Buildings and the Spectra of their Laplacians
| LIE GROUPS, REPRESENTATIONS AND DISCRETE MATH | |
| Topic: | Buildings and the Spectra of their Laplacians |
| Speaker: | Tim Steger |
| Affiliation: | IAS |
| Date: | Tuesday, November 1 |
| Time/Room: | 2:00pm - 3:15pm/S-101 |
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 $n=2$.
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Macdonald, I. G.
Spherical functions on a ${\germ p}$-adic Chevalley group.
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