Lie Groups, Representations and Discrete Mathematics

Consider an affine building of type $A_n$-tilde, which is a simplicial compex of dimension $n$. For $n=1$, this is a tree, which we will require to be homogeneous. Consider the space of complex valued functions on the vertices of the building, and then consider the algebra $A$ of invariant, finitely-supported difference operators. Here invariant can usually be taken to mean invariant with respect to the group of automorphisms on the building. (But for $n=2$, it can happen that there are not enough automorphisms. In that case one must define invariant differently.) The algebra $A$ is commutative. In fact, it is isomorphic to a polynomial algebra of degree $n$ over the complexes. As such, it has an algebraic spectrum corresponding to affine $n$-space. Since the elements of $A$ are finitely supported, they preserve the space of $\ell^2$ functions. Viewed this way, $A$ is a self-adjoint operator algebra, and may be completed to a $C^*$-algebra. One can identify concretely the spectrum of this operator algebra as a subset of the algebraic spectrum of $A$. The talk will be strictly expositional: most of what's to be discussed was published by Tamagawa in 1963, and the rest by MacDonald in 1968. Moreover, because I like to draw pictures, discussion will center on the cases $n=1$ (trees) and $n=2$. ===== REFERENCES ===== Tamagawa, Tsuneo On the $\zeta $-functions of a division algebra. Ann. of Math. (2) 77 1963 387--405. Satake, Ichir\^o Theory of spherical functions on reductive algebraic groups over ${\germ p}$-adic fields. Inst. Hautes\`Etudes Sci. Publ. Math. No. 18, 1963 5--69. Macdonald, I. G. Spherical functions on a ${\germ p}$-adic Chevalley group. Bull. Amer. Math. Soc. 74 1968 520--525. Spherical functions on a group of $p$-adic type. Publications of the Ramanujan Institute, No. 2. Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, 1971. vii+79 pp. Cartwright, Donald I.; M\l otkowski, Wojciech Harmonic analysis for groups acting on triangle buildings. J. Austral. Math. Soc. Ser. A 56 (1994), no. 3, 345--383. Cartwright, Donald I. Spherical harmonic analysis on buildings of type $A_n$-tilde. Monatsh. Math. 133 (2001), no. 2, 93--109. Cartwright, Donald I.; Sol\'e, Patrick; \.Zuk, Andrzej Ramanujan geometries of type $\sb n$. Discrete Math. 269 (2003), no. 1-3, 35--43.