Speaker
Hitoshi Konno, Tokyo University Marine
Abstract
We discuss some relationships between representations of elliptic quantum (toroidal) groups
and equivariant elliptic cohomology ${\mathrm E}_{\mathrm T}(X)$ of quiver varieties $X$.
Our main discussion is a conjecture on a certain equivalence between the vertex operators of
the elliptic quantum groups and the elliptic stable envelopes for ${\mathrm E}_{\mathrm T}(X)$
formulated by Aganagic and Okounkov. We show examples such as $U_{q,p}(\widehat{\mathfrak {sl}}_N)$
vs. $X$=the cotangent bundle of the partial flag variety, $U_{t_1,t_2,p}(\mathfrak {gl}_{1,tor})$ vs.
$X$=the Jordan quiver varieties (the instanton moduli spaces), $U_{t_1,t_2,p}(\mathfrak {gl}_{N,tor})$
vs. $X$=the $A^{(1)}_{N-1}$ quiver varieties. In particular, we show that the composition of the
vertex operators are equivalent to the shuffle product formula for the elliptic stable envelopes,
and the highest to highest expectation values of (the composition of) the vertex operators give
K-theoretic vertex functions counting quasimaps ${\mathbb P}^1$ to $X$.