Volume 19, Issue 5

We provide a new presentation for simply connected Kac-Moody groups of 2-spherical type and for their universal central extensions. Under mild local restrictions, these results extend to the more general class of groups of Kac-Moody type (i.e. groups endowed with a root datum).

In 1991, Bendersky and Davis used the BP -based unstable Novikov spectral sequence to study the 2-primary v 1 -periodic homotopy groups of SU(n) . Here we use a K -theoretic approach to add more detail to those results. In particular, whereas only the order of the groups was determined in the 1991 paper, here we determine the number of summands in these groups and much information about the orders of those summands. In addition, we give explicit conditions for certain differentials and extensions in a spectral sequence, which affect the homotopy groups. Finally, we give complete results for for n ≤ 13.

Let G be a real semisimple Lie group. We equip invariant presymplectic forms ω to some fibration X over the flag domain of G . By applying geometric quantization to ( X , ω), we obtain a unitary G -representation ℋ whose subrepresentations are infinitesimally equivalent to the Zuckerman modules. The occurence of the Zuckerman modules in ℋ are controlled by the image of the moment map of ω. This leads to our notion of Zuckerman model.

We define a weighted multiplicity function for closed geodesics of given length on a finite area Riemann surface. These weighted multiplicities appear naturally in the Selberg trace formula, and in particular their mean square plays an important role in the study of statistics of the eigenvalues of the Laplacian on the surface. In the case of the modular domain, E. Bogomolny, F. Leyvraz and C. Schmit gave a formula for the mean square, which was rigorously proved by M. Peter. In this paper we calculate the mean square of weighted multiplicities for some surfaces associated to congruence subgroups of the unit group of a rational quaternion algebra, in particular for congruence subgroups of the modular group. Remarkably, the result turns out to be a rational multiple of the mean square for the modular domain.

We deal here with the conjecture of Quantum Unique Ergodicity in a specific case. Rudnick and Sarnak showed in [Rudnick Z. and Sarnak P.: The Behaviour of Eigenstates of Arithmetic Hyperbolic Manifolds. Communications in Mathematical Physics 161 (1994), 195–213] that there is no strong scarring on closed geodesics for compact arithmetic congruence surfaces derived from quaternion division algebras (see theorem 1.1). We extend this theorem to a class ( K 2 S ) of Riemannian manifolds derived from quaternion division algebras, and show that there is no strong scarring on closed geodesics or on Γ R -closed imbedded totally geodesic surfaces in these manifolds (see theorem 4.1).

In this note we investigate a problem formulated by Pleijel in 1955. It asks for the cone over a convex plane domain 𝒦 having minimal surface among all cones over 𝒦 with the same given height h . For cones based on reflection symmetric polygonal 𝒦 we analyze the behaviour, as h → 0 and as h → ∞, of the position of the apex for the minimizing cone and characterize the coordinate of the limit points by necessary conditions. Furthermore, the question whether there are convex domains such that the minimal cone does not change with h is discussed. The results about the location of the optimal point in the limits h → ∞ and h → 0 presented here give the more or less explicit algebraic coordinates of the optimal apex. A complete (but implicit) characterization of this point was given by B. Cheng in [Cheng B. N.: On a problem of A. Pleijel. Geom. Dedicata 44 (1992), 139–145].