Determinantal Ideals and the Canonical Commutation Relations. Classically or Quantized

Abstract

We construct homomorphic images of $su(n,n)^{\mathbb C}$ in Weyl Algebras ${\mathcal H}_{2nr}$. More precisely, and using the Bernstein filtration of ${\mathcal H}_{2nr}$, $su(n,n)^{\mathbb C}$ is mapped into degree $2$ elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of ${\mathcal H}_{2nr}$, these homomorphisms give all unitary highest weight representations of $su(n,n)^{\mathbb C}$ thus reconstructing the Kashiwara--Vergne List for the Segal--Shale--Weil representation. Just as in the derivation of the their list, we construct a representation of $u(r)$ in the Fock space commuting with $su(n,n)^{\mathbb C}$, and this gives the multiplicities. The construction also gives an easy proof that the ideals of $(r+1)\times (r+1)$ minors are prime ($r\leq n-1)$. The quotients of all polynomials by such ideals carry the more singular of the representations. As a consequence, these representations can be realized in spaces of solutions to Maxwell type equations. We actually go one step further and determine exactly which representations from our list are missing some ${\mathfrak k}^{\mathbb C}$-types, thereby revealing exactly which covariant differential operators have unitary null spaces. We prove the analogous results for ${\mathcal U}_q(su(n,n)^{\mathbb C})$. The Weyl Algebras are replaced by the Hayashi--Weyl Algebras ${\mathcal H}{\mathcal W}_{2nr}$ and the Fock space by a $q$-Fock space. Further, determinants are replaced by $q$-determinants, and a commuting representation of ${\mathcal U}_q(u(r))$ in the $q$-Fock space is constructed. For this purpose a Drinfeld Double is used. We mention one difference: The quantized negative non-compact root spaces, while still of degree 2, are no longer given entirely by second order creation operators.