Motivated by a conjecture of Alday, Gaiotto and Tachikawa (AGT
conjecture), we construct an action of
a suitable $W$-algebra on the equivariant cohomology of the moduli
space $M_r$ of rank r instantons on $A^2$ (i.e.
on the moduli space of rank $r$ torsion free sheaves on $P^2$,
trivialized at the line at infinity). We show that
the resulting $W$-module is identified with a Verma module, and the
characteristic class of $M_r$ is the Whittaker vector
of that Verma module. One of the main ingredients of our construction
is the so-called cohomological Hall algebra of the
commuting variety, which is a certain associative algebra structure on
the direct sum of equivariant cohomology spaces
of the commuting varieties of $gl(r)$, for all $r$. Joint work with E. Vasserot.