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

Abstract

Finite traces and representations of the group of infinite matrices over a finite field. Vadim Gorin Massachusetts Institute of Technology, Cambridge, MA, USA and Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, RussiaSergei Kerov Anatoly Vershik St.Petersburg Department of V. A. Steklov Institute of Mathematics of Russian Academy of Sciences, Saint Petersburg, Russia AbstractThe article is devoted to the representation theory of locally compact infinitedimensional group 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 = ∞ analogue of general linear groups GL(n, q). It serves as an alternative to GL(∞, q), whose representation theory is poor. Our most important results are the description of semi-finite unipotent traces (characters) of the group GLB via certain probability measures on the Borel subgroup B and the construction of the corresponding von Neumann factor representations of type II ∞ .As a main tool we use the subalgebra A(GLB) of smooth functions in the group algebra L 1 (GLB). This subalgebra is an inductive limit of the finitedimensional group algebras 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 H q (∞) 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(∞) which Email addresses: vadicgor@gmail.com (Vadim Gorin), avershik@gmail.com (Anatoly Vershik) MSC 2010: 22D10, 22D25, 20G40. Keywords: Infinite-dimensional group; finite field; factor representation; Hecke algebra Preprint submitted to ElsevierDecember 19, 2013 had been intensively studied in numerous previous papers.To the memory of Andrei Zelevinsky. Contents Historical prefaceMy joint work with S. Kerov on the asymptotic representation theory of the matrix groups GL(n, q) over finite field as the rank n grows to infinity, 2 was started at the beginning of 80s as a continuation of our papers devoted to analogous problems for symmetric groups of growing ranks at the end of 70-th. It is a part of what I called "the asymptotic representation theory".The "trivial" embedding GL(n, q) ֒→ GL(n + 1, q) does not lead to an interesting or useful theory. However, another "true" (i.e. parabolic) embedding of the group algebras of GL(n, q) was well-known starting from the very first papers on the representation theory of GL(n, q) (see [Gr], [Zel], [F], etc). It was used by A. Zelevinsky and us (see [V82]) to define a natural limit object (i.e. inductive limit) which is the group (named GLB) of infinite matrices over finite field with finitely many non-zero elements below the main diagon...

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