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Minimal representations of simple real Lie groups of non Hermitian type | | Dehbia Achab
; | | Date: |
8 Mar 2011 | | Abstract: | In the recent paper [Achab-Faraut 2011], we introduced an analysis of the
Brylinski-Kostant model for the spherical minimal representations for simple
real Lie groups of non Hermitian type. We generalize here that analysis to all
the minimal representations. We start from a pair $(V,Q)$, where $V$ is a
complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. More
precisely, $V$ is a semi-simple Jordan algeba, $V=sumlimits_{i=1}^sV_i$ its
decomposition into simple ideals, $Delta_i$ is the determinant polynomial of
$V_i$ and $Q(z)=prodlimits_{i=1}^sDelta_i(z_i)^{k_i}$. The manifold
$ ildeXi_i $ is the orbit of $Delta_i$ under the action of a covering of
${
m Conf}(V_i,Delta_i)$, the conformal group of the pair $(V_i,Delta_i)$,
in a finite dimensional representation space of polynomials on $V_i$. By some
construction process, we obtain a complex simple Lie algebra $goth g$, and
furthermore a real form ${goth g}_{board R}$. The connected and simply
connected Lie group $G_{board R}$ with ${
m Lie}(G_{board R})={goth
g}_{board R}$ acts unitarily on a Hilbert space of holomorphic functions
defined on the manifold $prodlimits_{i=1}^sXi_i$, where $Xi_i$ is the set
of $xi^{k_i}$ for $xiin ildeXi_i$. | | Source: | arXiv, 1103.1614 | | Services: | Forum | Review | PDF | Favorites | |
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