Proposition 12.33.2A78CB65BF75E

Proof. Give $c_-,c_+$ weights $-1,1$, and $h,u,w$ weight zero. Using $c_-c_+=h^2$, the blocks of negative weight have the form $c_-^ kF(z,h,u,w)$, those of positive weight the form $c_+^ kF(z,h,u,w)$, and the block of weight zero consists of the polynomials $F$. These expressions are unique: the substitution $c_+=h^2/c_-$ embeds the algebra into $\mathcal O(\mathbb C)[h,u,w,c_-,c_-^{-1}]$, and distinct weights correspond to distinct powers of $c_-$ with nonzero polynomial coefficients. In tensor degree $r$, the remaining polynomial degree is $r-k$.

The coefficient bundle $U_3$ now has the inverse translation $u\mapsto u-m(q_0h+Kw)$. Thus the block of weight $-k$ is $H^0(E,A^ k\otimes \operatorname {Sym}^{r-k}U_3)$, the block of weight $k$ has $A^{-k}$ in place of $A^ k$, and the block of weight zero is $H^0(E,\operatorname {Sym}^ rU_3)$. Put $H=h+(K/q_0)w$. Since $q_0\ne 0$, this identifies $U_3$ with the bundle $U$ defined above, with $H$ in place of its Euler variable, plus the trivial line bundle generated by $w$.

By lemma 12.31.2, all positive weights vanish, and the algebra of weight zero is $\mathbb C[H,w]$. Expanding the negative blocks in powers of $w$ identifies them with the blocks for the tangent bundle in lemma 12.31.1, and the substitutions $h\mapsto H$ and $y^ k\mapsto c_-^ k$ give bijections in every degree and surviving weight. Products multiply the coefficient polynomials and add the powers of $c_-$, so these bijections preserve multiplication. This proves the isomorphism. Summing the dimensions for the tangent bundle and taking the quotient by $w$ gives the remaining assertions, exactly as in the proof of corollary 12.32.3. $\square$