Let $L$ be the unique even self-dual lattice of signature $(25,1)$. The automorphism group $\operatorname{ Aut}(L)$ acts on the hyperbolic space $\mathcal{ H}^{ 25}$. We study a Poincar\'e series $E(z,s)$ defined for $z$ in $\mathcal{ H}^{ 25}$, convergent for $\operatorname{ Re}(s) > 25$, invariant under $\operatorname{ Aut}(L)$ and having singularities along the mirrors of the reflection group of $L$. We compute the Fourier expansion of $E(z,s)$ at a "Leech cusp" and prove that it can be meromorphically continued to $\operatorname{ Re}(s) > 25/2$. Analytic continuation of Kloosterman sum zeta functions imply that the individual Fourier coefficients of $E(z,s)$ have meromorphic continuation to the whole $s$-plane.
Categories: math.RT math.NT