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 GLBGLB 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=∞n=∞ analogue of general linear groups GL(n,q)GL(n,q). It serves as an alternative to GL(∞,q)GL(∞,q), whose representation theory is poor.Our most important results are the description of semifinite unipotent traces (characters) of the group GLBGLB via certain probability measures on the Borel subgroup BB and the construction of the corresponding von Neumann factor representations of type II∞II∞.As a main tool we use the subalgebra A(GLB)A(GLB) of smooth functions in the group algebra L1(GLB)L1(GLB). This subalgebra is an inductive limit of the finite-dimensional group algebras C(GL(n,q))C(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 Hq(∞)Hq(∞) 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(∞)S(∞) which had been intensively studied in numerous previous papers.