Arbitrary-framing A1 Hecke cycles realize the highest-weight sl2 module

The top-dimensional Chow group of an irreducible $a_v$-dimensional variety is freely generated by its fundamental class: there is only one integral subvariety of dimension $a_v$, and rational equivalences in this degree would require an $(a_v+1)$-dimensional subvariety, which does not exist. Thus ${\mathcal{V}}_w$ has basis $b_0,\dots ,b_w$. By [F1]–[F3] the Hecke operators have exactly the required degree to preserve these top summands, and their actual push-pull actions are the raw coefficients of [F2]. Since $a_v-a_{v+1}=2v+1-w$, the reverse sign has parity $w-1$; multiplying it into the raw reverse coefficient gives $(-1{)}^{w-1+w-v-1}(w-v)=(-1{)}^v(w-v)$. This proves all displayed coefficient formulas.

For $0\le v\le w$, the formulas in step 1.1 give $EF{b}_v=(v+1)(w-v)b_v$ and $FE{b}_v=v(w-v+1)b_v$, with the same equalities at the endpoints by zero extension. Their difference is $(w-2v)b_v=H{b}_v$. Since $E$ raises the $H$-weight by $2$ and $F$ lowers it by $2$, the same formulas give $[H,E]=2E$ and $[H,F]=-2F$ on every basis vector. They are the defining brackets of the trace-zero two-by-two Lie algebra, so these operators extend to its enveloping-algebra representation. This is a correspondence action: [F3] identifies every product of its geometric kernels with the corresponding composite operator, so the computed relations are relations of that action rather than formal dimension matching.

The vector $b_0$ is killed by $E$ and has weight $w$. Successive lowering gives $F^v{b}_0=(-1{)}^{v(v-1)\mathrm{/}2}v!b_v$, which is nonzero for $0\le v\le w$; thus it generates the whole space. Both $E$ and $F$ are nilpotent since they move among finitely many summands. If a nonzero invariant subspace is given, a polynomial in the diagonal $H$ projects a nonzero vector onto one of its distinct weight components $\mathbb{Q}{b}_v$; applying $E^v$ gives a nonzero multiple of $b_0$, and then the powers of $F$ give all basis vectors. This proves irreducibility and integrability, including the trivial one-dimensional module at $w=0$. Replacing $b_v$ by $(-1{)}^{v(v-1)\mathrm{/}2}{b}_v$ gives the usual divided-power coefficients $F{c}_v=(v+1)c_{v+1}$ and $E{c}_v=(w-v+1)c_{v-1}$.

Equivariance follows from [F1]–[F3]; in fact each $h$ sends the entire $L_v$ to itself and hence fixes $b_v$, so it acts trivially on the ordinary top Chow representation. At $w=1$ all spaces and the Hecke correspondence are points and the formulas reduce to the existing fundamental two-dimensional example. This preserves that example as a special case and completes the arbitrary-framing construction.