The Capelli problem for the symmetric pairs $(mathfrak{gl}times mathfrak{gl},mathfrak{gl})$ $(mathfrak{gl},mathfrak{o})$, and $(mathfrak{gl},mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special
values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(mathfrak{g},mathfrak{g}):=(mathfrak{gl}(m|2n),mathfrak{osp}(m|2n))$ and $(mathfrak{gl}(m|n)timesmathfrak{gl}(m|n),mathfrak{gl}(m|n))$, acting on $W:=S^2(mathbb C^{m|2n})$ and $mathbb C^{m|n}otimes(mathbb C^{m|n})^*$. We also give an affirmative answer to the abstract Capelli problem for these cases.
For every finite dimensional Lie supergroup $(G,mathfrak g)$, we define a $C^*$-algebra $mathcal A:=mathcal A(G,mathfrak g)$, and show that there exists a canonical bijective correspondence between unitary representations of $(G,mathfrak g)$ and nond
egenerate $*$-representations of $mathcal A$. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations. For a broad class of Lie supergroups, which includes nilpotent as well as classical simple ones, we prove that the associated $C^*$-algebra is CCR. In particular, we obtain the uniqueness of direct integral decomposition for unitary representations of these Lie supergroups.
A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations $pi colon G to U(cH)$. In this paper we present a new approach
to host algebras for infinite dimensional Lie groups which is based on smoothing operators, i.e., operators whose range is contained in the space $cH^infty$ of smooth vectors. Our first major result is a characterization of smoothing operators $A$ that in particular implies smoothness of the maps $pi^A colon G to B(cH), g mapsto pi(g)A$. The concept of a smoothing operator is particularly powerful for representations $(pi,cH)$ which are semibounded, i.e., there exists an element $x_0 ing$ for which all operators $iddpi(x)$, $x in g$, from the derived representation are uniformly bounded from above in some neighborhood of $x_0$. Our second main result asserts that this implies that $cH^infty$ coincides with the space of smooth vectors for the one-parameter group $pi_{x_0}(t) = pi(exp tx_0)$. We then show that natural types of smoothing operators can be used to obtain host algebras and that, for every metrizable Lie group, the class of semibounded representations can be covered completely by host algebras. In particular, it permits direct integral decompositions.
