Provarieties and observable subgroups of pro-affine algebraic groups (Q2776801)

Given an affine algebraic group \(G\) (over an algebraically closed field \(k\)), a closed subgroup \(H\) of \(G\) is said to be `observable' if every rational \(H\)-module occurs as an \(H\)-submodule of a rational \(G\)-module. Investigations have led to a number of equivalent conditions under which \(H\) is observable. The authors are interested in studying this question for a pro-affine algebraic group \(G\) and algebraic subgroup \(H\). The main result of the paper is the identification of a set of representation-theoretic and geometric conditions equivalent to observability. These conditions are analogues of conditions for affine groups found by \textit{A. Bialynicki-Birula}, \textit{G. Hochschild}, and \textit{G. D. Mostow} [Am. J. Math. 85, 131-144 (1963; Zbl 0116.02302)] and \textit{F. Grosshans} [Am. J. Math. 95, 229-253 (1973; Zbl 0309.14039), Algebraic homogeneous spaces and invariant theory (Lect. Notes Math. 1673) (1997; Zbl 0886.14020)]. In the affine case, one particular geometric condition is that the subgroup is observable if and only if \(G/H\) is a quasi-affine variety. In the pro-affine case, the authors define the notion of a `provariety' (differing slightly from other definitions in the literature) and show analogously that a subgroup \(H\) is observable if and only if \(G/H\) is a quasi-affine provariety.