J Austral Math Soc Ser A 56 pp345--383, 1994.

Harmonic Analysis for Groups Acting on Triangle Buildings

Donald I. Cartwright and Wojciech Mlotkowski

Abstract

Let D be a thick building of type ~A₂, and let V be its set of vertices. We study a commutative algebra A of 'averaging' operators acting on the soace of complex valued functions on V. This algebra may be identified with a space of 'biradial functions' on V, or with a convolution algebra of bi-K-invariant functions on G, if G is a sufficiently large group of 'type-rotating' automorphisms of D, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on A and corresponding spherical functions. We consider the C*-algebra induced by A on l²(V), find its spectrum S, prove positive definiteness of a kernel k_z for each z Î S, find explicitly the spherical Plancheral formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representaions appearing therein. For the class of buildings D_J arising from the groups G_J introduced in [2], this involves proving that the weak closure of A is maximal abelian in the von Neumann algebra generated by the left regular representation of G_J.

1991 AMS Subject Classification: primary 51E24, 22E50l; secondary 43A35, 43A90

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Authors

Donald I. Cartwright

School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia.

Wojciech Mlotkowski

Institute of Mathematics, The University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland.