## Notes of Bartosz Trojan’s lecture

Heat kernel on affine buildings

Consider finite support probability distributions ${p(v,\cdot)}$ on an affinebuiding which are spherical : depends only on distance.

Plays the role of heat kernel on symmetric spaces. For such a kernel, one has rather sharp asymptotic estimates (Anker-Ji), used to dtermined the Martin boundary. There remains open questions about the asymptotic behaviour of Green function.

1. Case of ${{\mathbb Z}^r}$

Theorem 1 (Lawler-Limic)

$\displaystyle \begin{array}{rcl} p_n(v)\sim (det nB_s)^{-1/2}e^{-n\phi(\delta)}, \end{array}$

in some ball ${|v|<\leq\rho n}$. Here ${B_s =D^2 \log\kappa(s)}$, ${\kappa}$ is the Fourier transform of probability distribution ${p}$, ${\phi(\delta)=\delta\cdot s-\kappa(s)}$, ${s}$ is the unique solution of ${\Delta s=...}$.

${s}$ is real analytic on the interior of the convex hull ${M}$ of the support of ${p}$, but blows up at the boundary. Here is how.

Theorem 2

$\displaystyle \begin{array}{rcl} \frac{e^{s\cdot v}}{\kappa(s)}\geq c\,dist(\delta,\partial M)^\eta. \end{array}$

2. Generalization to affine buildings

We have analogous estimates, with extra terms: distance to boundary of Weyl chamber.

Eralier results on on-diagonal behaviour (local limit theorem) by Sawyer, Gerl, Woess (trees), Lindbauer, Voit, Cartwright, Tolli and finally Parkinson. Off-diagonal behaviour by Lalley (trees), Anker, Schapira, Trojan for ${\tilde{A}_n}$, general case here.

2.1. Spherical analysis

There is a spherical inversion formula reconstructing the probability distribution in terms of its Gelfand Fourier transform (goes back to Macdonald 1971 in special cases). Macdonald polynomial arise as multiplicative finctions on the algebra of finitely supported invariant operators.