018)] and after associating to \(G\) a so-called ‘building’ \(\mathcal X\) that commutative algebra appears to be isomorphic to an algebra \(\mathcal A\) of averaging operators defined on the space of all functions \(G/K\to\mathbb{C}\).
Later [D. I. Cartwright, Monatsh. Math. 133, No. 2, 93-109 (2001; Zbl 1008.51019)] it was shown that \(\mathcal A\) is commutative and that the algebra homomorphisms \(h\colon{\mathcal A}\to\mathbb{C}\) can be expressed as classical Hall-Littlewood polynomials [I. G. Macdonald, Symmetric functions and Hall polynomials. 2nd ed. Oxford: Clarendon Press (1995; Zbl 0824.05059)].
The aim of this paper is to put everything in a more general setting by a combinatorial study of two algebras of averaging operators associated to buildings, providing the close connection between those buildings and Hecke algebras.
It is outside the scope of this review to go into detail: the paper develops the background needed step by step, building up to the ‘climax’ given by the proof that the aforementioned algebra \(\mathcal A\) is isomorphic to the center of an appropriately parameterized affine Hecke algebra.

MSC:
20C08 Hecke algebras and their representations
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
05E30 Association schemes, strongly regular graphs
20E42 Groups with a \(BN\)-pair; buildings
43A90 Harmonic analysis and spherical functions
43A62 Harmonic analysis on hypergroups
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