| Résume||Semi-infinite flag manifolds, that is a variant of an affine flag manifolds, essentially appears in the considerations of Lusztig and Drinfeld in the late 1970s to early 1980s. It encodes representation theory of affine Lie algebras at the critical level as exhibited by Feigin and Frenkel. As typical in infinite-dimensional objects, it has several disguises. One of its incarnation (that we call the ind-model), realization as the space of quasi-maps, was pursued in detail by Braverman, Finkelberg, Mirkovic and their collaborators.
There is another incarnation (that we call the proj-model), that directly consider it as an ind-scheme of infinite type, existed from the beginning. However, it was not extensively studied as it denied ad hoc approaches to implement geometric objects on that.
In the first part of this talk, we begin by recalling representation theory of loop algebras and show that it captures the structure of the proj-models of semi-infinite flag manifolds (and some of the basic theorems analogous to those in the usual flag manifolds).
In the second part of my talk, we briefly explain how to make sense of equivariant $K$-groups of them. Using this, we construct an isomorphism between the equivariant $K$-groups of semi-infinite flag manifolds and the (equivariant small) quantum $K$-groups of the (finite-dimensional) flag manifold that respects natural bases offered by Schubert subvarieties.|