We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichmuller space. We then use this formula to determine the asymptotic behavior as $text{Re} (s) to infty$ of the second variation. As a c
onsequence, for $m in mathbb{N}$, we obtain the complete expansion in $m$ of the curvature of the vector bundle $H^0(X_t, mathcal K_t)to tin mathcal T$ of holomorphic m-differentials over the Teichmuller space $mathcal T$, for $m$ large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $O(m^2 e^{-l_0 m}),$ where $l_0$ is the length of the shortest closed hyperbolic geodesic.
Let $G_{n,r}(bbK)$ be the Grassmannian manifold of $k$-dimensional $bbK$-subspaces in $bbK^n$ where $bbK=mathbb R, mathbb C, mathbb H$ is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, $mathcal
R_{r^prime, r}$, $mathcal C_{r^prime, r}$ and $mathcal S_{r^prime, r}$, from the $L^2$ space $L^2(G_{n,r}(bbK))$ to the space $L^2(G_{n,r^prime}(bbK))$, for $r, r^prime le n-1$. The $L^2$ spaces are decomposed into irreducible representations of $G$ with multiplicity free. We compute the spectral symbols of the transforms under the decomposition. For that purpose we prove two Bernstein-Sato type formulas on general root systems of type BC for the sine and cosine type functions on the compact torus $mathbb R^r/{2pi Q^vee}$ generalizing our recent results for the hyperbolic sine and cosine functions on the non-compact space $mathbb R^r$. We find then also a characterization of the images of the transforms. Our results generalize those of Alesker-Bernstein and Grinberg. We prove further that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we give the unitary structure of the Steins complementary series in the compact picture.
