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An Analogue of a Reductive Algebraic Monoid Whose Unit Group Is a Kac-Moody Group
By an easy generalization of the Tannaka-Krein reconstruction we associate to the category of admissible representations of the category ${\mathcal O}$ of a Kac-Moody algebra, and its category of admissible duals, a monoid with a coordinate ring. The Kac-Moody group is the Zariski open dense unit group of this monoid. The restriction of the coordinate ring to the Kac-Moody group is the algebra of strongly regular functions introduced by V. Kac and D. Peterson. This monoid has similar structural properties as a reductive algebraic monoid. In particular it is unit regular, its idempotents related to the faces of the Tits cone. It has Bruhat and Birkhoff decompositions. The Kac-Moody algebra is isomorphic to the Lie algebra of this monoid.
Representation Type of Commutative Noetherian Rings III: Global Wildness and Tameness
This memoir completes the series of papers beginning with [KL1,KL2], showing that, for a commutative noetherian ring $\Lambda$, either the category of $\Lambda$-modules of finite length has wild representation type or else we can describe the category of finitely generated $\Lambda$-modules, including their direct-sum relations and local-global relations. (There is a possible exception to our results, involving characteristic 2.)
Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor $D$-Modules, Part 2
The author studies the asymptotic behaviour of tame harmonic bundles. First he proves a local freeness of the prolongment of deformed holomorphic bundle by an increasing order. Then he obtains the polarized mixed twistor structure from the data on the divisors. As one of the applications, he obtains the norm estimate of holomorphic or flat sections by weight filtrations of the monodromies. As another application, the author establishes the correspondence of semisimple regularholonomic $D$-modules and polarizable pure imaginary pure twistor $D$-modules through tame pure imaginary harmonic bundles, which is a conjecture of C. Sabbah. Then the regular holonomic version of M. Kashiwara's conjecture follows from the results of Sabbah and the author.
A Categorical Approach to Imprimitivity Theorems for $C^*$-Dynamical Systems
It has become apparent that studying the representation theory and structure of crossed-product C*-algebras requires imprimitivity theorems. This monograph shows that the imprimitivity theorem for reduced algebras, Green's imprimitivity theorem for actions of groups, and Mansfield's imprimitivity theorem for coactions of groups can all be understoo
Fermionic Expressions for Minimal Model Virasoro Characters
Fermionic expressions for all minimal model Virasoro characters $\chi DEGREES{p, p'}_{r, s}$ are stated and proved. Each such expression is a sum of terms of fundamental fermionic f
A Generating Function Approach to the Enumeration of Matrices in Classical Groups over Finite Fields
Generating function techniques are used to study the probability that an element of a classical group defined over a finite field is separable, cyclic, semisimple or regular. The limits of these probabilities as the dimension tends to infinity are calculated in all cases, and exponential convergence to the limit is proved. These results complement and extend earlier results of the authors, G. E. Wall, and Guralnick & Lubeck.
Integral Transformations and Anticipative Calculus for Fractional Brownian Motions
A paper that studies two types of integral transformation associated with fractional Brownian motion. They are applied to construct approximation schemes for fractional Brownian motion by polygonal approximation of standard Brownian motion. This approximation is the best in the sense that it minimizes the mean square error.
Twisted Tensor Products Related to the Cohomology of the Classifying Spaces of Loop Groups
Let $G$ be a compact, simply connected, simple Lie group. By applying the notion of a twisted tensor product in the senses of Brown as well as of Hess, we construct an economical injective resolution to compute, as an algebra, the cotorsion product which is the $E_2$-term of the cobar type Eilenberg-Moore spectral sequence converging to the cohomology of classifying space of the loop group $LG$. As an application, the cohomology $H^*(BLSpin(10); \mathbb{Z}/2)$ is explicitly determined as an $H^*(BSpin(10); \mathbb{Z}/2)$-module by using effectively the cobar type spectral sequence and the Hochschild spectral sequence, and further, by analyzing the TV-model for $BSpin(10)$.
Holder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients
This mathematical monograph is a study of interior regularity of weak solutions of second order linear divergence form equations with degenerate ellipticity and rough coefficients. The authors show that solutions of large classes of subelliptic equations with bounded measurable coefficients are H lder continuous. They present two types of results f
Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces
Intends to prove a functional equation for a local zeta function attached to the minimal spherical series for a class of real reductive symmetric spaces.
