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The infinite-dimensional Iwahori--Hecke algebras $\mathcal{H}_\infty(q)$ aredirect limits of the usual finite-dimensional Iwahori--Hecke algebras. Theyarise in a natural way as convolution algebras of bi-invariant functions ongroups $\mathrm{GLB}(\mathbb{F}_q)$ of infinite-dimensional matrices overfinite-fields having only finite number of non-zero matrix elements under thediagonal. In 1988 Vershik and Kerov classified all indecomposable positivetraces on $\mathcal{H}_\infty(q)$. Any such trace generates a representation ofthe double $\mathcal{H}_\infty(q)\otimes \mathcal{H}_\infty(q)$ and of thedouble $\mathrm{GLB}(\mathbb{F}_q)\times \mathrm{GLB}(\mathbb{F}_q)$. Wepresent constructions of such representations; the traces are somedistinguished matrix elements. We also obtain some (simple) general statementson relations between unitary representations of groups and representations ofconvolution algebras of measures bi-invariant with respect to compactsubgroups.

Some remarks on traces on the infinite-dimensional Iwahori--Hecke algebra