Modified series of integrable discrete equations on a quadratic lattice with a nonstandard symmetry structure

Theoretical and Mathematical Physics - We recently constructed a series of integrable discrete autonomous equations on a quadratic lattice with a nonstandard structure of higher symmetries. Here,...     

On the set of discrete subgroups of bounded covolume in a semisimple group

In this noteG is a locally compact group which is the product of finitely many groups G_s(k_s)(s∈S), where k_s is a local field of characteristic zero and G_s an absolutely almost simplek _s-group, ofk _s-rank ≥1. We assume that the sum of the r_s is ≥2 and fix a Haar measure onG. Then, given a constantc > 0, it is shown that, up to conjugacy,G contains only finitely many irreducible discrete subgroupsL of covolume ≥c (4.2). This generalizes a theorem of H C Wang for real groups. His argument extends to the present case, once it is shown thatL is finitely presented (2.4) and locally rigid (3.2).     

On the integrable discrete versions of the Leznov lattice: Determinant solutions and pfaffianization

In this paper, we present the Casorati form of the N-soliton solution for an integrable fully-discrete version and two integrable semi-discrete versions of the Leznov lattice, which arise from the integrable discretization of the two-dimensional Leznov lattice. By using the pfaffianization procedure of Hirota and Ohta, a new integrable coupled system is generated from the semi-discrete version of the Leznov lattice in the y-direction.     

On the Brauer group of a semisimple algebraic group

Let k be a field with algebraic closure , G a semisimple algebraic k-group, and a maximal torus with character group X(T). Denote Λ the abstract weight lattice of the roots system of G, and by and the n-torsion subgroup of the Brauer group of k and G, respectively. We prove that if chark does not divide n and n is prime to the order of Λ/X(T) then the natural homomorphism is an isomorphism.     

The joint distribution of sojourn times for a customer traversing an overtake-free series of queues: The discrete time case

We investigate a cycle of Bernoulli servers in discrete time. We show that the joint distribution of a customer’s sojourn times at the nodes during a round trip is of product form and can explicitly be written down. The results are used to prove independence of sojourn times in an open series of Bernoulli servers. This revised version was published online in June 2006 with corrections to the Cover Date.     

A series of unitary irreducible representations induced from a symmetric subgroup of a semisimple Lie group

LetG/H be a semisimple symmetric space. Generalizing results of Flensted-Jensen we give a sufficient condition for the existence of irreducible closed invariant subspaces of the unitary representations ofG induced from unitary finite dimensional representations ofH. This provides a method of constructing unitary irreducible representations ofG, and we show by examples that for some irreducible admissible representations ofG, this method exhibits not previously known unitarity.This work was supported by the Danish Natural Science Research Council.     

Theta lifting of holomorphic discrete series: The case of

Let be a reductive dual pair in the stable range. We investigate theta lifts to of unitary characters and holomorphic discrete series representations of , in relation to the geometry of nilpotent orbits. We give explicit formulas for their -type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme -type, we show that the structure of the resulting representations of is almost identical to the -module structure of the regular function rings on the closure of the associated nilpotent -orbits in , where is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.

     

The product of lattice covolume and discrete series formal dimension: p-adic GL(2)

Let $F$ be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in $PGL(2,F)$ and the formal dimension of a discrete series representation of $GL(2,F)$. The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.     

Composition series of generalized principal series; the case of strongly positive discrete series

In this paper we determine the composition series of the generalized principal seriesδ⋊σ assuming thatσ is strongly positive discrete series.     

A series of a discrete group and its application to the Dirichlet series associated with automorphic forms

Starting from the Poincaré-Selberg series, another series is defined for a discrete group; the relationship is established between this series and the Dirichlet series for which the coefficients are products of the Fourier coefficients of the automorphic forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 3–7, 1976.     

Spherical functions on a real semisimple Lie group. A method of reduction to the complex case

The spherical functions on a real semisimple Lie group (w.r.t. a maximal compact subgroup) are characterized as joint eigenfunctions of certain differential operators on the corresponding complex group. Using this, several results concerning the spherical Fourier transform on the real group are reduced to the corresponding results for the complex group.When the group in question is a normal real form, this leads to new and simpler proofs of such results as the Plancherel formula, the Paley-Wiener theorem and the characterization of the image under the spherical Fourier transform of the L¹- and L²-Schwartz spaces. In these proofs neither any knowledge of Harish-Chandras c-function nor the series expansion for the spherical function are used.For the proof of the main result some analysis of independent interest on pseudo-Riemannian symmetric spaces is developed. Such as a generalized Cartan decomposition and a method of analytic continuation between two “dual” pseudo-Riemannian symmetric spaces.