# Invariant differential operators and an infinite dimensional Howe-type
correspondence. Part I: Structure of the associated algebras of differential
operators

Research paper by **Hubert Rubenthaler**

Indexed on: **04 Feb '08**Published on: **04 Feb '08**Published in: **Mathematics - Representation Theory**

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#### Abstract

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 the study of the case where $Q(x)$ is replaced
by $\Delta_{0}(x)$, where $\Delta_{0}(x)$ is the relative invariant of a
prehomogeneous vector space of commutative parabolic type ($ {\go g},V $), or
equivalently where $\Delta_{0}$ is the "determinant" function of a simple
Jordan algebra $V$ over ${\bb C}$. In this Part I we show several structure
results for the associative algebra generated by $X=\Delta_{0}(x)$,
$Y=\Delta_{0}(\partial)$. Our main result shows that if we consider this
algebra as an algebra over a certain commutative ring ${\bf A}$ of invariant
differential operators it is isomorphic to the quotient of what we call a
generalized Smith algebra $S(f, {\bf A}, n)$ where $f\in {\bf A}[t]$. The Smith
algebras (over ${\bb C}$) were introduced by P. Smith as "natural"
generalizations of ${\cal U}({\go sl}_{2})$.