arXiv subjects

Pascal Lavaud

Publications and source records attributed to Pascal Lavaud.

3 recordsLinked to original sources

arXiv2004

Invariant generalized functions on $sl(2,R)$ with values in a $sl(2,R)$-module

Let $g$ be a finite dimensional real Lie algebra. Let $r:g\to End(V)$ be a representation of $g$ in a finite dimensional real vector space. Let $C_{V}=(End(V)\tens S(g))^{g}$ be the algebra of $End(V)$-valued invariant differential operators with constant coefficients on $g$. Let $U$ be an open subset of $g$. We consider the problem of determining the space of generalized functions $ϕ$ on $U$ with values in $V$ which are locally invariant and such that $C_{V}ϕ$ is finite dimensional. In this article we consider the case $g=sl(2,R)$. Let $N$ be the nilpotent cone of $sl(2,R)$. We prove that when $U$ is $SL(2,R)$-invariant, then $ϕ$ is determined by its restriction to $U\setminus N$ where $ϕ$ is analytic. In general this is false when $U$ is not $SL(2,R)$-invariant and $V$ is not trivial. Moreover, when $V$ is not trivial, $ϕ$ is not always locally $L^{1}$. Thus, this case is different and more complicated than the situation considered by Harish-Chandra where $g$ is reductive and $V$ is trivial. To solve this problem we find all the locally invariant generalized functions supported in the nilpotent cone $N$. We do this locally in a neighborhood of a nilpotent element $Z$ of $g$ and on an $SL(2,R)$-invariant open subset $U\subset sl(2,R)$. Finally, we also give an application of our main theorem to the Superpfaffian.

math.RT↗

arXiv2004

Superpfaffian

Let V=V_0+V_1 be a real finite dimensional supervector space provided with a non-degenerate antisymmetric even bilinear form B. Let spo(V) be the Lie superalgebra of endomorphisms of V which preserve B. We consider spo(V) as a supermanifold. We show that a choice of an orientation of V_1 and of a square root i of -1 determines a very interesting generalized function on the supermanifold spo(V), the superPfaffian. When V=V_1, spo(V) is the orthogonal Lie algebra so(V_1) and the superPfaffian is the usual Pfaffian, a square root of the determinant. When V=V_0, spo(V) is the symplectic Lie algebra sp(V_0) and the superPfaffian is a constant multiple of the Fourier transform of one the two minimal nilpotent orbits in the dual of the Lie algebra sp(V_0), and is an analytic square root of the inverse of the determinant in the open subset of invertible elements of spo(V). Our opinion is that the superPfaffians (there are four of them, corresponding to the two orientations on V_1, and to the two square roots of -1) are fundamental objects. At least, they occur in the study of equivariant cohomology of supermanifolds, and in the study of the metaplectic representation of the metaplectic group with Lie algebra spo(V). In this article, we present the definition and some basic properties of the superPfaffians.

math.GR↗

arXiv2004

Equivariant Cohomology and Localization Formula in Supergeometry

Let G be a compact Lie group. Let M be a smooth G-manifold and V --> M be an oriented G-equivariant vector bundle. One defines the spaces of equivariant forms with generalized coefficients on V and M. An equivariant Thom form $θ$ on V is a compactly supported closed equivariant form such that its integral along the fibres is the constant function 1 on M. Such a Thom form was constructed by Mathai and Quillen. Its restriction to M gives a representative of the equivariant Euler class of V. In the supergeometric situation we give proper definitions of all the objects involved. But, in this case a Thom form doesn't always exist. In this article, when the action of G on V is sufficiently non-trivial, we construct such a Thom form with generalized coefficients. We use it to construct an equivariant Euler form of V and to generalize Berline-Vergne's localization formula to the supergeometric situation.

math.DG↗