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Let $(\tau , V_{\tau })$ be a finite dimensional representation of a maximal compact subgroup K of a connected non-compact semisimple Lie group G, and let $\Gamma $ be a uniform torsion-free lattice in G. We obtain an infinitesimal version of the celebrated Matsushima–Murakami formula, which relates the dimension of the space of automorphic forms associated to $\tau $ and multiplicities of irreducible $\tau ^\vee $-spherical spectra in $L^2(\Gamma \backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $\tau $-isospectrality. Along with this, we prove that the almost equality of $\tau $-spherical spectra of two lattices assures the equality of their $\tau $-spherical spectra.
The starting point for string theory is the idea that the elementary constituents of the theory, which in quantum field theory are assumed to be point-like, are in fact one-dimensional objects, namely strings. As time evolves, a string sweeps out a Riemann surface whose topology governs the interactions that result from joining and splitting strings. The Feynman–Polyakov prescription for quantum mechanical string amplitudes amounts to summing over all topologies of the Riemann surface, for each topology integrating over the moduli of the Riemann surface, and for each value of the moduli solving a conformal field theory. Modular invariance plays a key role in the reduction of the integral over moduli to an integral over a single copy of moduli space and, in particular, is responsible for rendering string amplitudes well behaved at short distances. In this chapter, we present a highly condensed introduction to key ingredients of string theory and string amplitudes, relegating the important aspects of toroidal compactification and T-duality to Chapter 13 and a discussion of S-duality in Type IIB string theory to Chapter 14.
The purpose of this paper is to extend the explicit geometric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of the enveloping algebra.
The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.
Let $\Gamma $ be a rank-one arithmetic subgroup of a semisimple Lie group $G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of $G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma $-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Gamma $ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma $ follows from results of DeGeorge-Wallach and Delorme.
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