The Ismagilov conjecture over a finite field ${\mathbb F}_p$

2016

We construct the so-called quasiregular representations of the group B_0^ N( F_p) of infinite upper triangular matrices with coefficients in a finite field and give the criteria of theirs irreducibility in terms of the initial measure. These representations are particular case of the Koopman representation hence, we find new conditions of its irreducibility. Since the field F_p is compact some new operators in the commutant emerges. Therefore, the Ismagilov conjecture in the case of the finite field should be corrected.

The notion of quasiregular representation is well known for a locallycompact groups. We construct here an analog of the quasiregular representation for infinite-dimensional group G = B N 0 of finite uppertriangular matrices and give a criterion of the irreducibility and the equivalence of constructed representations. Conjecture 2 (Ismagilov, 1985) The right regular representation T R,µ : G → U (L 2 (G, µ) is irreducible if and only if 1) µ Ls ⊥ µ ∀s ∈ G\{e}, 2) the measure µ is G-right ergodic. Remark 1. In the case of the right regular representation the group α(G) = R(G) ⊂ Aut(G) contains obviously the group L(G), the image of the group G with respect to the left action L : G → Aut(G), L s (x) = sx.

The notion of quasiregular representation is well known for a locallycompact groups. We construct here an analog of the quasiregular representation for infinite-dimensional group G = B N 0 of finite uppertriangular matrices and give a criterion of the irreducibility and the equivalence of constructed representations. Conjecture 2 (Ismagilov, 1985) The right regular representation T R,µ : G → U (L 2 (G, µ) is irreducible if and only if 1) µ Ls ⊥ µ ∀s ∈ G\{e}, 2) the measure µ is G-right ergodic. Remark 1. In the case of the right regular representation the group α(G) = R(G) ⊂ Aut(G) contains obviously the group L(G), the image of the group G with respect to the left action L : G → Aut(G), L s (x) = sx.

Advances in Mathematics, 2014

The 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

Journal of Mathematical Sciences, 2007

Functional Analysis and Its Applications, 1998

Introduction. The asymptotic theory of representations studies the behavior of representations of classical groups of high degrees and their infinite-dimensional analogs. One of the key examples is the series of linear groups GL,~(Fq) over a finite field k = Fq, and the inductive limit GLc~(Fq) of these groups, the group of infinite invertible matrices over the field k that differ from the identity matrix by only finitely many nonzero entries. In this note we introduce a new locally compact group GLB and study its structure, characters, and continuous unitary representations. The group contains GL~(k) as a countable dense subgroup. The importance of the group GLB relies on its close connections with the theory of parabolic induction, which is the core of the representation theory of the groups GL,(k). The description of irreducible representations of the groups GL,(k) was first obtained by Green [6]. The main tool of his theory, which was developed later in the papers [5, 10], is the operation of parabolic induction of representations. Just this operation determines the inclusions of group algebras C(GL,,(k)) that lead to the group algebra A of the group GLB to be studied in this paper. The algebra A can be represented as an inductive limit of the algebras C(GLn(k)), though the inclusions of these group algebras are determined by averaging operators over the cosets of a subgroup, rather than generated by group inclusions. The group GLB has been defined by the authors of the present paper together with A. V. Zelevinsky (see the supplement by the editor of the Russian translation of [3]) in 1981 in the course of discussing the relationships of his paper [10] with the theory of representations and characters of locally finite groups (similar to GLoo(k) and to the infinite symmetric group 6~). Our approach relies on the general theory of inductive limits of finlte-dimensional semi.qimple algebras. Note that, for q-1, we obtain the representation theory of the infinite symmetric group; this theory was studied in a n11rnber of papers by the authors (see, e.g., the references in [1]). The remarkable fact that our infinlte-dimensional nondiscrete group GLB is locally compact allows one to use the full power of classical methods of representation theory and, in particular, the method of induced representations. We intend to study the unipotent characters of the group GLB, and give realizations of factor representations of the groups GLB and GLoo (k) in detail in forthcoming papers. 1. The group GLB. The objects defined below depend on the cardinality q of the field but, for convenience of the notation, we do not always show this dependence explicitly. Consider the k-linear space V-k~ of all finite vectors with coordinates in a finite field k = Fq. Fix a basis of vectors {ei, i = 1, 2,... } in the space V and denote by Vn the subspace generated by the first n vectors. Definition. The group GLB consists of k-linear transformations of the space V that preserve all subspaces Vn except for finitely many of them. Let us reformulate the definition in terms of matrices of operators g E GLB with respect to the distinguished basis. An infinite matrix g = (g~j), i, j = 1,2,..., is said to be almost triangular if the number of its nonzero subdiagonal elements a~i r 0, i > j, is finite. The group GLB consists precisely of operators with almost triangular matrices.

We define the analog of the regular representations of the group of finite upper-triangular matrices of infinite order corresponding to quasi-invariant product measures on the group of all upper-triangular matrices and give the criterium of irreducibility of constructed representations under some technical conditions. In [1], [2] the criterium was proved for an arbitrary centered Gaussian product-measures on the same group and in [3] for groups of the interval and circle diffeomorphisms.

Journal of Functional Analysis

Our aim is to find the irreducibility criteria for the Koopman representation, when the group acts on some space with a measure (Conjecture 1.5). Some general necessary conditions of the irreducibility of this representation are established. In the particular case of the group GL 0 (2∞, R) = lim − →n GL(2n − 1, R), the inductive limit of the general linear groups we prove that these conditions are also the necessary ones. The corresponding measure is infinite tensor products of one-dimensional arbitrary Gaussian noncentered measures. The corresponding G-space X m is a subspace of the space Mat(2∞, R) of infinite in both directions real matrices. In fact, X m is a collection of m infinite in both directions rows. This result was announced in [20]. We give the proof only for m ≤ 2. The general case will be studied later.

Functional Analysis and Its Applications, 1991

Journal of Mathematical Sciences

Let G be an infinite-dimensional real classical group containing the complete unitary group (or complete orthogonal group) as a subgroup. Then G generates a category of double cosets (train) and any unitary representation of G can be canonically extended to the train. We prove a technical lemma about the complete group GL of infinite p-adic matrices with integer coefficients, this lemma implies that the phenomenon of automatic extension of unitary representations to trains is valid for infinite-dimensional p-adic groups.