Mathematics – Representation Theory
Scientific paper
2011-03-09
Mathematics
Representation Theory
Scientific paper
If $Q$ is a non degenerate quadratic form on ${\bb C}^n$, it is well known that the differential operators $X=Q(x)$, $Y=Q(\partial)$, and $H=E+\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebra isomorphic to ${\go sl}_{2}$. Therefore the associative algebra they generate is a quotient of the universal enveloping algebra ${\cal U}({\go sl}_{2})$. This fact is in some sense the foundation of the metaplectic representation. The present paper is devoted to a generalization where $Q(x)$ is replaced by $\Delta_{0}(x)$ which is a relative invariant of a multiplicity free representation $(G,V)$ with a one dimensional quotient (see definition below). Over these spaces we study various algebras of differential operators. In particular if $G'=[G,G]$ is the derived group of the reductive group $G$, we prove that the algebra $D(V)^{G'}$ of $G'$-invariant differential operators with polynomial coefficients on $V$, is a quotient of a Smith algebra over its center. Over ${\bb C}$ this class of algebras was introduced by S.P. Smith as a class of algebras similar to ${\cal U}({\go s}{\go l}_{2})$. This allows us to describe by generators and relations the structure of $D(V)^{G'}$. As a corollary we obtain that various "algebras of radial components" are quotients of ordinary Smith algebras over ${\bb C}$. We also give the complete classification of the multiplicity free spaces $(G,V)$ with a one dimensional quotient, and pay particular attention to the subclass of prehomogeneous vector spaces of commutative parabolic type, for which further results are obtained.
No associations
LandOfFree
Invariant differential operators on a class of multiplicity free spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Invariant differential operators on a class of multiplicity free spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Invariant differential operators on a class of multiplicity free spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-643959