Spherical and Symmetric Supervarieties

We develop and study the notion of a spherical supervariety, which is a generalization of the classical notion of a spherical variety in algebraic geometry. Spherical supervarieties are supervarieties admitting an action of a quasi-reductive group with an open orbit of a hyperborel subgroup. Three characterizations of spherical supervarieties are given: one which generalizes the Vinberg-Kimelfeld characterization of affine spherical varieties, another that extends the ideas of the affine case to the quasi-projective case, and finally one in terms of invariant rational functions which applies to any supervariety. Our characterization of affine spherical supervarieties leads to (non-constructive) existence theorems for finite-dimensional highest weight representations admitting certain coinvariants under spherical quasireductive subgroups.Several interesting examples of spherical supervarieties are given. We present a classification of indecomposable spherical representations (for certain supergroups) and for each the description of its algebra of functions. Adjoint orbits of odd self-commuting elements are shown to be spherical in many cases, in particular for basic simple Lie superalgebras. We study group-graded supergroups and their spherical homogeneous supervarieties, showing in particular that the algebra of functions on an affine homogeneous supervariety is almost never completely reducible for such supergroups.Finally we study the case of symmetric supervarieties and show that, despite their not always being spherical (in contrast to the classical case), we may under some circumstances guarantee the existence of an Iwasawa decomposition, which implies sphericity. The fixed points of automorphisms of generalized root systems coming from supersymmetric pairs are studied along the way. We use the Iwasawa decomposition to gain partial understanding of the structure of the space of functions as a representation. Finally, the case of a supergroup as a symmetric supervariety is discussed in detail, culminating in a description of the socle filtration and the Loewy layers of its space of functions