On modality of representations

For connected simple algebraic groups defined over an algebraically closed field of characteristic zero, the classifications of irreducible algebraic representations of modalities 0, 1, and 2 are obtained.     

On primitive spinor exceptional representations

We establish a relationship between primitive representations of certain n-ary quadratic form g by a spinor genus of an (n + 2)-ary quadratic form f and primitive representations of some n-ary subform g′ of g by f itself. We discuss this in the general context of an algebraic number field.     

On linear representations of quasigroups

For homotopies of quasigroups, an analog of the fundamental theorem on homomorphisms does not hold in general. In this paper, we consider two approaches that allow one to obtain an analog of this theorem: the introduction of strict homotopies and the move from quasigroups to three-sorted quasigroups.     

On p-rank representations

The p-rank of an algebraic curve X over an algebraically closed field k of characteristic p>0 is the dimension of the vector space H¹(X_et,F_p). We study the representations of finite subgroups GAut(X) induced on H¹(X_et,F_p)k, and obtain two main results.First, the sum of the nonprojective direct summands of the representation, i.e., its core, is determined explicitly by local data given by the fixed point structure of the group acting on the curve. As a corollary, we derive a congruence formula for the p-rank.Secondly, the multiplicities of the projective direct summands of quotient curves, i.e., their Borne invariants, are calculated in terms of the Borne invariants of the original curve and ramification data. In particular, this is a generalization of both Nakajima's equivariant Deuring–Shafarevich formula and a previous result of Borne in the case of free actions.     

On some representations of the Bootstrap

Summary The bootstrap statistic is represented as the sum of two terms, one of which has the same distribution as and the other is a relatively negligible remainder. The representation is then applied to the problems of estimating a distribution and a quantile. The study includes the tail part of the distribution and quantiles for which the convergence rates could be faster than the usual n ^-1/2 rate.     

On representations of the Weil-Deligne group

We study admissible orthogonal and symplectic representations of the Weil-Deligne group W′(\(\overline K \)/K) of a local non-Archimedean field K. As an application of the obtained results we show that the root number of the tensor product of two admissible symplectic representations of W′(\(\overline K \)/K) is 1.     

On twisted representations of vertex algebras

In this paper, we develop a formalism for working with representations of vertex and conformal algebras by generalized fields—formal power series involving non-integer powers of the variable. The main application of our technique is the construction of a large family of representations for the vertex superalgebra corresponding to an integer lattice Λ. For an automorphism coming from a finite-order automorphism we find the conditions for existence of twisted modules of . We show that the category of twisted representations of is semisimple with finitely many isomorphism classes of simple objects.     

On integral representations of linear forms

Starting from a given *-algebra, we consider integral representations of positive linear forms on the hermitian spectrum of the algebra, providing necessary and sufficient conditions theorem. This specializes to previous results of R. S. Bucy—G. Maltese and G. Maltese for Banach *-algebras, and M. Fragoulopoulou for Imc *-algebras.     

On representations of real Jacobi groups

We consider a category of continuous Hilbert space representations and a category of smooth Fr’echet representations,of a real Jacobi group G.By Mackey’s theory,they are respectively equivalent to certain categories of representations of a real reductive group L.Within these categories,we show that the two functors that take smooth vectors for G and for L are consistent with each other.By using Casselman-Wallach’s theory of smooth representations of real reductive groups,we define matrix coefficients for distributional vectors of certain representations of G.We also formulate Gelfand-Kazhdan criteria for real Jacobi groups which could be used to prove multiplicity one theorems for Fourier-Jacobi models.