The unitary group over the integers of a quaternion algebra

Let L be a lattice over the integers of a quaternion algebra with center K which is a $\text{B}$-adic field. Then the unitary group U(L) equals its own commutator subgroup $\text{Ω(L)}$ and is generated by the unitary transvections and quasitransvections contained in it. Let g be a tableau, U(g), U⁺(g), $\text{Ω(g)}$, T(g) be the corresponding congruence subgroups of order g. Then $\text{U(g)}\text{U}{}^{\mathrm{+}}\text{(g)}\text{? X}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{\tau }}\text{Z}{}_2$, and $\text{Ω(g) = T(g)}$ (the subgroup generated by the unitary transvections and quasitransvections with order ≤ g). Let G be a subgroup of U(L) with o(G) = g, then G is normal in U(L) if and only if U(g) ? G ? T(g).     

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.     

Normal structure of the unitary group over a local ring

A new approach is used to describe the normal subgroups of unitary groups over the ring of integers of an unramified quadratic extension of a local field of characteristic 2. This method is based on the author's results concerning subgroups of unitary groups normalized by diagonal unitary matrices. By means of this method it has been possible to remove restrictions imposed by other authors in a number of papers pertaining to the same question.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 150–163, 1982.The results in this paper were obtained by the author while he was a post-graduate student under the direction of Z. I. Borevich. The author would like to express his sincere gratitude to Professor Borevich.     

On orthogonal groups over the integers of a non-dyadic local field

Let L be a lattice in a quadratic space over a non-dyadic local field. We shall answer the question: What are the lattices whose unit groups coincide with that of L? If the residue class field has more than three elements the question is easy. In this case such a lattice must be $\text{a}$L or $\text{a}$L^# with a fractional ideal $\text{a}$ and the dual lattice L^# by Satz 2 of A. Kallmann, M. Kneser, and U. Stuhler (J. Reine Angew. Math.258 (1978), 51–54) or Theorem 5.2 of C. R. Riehm (Amer. J. Math.89 (1967), 549–577). But it is not easy in the case of the residue class field of three elements.     

The reflection length of a transformation in the unitary group over a finite field

Let U be the unitary group over a finite field K where [K]≠3 and char K≠2. Every transformation π in U with detπ = ±1 is a product of reflections. We determine the length t of every such transformation π i.e. for each π we find reflections σ_i in U such that π=σ₁ σ_t and so that no factorization of π into fewer than t reflections exists.     

On Hardy-Bessel potential spaces over the ring of integers in a local field

Штрихарц [3] дал характе ристику пространствL ^p (R ⁿ ) бесселевых потенциа лов порядка α функций из п ространстваL ^p (R ⁿ ) с пом ощьюL ^p -норм функционалов $$D_\alpha f(x) = \left( {\smallint _0^\infty \left( {\smallint _{\rm B} |f(x + rt) - f(x)|dt} \right)^2 r^{ - 2\alpha - 1} dr} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$ для 0<α<1, гдеB обозначает ед иничный шар. Целью нас тоящей статьи является изучение пр остранств потенциал а Харди-БесселяF _α ^p (P ⁰) (0<p<1, α>1/р?1) в терминах функц ионаловS _r ^α f(x) (1τ≤2), которые в случаеR ⁿ } соответствуютD _α f (x), гдеР ⁰ - кольцо целых в локальном поле. Получ ено приложение, относяще еся к регулярности бе сселевых потенциалов.     

The unitary dual of GL(n) over an archimedean field

Inventiones mathematicae -     

Abelian p-extensions of an absolutely unramified field

In this paper the characteristic of the Abelian p-extensions of a field is given using valuations from its cyclotomic extensions. On the basis of this the character group of the Galois groups of the maximal Abelian p-extension of an absolutely unramified complete field is calculated. The results are applied to the case of a two-dimensional local field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 80–90, 1991.     

Isomorphisms of the unitary group over a ring

We describe isomorphisms of unitary groups over associative rings with 1/2 that contain a compact system of idempotents. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 55–70, 2006.