C.A. Pallikaros

On point transformations of generalized nonlinear diffusion equations

This paper classifies all finite point transformations of a general class between generalized diffusion equations of the form ut=x1-M [xN-1f(u)ux]x. These transformations may be divided into three cases, depending on the functional form of f(u): (i) f arbitrary, (ii) f=un and (iii) f=eu. In particular, these transformations include all the invariant infinitesimal transformations and, in addition, they include a number of point transformations which relate different equations of the above form. Many exact solutions are already known and the transformations which are derived may be used to obtain new solutions from these.

Weil Representations and Some Nonreductive Dual Pairs in Symplectic and Unitary Groups

We study the restriction of the Weil representations of symplectic and unitary groups to subgroups that are the centralizers of certain elements, and show that these are multiplicity free. This work is along the line of the Howe philosophy for so-called dual pairs in the symplectic and unitary groups, but our dual pairs are not reductive. We also obtain an explicit formula for the character of the restriction to the centralizer of a regular unipotent element.

Commutative Nilpotent Closed Algebras and Weil Representations

Let 𝔽 be the field GF(q 2) of q 2 elements, q odd, and let V be an 𝔽-vector space endowed with a nonsingular Hermitian form ϕ. Let σ be the adjoint involutory antiautomorphism of End𝔽 V associated to the form, and let U(ϕ) be the corresponding unitary group. We ask whether the restrictions of the Weil representation of U(ϕ) to certain subgroups are multiplicity-free. These subgroups consist of the members of U(ϕ) in subalgebras of the form 𝔽I + N, where N is a σ-stable commutative nilpotent subalgebra of End𝔽 V with the further property that N contains its annihilator. We give a necessary condition for multiplicity-freeness that depends on the dimensions of N and that annihilator. Moreover, the case that N is conjugate to its regular representation is completely settled. Several other classes of subalgebra are discussed in detail.