We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles $F_{p}$ . For $p\in \mathbf{N}$ , the flat vector bundle $F_{p}$ is the direct image of $L^{p}$ , where $L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

Let $F$ be a non-Archimedean local field, and let $G^{\sharp }$ be the group of $F$-rational points of an inner form of $\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of $G^{\sharp }$, via restriction from an inner form $G$ of $\text{GL}_{n}(F)$.

For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth $G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of $G^{\sharp }$, and the idempotent is derived from a type for $G$. We show that the Hecke algebras for Bernstein components of $G^{\sharp }$ are similar to affine Hecke algebras of type $A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounded non-unitarisable groups on Hilbert space. In particular, we provide structural information about bounded groups with an essentially unique invariant complemented subspace. This is subsequently combined with rigidity results for the unitary representation of $\text{Aut}(T)$ on $\ell _{2}(T)$, where $T$ is the countably infinite regular tree, to describe the possible bounded subgroups of $\text{GL}({\mathcal{H}})$ extending a well-known non-unitarisable representation of $\mathbb{F}_{\infty }$.

As a related result, we also show that a transitive norm on a separable Banach space must be strictly convex.