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The article is devoted to the representation theory of locally compactinfinite-dimensional group $\mathbb{GLB}$ of almost upper-triangular infinitematrices over the finite field with $q$ elements. This group was defined byS.K., A.V., and Andrei Zelevinsky in 1982 as an adequate $n=\infty$ analogue ofgeneral linear groups $\mathbb{GL}(n,q)$. It serves as an alternative to$\mathbb{GL}(\infty,q)$, whose representation theory is poor.  Our most important results are the description of semi-finite unipotenttraces (characters) of the group $\mathbb{ GLB}$ via certain probabilitymeasures on the Borel subgroup $\mathbb{B}$ and the construction of thecorresponding von Neumann factor representations of type $II_\infty$.  As a main tool we use the subalgebra $\mathcal A(\mathbb{ GLB})$ of smoothfunctions in the group algebra $L_1(\mathbb{GLB})$. This subalgebra is aninductive limit of the finite--dimensional group algebras ${\mathbbC}(\mathbb{GL}(n,q))$ under parabolic embeddings.  As in other examples of the asymptotic representation theory we discoverremarkable properties of the infinite case which does not take place for finitegroups, like multiplicativity of indecomposable characters or connections toprobabilistic concepts.  The infinite dimensional Iwahori-Hecke algebra $\mathcal H_q(\infty)$ plays aspecial role in our considerations and allows to understand the deep analogy ofthe developed theory with the representation theory of infinite symmetric group$S(\infty)$ which had been intensively studied in numerous previous papers.

Finite traces and representations of the group of infinite matrices over a finite field