Invariant theory for the orthogonal group via star products (Q2730491)

Let \(K\) be a compact Lie group acting unitarily on a complex vector space \(V\) with a finite dimension and a positive definite Hermitian inner product. Let \(\mathbb{C}[V]\) be the space of holomorphic polynomials on \(V\) equipped with the Fock inner product. One says that the action of \(K\) on \(V\) is multiplicity free if the space \(\mathbb{C}[V]\) decomposes into pairwise inequivalent irreducible \(K\)-modules. Let \(\mathbb{C}[V]=\sum_{\lambda\in\Lambda}P_\lambda\) denote the decomposition of \(\mathbb{C}[V]\). Here \(\Lambda\) is a countably infinite set of indices. We consider \(K\)-invariants in \(\mathbb{C}[V_\mathbb{R}]\), the ring of polynomial functions on the underlying real space \(V_\mathbb{R}\). In this context, we find two canonical vector space bases for the algebra \(\mathbb{C}[V_\mathbb{R}]^K\) of \(K\)-invariants, each indexed by \(\Lambda\): a base \(\{p_\lambda\}_{\lambda\in\Lambda}\) consisting of homogeneous polynomials and a base \(\{q_\lambda\}_{\lambda\in\Lambda}\) consisting of polynomials orthogonal with respect to the Fock inner product on \(\mathbb{C}[V_\mathbb{R}]\). The \(p_\lambda\)'s are orthogonal with respect to the star inner product on \(\mathbb{C}[V_\mathbb{R}]\) which is obtained from the Fock inner product on \(\mathbb{C}[V]\) using \(\mathbb{C}[V_\mathbb{R}]\simeq\mathbb{C}[V]\otimes\mathbb{C}[\overline V]\). The star inner product arises naturally in the context of the (Berezin) star product. The polynomials \(\{q_\lambda\}\) are obtained from the \(p_\lambda\)'s via Gram-Schmidt orthogonalization using the Fock inner product.NEWLINENEWLINENEWLINEThe authors show how the theory of star products can be used as a tool to study invariant theory for multiplicity free actions. In a specific multiplicity free action, that of \(K=SO(n,\mathbb{R})\times\mathbb{T}\) on \(V=\mathbb{C}^n\) \((n\geq 3)\), the \(p_\lambda\)'s satisfy recurrence relations. Using this result they give explicit formulae for the \(p_\lambda\)'s and \(q_\lambda\)'s by using properties of the star inner product. Moreover, they give product formulas for the \(p_\lambda\)'s by using the same techniques of the star inner product.NEWLINENEWLINENEWLINEIf one extends \(V\) to form the Heisenberg group \(H_V=V\times\mathbb{R}\), then \(K\) acts via automorphisms on \(H_V\) which fix the center \(\mathbb{R}\). One says that the action of \(K\) on \(H_V\) yields a Gelfand pair if the \(K\)-invariant integrable functions \(L^1_K(H_V)\) form a commutative algebra under convolution. This is the case if and only if the action of \(K\) on \(V\) is multiplicity free. There is, moreover, a well developed theory of spherical functions associated to such Gelfand pairs. The polynomials \(\{q_\lambda\}_{\lambda\in\Lambda}\) determine a dense set of full measure in the space of bounded spherical functions.