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

The article is devoted to the representation theory of locally compact infinite-dimensional group $\mathbb{GLB}$ of almost upper-triangular infinite matrices over the finite field with $q$ elements. This group was defined by S.K., A.V., and Andrei Zelevinsky in 1982 as an adequate $n=\infty$ analogue of general 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 unipotent traces (characters) of the group $\mathbb{ GLB}$ via certain probability measures on the Borel subgroup $\mathbb{B}$ and the construction of the corresponding von Neumann factor representations of type $II_\infty$. As a main tool we use the subalgebra $\mathcal A(\mathbb{ GLB})$ of smooth functions in the group algebra $L_1(\mathbb{GLB})$. This subalgebra is an inductive limit of the finite--dimensional group algebras ${\mathbb C}(\mathbb{GL}(n,q))$ under parabolic embeddings. As in other examples of the asymptotic representation theory we discover remarkable properties of the infinite case which does not take place for finite groups, like multiplicativity of indecomposable characters or connections to probabilistic concepts. The infinite dimensional Iwahori-Hecke algebra $\mathcal H_q(\infty)$ plays a special role in our considerations and allows to understand the deep analogy of the developed theory with the representation theory of infinite symmetric group $S(\infty)$ which had been intensively studied in numerous previous papers.