Automorphic forms constructed from Whittaker vectors

AbstractLet G be a semi-simple Lie group of split rank 1 and Γ a discrete subgroup of G of cofinite volume. If P is a percuspidal parabolic of G with unipotent radical N and if χ is a non-trivial unitary character of N such that χ(Γ ∩ N) = 1 then a meromorphic family of functions M(v) on gG / G that satisfy all of the conditions in the definition of automorphic form except for the condition of moderate growth is constructed. It is shown that the principal part of M(v) at a pole v0 with Re v0 ⩾ 0 is square integrable and that “essentially” all square integrable automorphic forms with non-zero χ-Fourier coefficient can be constructed using the principal parts of the M-series. For square integrable automorphic forms that are fixed under a maximal compact subgroup the proviso “essentially” can be dropped. The Fourier coefficients of the M-series are computed. A specific term in the χ-Fourier coefficient is shown to determine the structure of the singularities of the M-series. This term is related to Selberg's “Kloosterman-Zeta function.” A functional equation for the M-series is derived. For the case of SL(2, R) the results are made more explicit and a complete family of square integrable automorphic forms is constructed. Also the paper introduces the conjecture that for semi-simple Lie groups of split rank > 1 and irreducible Γ the condition of moderate growth in the definition of automorphic form is redundant. Evidence for this conjecture is given for SO(n, 1) over a number field

System of Complex Brownian Motions Associated with the O'Connell Process

The O'Connell process is a softened version (a geometric lifting with a parameter $a>0$) of the noncolliding Brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length $a$. This process is not determinantal. Under a special entrance law, however, Borodin and Corwin gave a Fredholm determinant expression for the expectation of an observable, which is a softening of an indicator of a particle position. We rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and show, if the number of particles is $N$, the rank of the matrix of the Fredholm determinant is $N$. Then we give a representation for the quantity by using an $N$-particle system of complex Brownian motions (CBMs). The complex function, which gives the determinantal expression to the weight of CBM paths, is not entire, but in the combinatorial limit $a \to 0$ it becomes an entire function providing conformal martingales and the CBM representation for the noncolliding Brownian motion is recovered.Comment: v3: AMS_LaTeX, 25 pages, no figure, minor corrections made for publication in J. Stat. Phy

Higher dimensional geometries related to fuzzy odd-dimensional spheres

We study $SO(m)$ covariant Matrix realizations of $\sum_{i=1}^{m} X_i^2 = 1$ for even $m$ as candidate fuzzy odd spheres following hep-th/0101001. As for the fuzzy four sphere, these Matrix algebras contain more degrees of freedom than the sphere itself and the full set of variables has a geometrical description in terms of a higher dimensional coset. The fuzzy $S^{2k-1}$ is related to a higher dimensional coset ${SO(2k) \over U(1) \times U(k-1)}$. These cosets are bundles where base and fibre are hermitian symmetric spaces. The detailed form of the generators and relations for the Matrix algebras related to the fuzzy three-spheres suggests Matrix actions which admit the fuzzy spheres as solutions. These Matrix actions are compared with the BFSS, IKKT and BMN Matrix models as well as some others. The geometry and combinatorics of fuzzy odd spheres lead to some remarks on the transverse five-brane problem of Matrix theories and the exotic scaling of the entropy of 5-branes with the brane number.Comment: 32 pages, v2 : ref and acknowledgment adde

Resolvent and lattice points on symmetric spaces of strictly negative curvature

We study the asymptotics of the lattice point counting function N(x,y;r)=#{γ∈Γ:d(x,γy)} for a Riemannian symmetric space X obtained from a semisimple Lie group of real rank one and a discontinuous group Γ of motions in X, such that Γ∖X has finite volume. We show that as r→∞ , for each ε>0 . The constant 2ρ corresponds to the sum of the positive roots of the Lie group associated to X, and n = dimX. The sum in the main term runs over a system of orthonormal eigenfunctions φj∈L2(Γ∖X) of the Laplacian, such that the eigenvalues ρ2−ν2j are less than 4nρ2/(n+1)2