Automorphic Plancherel density theorem

Let F be a totally real field, G a connected reductive group over F, and S a finite set of finite places of F. Assume that G(F ?_? ?) has a discrete series representation. Building upon work of Sauvageot, Serre, Conrey-Duke-Farmer and others, we prove that the S-components of cuspidal automorphic representations of $G\left( {\mathbb{A}_F } \right)$ are equidistributed with respect to the Plancherel measure on the unitary dual of G(F _S ) in an appropriate sense. A few applications are given, such as the limit multiplicity formula for local representations in the global cuspidal spectrum and a quite flexible existence theorem for cuspidal automorphic representations with prescribed local properties. When F is not a totally real field or G(F ?_? ?) has no discrete series, we present a weaker version of the above results.     

Abstract Plancherel equation and inversion formula

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 8, pp. 1041–1047, August, 1989.     

Fibred Frobenius theorem

We give a version of Frobenius Theorem for fibred manifolds whose proof is shorter than the “short proofs” of the classical Frobenius Theorem. In fact, what shortens the proof is the fibred form of the statement, since it permits an inductive process which is not possible from the standard statement.     

A reciprocity law for certain Frobenius extensions

Let be a finite Galois extension of algebraic number fields with Galois group . Assume that is a Frobenius group and is a Frobenius complement of . Let be the maximal normal nilpotent subgroup of . If is nilpotent, then every Artin L-function attached to an irreducible representation of arises from an automorphic representation over , i.e., the Langlands' reciprocity conjecture is true for such Galois extensions.

     

Induced representation and Frobenius reciprocity for compact quantum groups

Unitary representations of compact quantum groups have been described as isometric comodules. The notion of an induced representation for compact quantum groups has been introduced and an analogue of the Frobenius reciprocity theorem is established.     

Generalized Weil's reciprocity law and multiplicativity theorems

Fix a one-dimensional group variety with Euler-characteristic , and a quasi-projective variety , both defined over . For any and constructible sheaf on , we construct an invariant , which provides substantial information about the topology of the fiber-structure of and the structure of along the fibers of . Moreover, is a group homomorphism.

     

A formal Frobenius theorem and argument shift

A formal Frobenius theorem, which is an analog of the classical integrability theorem for smooth distributions, is proved and applied to generalize the argument shift method of A. S. Mishchenko and A. T. Fomenko to finite-dimensional Lie algebras over any field of characteristic zero. A completeness criterion for a commutative set of polynomials constructed by the formal argument shift method is obtained.     

Abstract Kuhn-Tucker theorems

In this note, we discuss abstract versions of the Kuhn-Tucker theorem for constrained convex minimization problems, in which we relax both the conditions of convexity and also the requirement that the number of constraints be finite.     

Frobenius properties and Maschke-type theorems for entwined modules

Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems known for Doi-Hopf modules are extended to the case of entwined modules.

     

Abstract Lebesgue-Radon-Nikodym theorems

Summary In this paper generalizations of the classical Lebesgue-Radon-Nikodym type decomposition of additive set functions are obtained for pairs of vector measures when both measures take values in possibly different Banach spaces. Some applications of these results are made to (i) the representation of wearly compact operators on the spaces of integrable scalar functions relative to a vector measure to an arbitrary Banach space, and (ii) a problem of comparison of measures in inference theory. The abstract conditional expectations of operator valued strongly measurable and integrable random variables on a σ-finite space are briefly treated. Supported, in part, under the NSF Grants GP-1349 and GP-5921.