  The infinite-dimensional Iwahori--Hecke algebras $\mathcal{H}_\infty(q)$ are
direct limits of the usual finite-dimensional Iwahori--Hecke algebras. They
arise in a natural way as convolution algebras of bi-invariant functions on
groups $\mathrm{GLB}(\mathbb{F}_q)$ of infinite-dimensional matrices over
finite-fields having only finite number of non-zero matrix elements under the
diagonal. In 1988 Vershik and Kerov classified all indecomposable positive
traces on $\mathcal{H}_\infty(q)$. Any such trace generates a representation of
the double $\mathcal{H}_\infty(q)\otimes \mathcal{H}_\infty(q)$ and of the
double $\mathrm{GLB}(\mathbb{F}_q)\times \mathrm{GLB}(\mathbb{F}_q)$. We
present constructions of such representations; the traces are some
distinguished matrix elements. We also obtain some (simple) general statements
on relations between unitary representations of groups and representations of
convolution algebras of measures bi-invariant with respect to compact
subgroups.
