On finite groups isospectral to simple linear and unitary groups

Let L be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic p. We deal with the class of finite groups isospectral to L. It is known that a group of this class has a unique nonabelian composition factor. We prove that if L ≠ U ₄(2), U ₅(2) then this factor is isomorphic to either L or a group of Lie type over a field of characteristic different from p.     

A reciprocity theorem for unitary representations of Lie groups

LetG be a Lie group,H a closed subgroup,L a unitary representation ofH andU ^L the corresponding induced representation onG. The main result of this paper, extending Ol’ŝanskii’s version of the Frobenius reciprocity theorem, expresses the intertwining number ofU ^L and an irreducible unitary representationV ofG in terms ofL and the restriction ofV _∞ toH.     

The local theta correspondence for unramified unitary groups

We study the local theta correspondences for dual reductive pairs consisting of quasi-split unitary groups defined over a non-archimedean local field. We construct Howe’s correspondence between the set of spherical representations of the one group and that of the other group by using the Whittaker model.     

Spectra of finite linear and unitary groups

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups. Supported by RFBR (grant Nos. 08-01-00322 and 06-01-39001) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 157–173, March–April, 2008.     

On the Grassmann modules for the unitary groups

Let V be 2n-dimensional vector space over a field 𝕂 equipped with a nondegenerate skew-ψ-Hermitian form f of Witt index n ≥ 1, let 𝕂₀ ? 𝕂 be the fix field of ψ and let G denote the group of isometries of (V, f). For every k ∈ {1, …, 2n}, there exist natural representations of the groups G ? U(2n, 𝕂/𝕂₀) and H = G ∩ SL(V) ? SU(2n, 𝕂/𝕂₀) on the k-th exterior power of V. With the aid of linear algebra, we prove some properties of these representations. We also discuss some applications to projective embeddings and hyperplanes of Hermitian dual polar spaces.     

On the Hahn-Banach theorem for groups

Let be the family of all groups such that under each subadditive functional there exists an additive functional. We show that the class is between the class of all amenable groups and the family of all groups for which the Hyers stability theorem for homomorphisms holds true. Next, we generalize the classical Hahn-Banach theorem to the class . Received: 6 May 2005     

On wandering vector multipliers for unitary groups

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system acting on a separable Hilbert space into itself. It is proved that the wandering vector multipliers for a unitary group form a group, which gives a positive answer for a problem of Han and Larson. Furthermore, non-abelian unitary groups of order 6 are considered. We prove that the wandering vector multipliers of such a unitary group can not generate . This negatively answers another of their problems.

     

Dade's invariant conjecture for general linear and unitary groups in non-defining characteristics

This paper is part of a program to study the conjecture of E. C. Dade on counting characters in blocks for several finite groups.

The invariant conjecture of Dade is proved for general linear and unitary groups when the characteristic of the modular representation is distinct from the defining characteristic of the groups.

     

Elementary equivalence of unitary linear groups over rings

We find necessary and sufficient conditions for two unitary linear groups over rings with involution with forms of hyperbolic rank at least 2 to be elementarily equivalent.     

A 2-adic automorphy lifting theorem for unitary groups over CM fields

We prove a ‘minimal’ type automorphy lifting theorem for 2-adic Galois representations of unitary type, over imaginary CM fields. We use this to improve an automorphy lifting theorem of Kisin for \({{\mathrm{GL}}}_2\).     

On the Siegel–Weil formula for unitary groups

Following S. S. Kudla and S. Rallis, we extend the Siegel–Weil formula for unitary groups, which relates a value of a Siegel Eisenstein series to the convergent integral of a theta function.     

On the Schur theorem for n-ary groups

We prove n-ary analogs of the well-known Schur theorem on the finiteness of a commutator subgroup of a group whose center is of finite index. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 730–741, June, 2006.     

Topological flatness of local models for ramified unitary groups. I. The odd dimensional case

Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which give étale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Qp are ramified, quasi-split GUn, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.