×

On an infinite-dimensional group over a finite field. (English. Russian original) Zbl 1089.22006

Funct. Anal. Appl. 32, No. 3, 147-152 (1998); translation from Funkts. Anal. Prilozh. 32, No. 3, 3-10 (1998).
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.

MSC:

22D10 Unitary representations of locally compact groups
19A31 \(K_0\) of group rings and orders
20B30 Symmetric groups
22D15 Group algebras of locally compact groups

Citations:

Zbl 0723.20005
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. M. Vershik and S. V. Kerov, ”Locally semisimple algebras. Combinatorial theory and theK 0-functor,” in: Contemporary Problems in Mathematics. Newest Results [in Russian], Vol. 26, 3–56, Itogi Nauki i Tekhniki, VINITI, Moscow, 1985; English transl. in J. Soviet Math.,38, 1701–1733 (1987).
[2] A. M. Vershik and S. V. Kerov, ”Characters and realizations of the infinite-dimensional Hecke algebra, and knot invariants,” Dokl. Akad. Nauk SSSR,301, No. 4, 777–780 (1988); English transl. in Soviet Math. Dokl.,39, 134–137 (1989).
[3] G. D. James, The Representation Theory of the Symmetric Groups, Lect. Notes in Math., Vol. 682, Springer-Verlag, Berlin-New York, 1978; Russian transl.: Mir, Moscow, 1982. · Zbl 0393.20009
[4] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979. · Zbl 0487.20007
[5] D. K. Faddeev, ”Complex representations of the general linear group over a finite field. Modules and representations,” Zap. Nauchn. Sem. LOMI,46, 64–88 (1974); English transl. in J. Soviet Math.,9, No. 3, 64–88 (1978).
[6] J. A. Green, ”The characters of the finite general linear groups,” Trans. Amer. Math. Soc.,80, 402–447 (1955). · Zbl 0068.25605
[7] H. L. Skudlarek, ”Die unzerlegbaren Charaktere einiger diskreter Gruppen,” Math. Ann.,223, 213–231 (1976). · Zbl 0325.43014
[8] E. Thoma, ”Die Einschränkung der Charactere vonGL(n, q) aufGL(n, q),” Math. Z.,119, 321–338 (1971). · Zbl 0205.32602
[9] S. Kerov and A. Vershik, ”The Grothendieck group of the infinite symmetric group and symmetric functions with the elements of theK 0-functor theory ofAF-algebras,” in: Adv. Stud. Contemp. Math., Gordon and Breach, Vol. 7, 1990, pp. 36–114. · Zbl 0723.20005
[10] A. V. Zelevinsky, Representations of finite classical groups, Lect. Notes in Math., Vol. 869, Springer-Verlag, 1981. · Zbl 0465.20009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.