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Harmonic analysis for groups acting on triangle buildings
- Part of
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- Journal:
- Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 56 / Issue 3 / June 1994
- Published online by Cambridge University Press:
- 09 April 2009, pp. 345-383
- Print publication:
- June 1994
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Let Δ be a thick building of type Ã2, and let
be its set of vertices. We study a commutative algebra 
of ‘averaging’ operators acting on the space of complex valued functions on . This algebra may be identified with a space of ‘biradial functions’ on , or with a convolution algebra of bi-K-invariant functions on G, if G is a sufficiently large group of ‘type-rotating’ automorphisms of Δ, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on and the corresponding spherical functions. We consider the C*-algebra induced by on l2, find its spectrum Σ, prove positive definiteness of a kernel kz for each z ∈ Σ, find explicity the spherical Plancherel formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representations appearing therein. For the class of buildings ΔJ arising from the groups ΓJ introduced in [2], this involves proving that the weak closure of is maximal abelian in the von Neumann algebra generated by the left regular representation of ΓJ.
be its set of vertices. We study a commutative algebra 
of ‘averaging’ operators acting on the space of complex valued functions on 