On an infinite-dimensional group over a finite field
The paper is devoted to the study of the structure, characters and unitary representations of a locally compact group called GLB which consists of all infinite invertible matrices with entries in a finite field \(\mathbb{F}_q\) which are ``almost triangular'', that is, have a finite number of nonzero elements under the diagonal. The group GLB contains the group GL\(_\infty(\mathbb{F}_q)\) of finite invertible matrices over the field \(\mathbb{F}_q\) as a denumerable dense subgroup. The results are presented without proofs. An important step in the study of the group and its representations is the construction of a dense subalgebra \(A\) of the group algebra \(L^1(\text{GLB},\mu)\) which is analogous to the Bruhat-Schwartz algebra known in the theory of \(p\)-adic linear groups. The algebra \(A\) is locally semisimple and can be represented as the union of an increasing sequence of subalgebras \(A_n\) which are finite-dimensional and isomorphic to the group algebras of the groups \(\text{GL}_n(\mathbb{F}_q)\). The method of describing the structure and the representations of this class of algebras was developed by the authors [in Representation of Lie groups and related topics, Adv. Stud. Contemp. Math. 7, 39--117, (New York 1990; Zbl 0723.20005)]. The authors prove that the algebra \(A\) is a denumerable direct union of two-sided ideals \(I(\phi)\) parametrized by families of Young diagrams. Every ideal \(I(\phi)\) is isomorphic to the algebra of finite functions on the infinite symmetric group \(G_\infty\). The relation between the \(K_0\)-functor of the algebra \(A\) and the Hopf algebra \(R=\bigoplus^\infty_{n=0}K_0(\text{GL} _n(\mathbb{F}_q))\) is also established.
scientific article; zbMATH DE number 1292495