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Register a new userGeneralized zeta integrals on certain real prehomogeneous vector spaces
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Wen-Wei Li
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Let $X$ be a real prehomogeneous vector space under a reductive group $G$, such that $X$ is an absolutely spherical $G$-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz-Bruhat functions on $X$ against generalized matrix coefficients of admissible representations of $G(mathbb{R})$, twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation as well as a functional equation in terms of abstract $gamma$-factors. This subsumes the Archimedean zeta integrals of Godement-Jacquet, those of Sato-Shintani (in the spherical case), and the previous works of Bopp-Rubenthaler. The proof of functional equations is based on Knops results on Capelli operators.
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Let $G$ be a reductive group over a local field $F$ of characteristic zero, Archimedean or not. Let $X$ be a $G$-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on $X$, considered as a representation of $G$. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map, and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis. Applying our theorems to symmetric pairs $(G,H)$ we show that there exists an infinite-dimensional $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is non-compact. For quasi-split $G$ we also extend to the Archimedean case the theorem of Prasad stating that there exists a generic $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is quasi-split. In the non-Archimedean case our result also gives bounds on the wave-front sets of distinguished representations.
We aim to introduce the generalized multiindex Bessel function $J_{left( beta _{j}right) _{m},kappa ,b}^{left( alpha _{j}right)_{m},gamma ,c}left[ zright] $ and to present some formulas of the Riemann-Liouville fractional integration and differentiation operators. Further, we also derive certain integral formulas involving the newly defined generalized multiindex Bessel function $J_{left( beta _{j}right) _{m},kappa ,b}^{left( alpha _{j}right)_{m},gamma ,c}left[ zright] $. We prove that such integrals are expressed in terms of the Fox-Wright function $_{p}Psi_{q}(z)$. The results presented here are of general in nature and easily reducible to new and known results.
A translation of Kummer`s paper On certain definite integrals and infinite series
We present a summary of recent and older results on Bessel integrals and their relation with zeta numbers.
We calculate the ku-homology of the groups Z/p^n X Z/p and Z/p^2 X Z/p^2. We prove that for this kind of groups the ku-homology contains all the complex bordism information. We construct a set of generators of the annihilator of the ku-toral class. These elements also generates the annihilator of the BP-toral class.
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