Torus period coefficients on PGL(2) and Dirichlet series

We fix a quadratic number field E, and the corresponding torus in $\mathrm{PGL}(2)$. We consider twisted (by a Hecke character of the field E) torus periods of automorphic functions. We prove meromorphic continuation for a Dirichlet series generated by these twisted periods.     

The covering radius of PGL2(q)

The covering radius of a subset $C$ of the symmetric group ${\mathrm{S}}_n$ is the maximal Hamming distance of an element of ${\mathrm{S}}_n$ from $C$. This note determines the covering radii of the finite $2$-dimensional projective general linear groups. It turns out that the covering radius of ${\mathrm{PGL}}_2\left(q\right)$ is $q?2$ if $q$ is even, and is $q?3$ if $q$ is odd.     

Tetravalent Edge-transitive Cayley Graphs of PGL （2, p）

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.     

Approximating Fourier transformation of orbital integrals

In this paper, we are concerned with orbital integrals on a class of real reductive Lie groups with non-compact Iwasawa K-component. The class contains all connected semisimple Lie groups with infinite center. We establish that any given orbital integral over general orbits with compactly supported continuous functions for a group G in is convergent. Moreover, it is essentially the limit of corresponding orbital integrals for its quotient groups in Harish-Chandra's class. Thus the study of orbital integrals for groups in class reduces to those of Harish-Chandra's class. The abstract theory for this limiting technique is developed in the general context of locally compact groups and linear functionals arising from orbital integrals. We point out that the abstract theory can be modified easily to include weighted orbital integrals as well. As an application of this limiting technique, we deduce the explicit Plancherel formula for any group in class .     

The Uniformly 3-Homogeneous Subsets of PGL(2, q)

We use the character-table of PGL(2, q) to determine the subsets of that group acting uniformly 3-homogeneously on the projective line.     

Chern-simons theory in the SO(5)/U(2) harmonic superspace

We consider the superspace of D=3, N=5 supersymmetry using SO(5)/U(2) harmonic coordinates. Three analytic N=5 gauge superfields depend on three vector and six harmonic bosonic coordinates and also on six Grassmann coordinates. Decomposing these superfields in Grassmann and harmonic coordinates yields infinite-dimensional supermultiplets including a three-dimensional gauge Chern-Simons field and auxiliary bosonic and fermionic fields carrying SO(5) vector indices. The superfield action of this theory is invariant with respect to the D=3, N=6 conformal supersymmetry realized on N=5 superfields. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 217–234, November, 2008.     

On the sharply 1-transitive subsets of PGL(2,pm)

If p is an odd prime and R is a sharply 1-transitive subset of PGL(2,p^m) which contains the identity but is not a group, then the subgroup generated by R is either PSL(2,p^m) or PGL(2,p^m).work done within the activity of G.N.S.A.G.A. and supported by the Italian Ministry of Public EducationDedicated to Professor Helmut Karzel on his 60^th birthday     

Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

We prove that if ${\mu _{a}\,{=}\,m_{K}*\delta _{a}*m_{K}}$ is the K-bi-invariant measure supported on the double coset ${KaK\subseteq SU(n)}$ , for K = SO(n), then ${\mu _{a}^{k}}$ is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μ _a , supported on the minimal dimension double coset has the property that ${\mu _{a}^{n-1}}$ is singular to the Haar measure.