A Global version of Grozman’s theorem

Abstract

Let $X$ be a manifold. The classification of all equivariant bilinear maps between tensor density modules over $X$ has been investigated by Grozman (Funct Anal Appl 14(2):58–59, 1980), who has provided a full classification for those which are differential operators. Here we investigate the same question without the hypothesis that the maps are differential operators. In our paper, the geometric context is algebraic geometry and the manifold $X$ is the circle $\text{ Spec}\, \mathbb{C }[z,z^{-1}]$. Our main motivation comes from the fact that such a classification is required to complete the proof of the main result of Iohara and Mathieu (Proc Lond Math Soc, 2012, in press). Indeed it requires to also include the case of deformations of tensor density modules.

Appendix: Complete expression for \(\mathbf{D}_{\delta _1,\delta _2,\gamma }(x,y)\)

Recall that $\mathbf{D}_{\delta _1,\delta _2,\gamma }=\det \mathbf{M}$, where $\mathbf{M}$ is a $6\times 6$ matrix whose entries are polynomials of degree $2$ in $x,y,\delta _1,\delta _2$ and $\gamma$. The Appendix provides the explicit formula for $\mathbf{D}$.

It is quite obvious that $\mathbf{D}_{\delta _1,\delta _2,\gamma }(x,y)=\mathbf{D}_{\delta _1,\delta _2,\gamma }(-x-7,-y+7)$. Thus, in order to provide more compact formulas, it is better to list the polynomials $\tilde{q}_{i,j}$ defined by

$$\begin{aligned} \mathbf{D}_{\delta _1,\delta _2,\gamma }(x-7/2,y+7/2)=C(\delta _1,\delta _2,\gamma ) \sum \nolimits _{i,j}\tilde{q}_{i,j}(\delta _1,\delta _2,\gamma ) x^iy^j. \end{aligned}$$

The polynomials $\tilde{q}_{i,j}$ are calculated with MAPLE. It turns out that the only non-zero polynomials are $\tilde{q}_{0,0}, \tilde{q}_{0,2},\tilde{q}_{1,1},\tilde{q}_{2,0}, \tilde{q}_{1,3},\tilde{q}_{2,2}$ and $\tilde{q}_{3,1}$. It follows that $\mathbf{D}_{\delta _1,\delta _2,\gamma }$ is a polynomial of degree 4 in $x$ and $y$ and therefore $\tilde{q}_{1,3}=q_{1,3},\,\tilde{q}_{3,1}=q_{3,1}$ and $\tilde{q}_{2,2}=q_{2,2}$, where the polynomials $q_{1,3}, q_{3,1}$ and $q_{2,2}$ are given in Sect. 8.3. The other non-zero polynomials $\tilde{q}_{i,j}$ are given by the following formulas:

$$\begin{aligned} 16\tilde{q}_{0,0}&= [(4\delta _1+1)(4\delta _2+1)\gamma (1-\gamma )+16(\delta _1^2+\delta _2^2-\delta _1\delta _2)\delta _1\delta _2 \\&\quad +\,4(\delta _1^2+\delta _2^2-3\delta _1\delta _2(\delta _1+\delta _2)) +(13(\delta _1^2+\delta _2^2)-50\delta _1\delta _2)+11(\delta _1+\delta _2)+2] \\&\quad \times [(4\delta _1+1)(4\delta _2+1)\gamma (1-\gamma )++16(\delta _1^2+\delta _2^2-\delta _1\delta _2)\delta _1\delta _2 \\&\quad +\,4(\delta _1^2+\delta _2^2)(\delta _1+\delta _2) -3(\delta _1^2+\delta _2^2+6\delta _1\delta _2)-7(\delta _1+\delta _2)+6],\\ -4\tilde{q}_{0,2}&= (4\delta _1+1)^2\gamma ^2(1-\gamma )^2 +2(4\delta _1+1)((4\delta _1+1)\delta _2^2\\&\quad -\,2(4\delta _1+1)(\delta _1+1)\delta _2+4\delta _1^3+5\delta _1^2+2\delta _1+4)\gamma (1-\gamma ) +(4\delta _1+1)^2\delta _2^4 \\&\quad -\,4(4\delta _1+1)^2(\delta _1+1)\delta _2^3 +(32\delta _1^4+112\delta _1^3+142\delta _1^2+52\delta _1-13)\delta _2^2 \\&\quad -\, (64\delta _1^5+32\delta _1^4-92\delta _1^3+68\delta _1^2+82\delta _1-4)\delta _2 \\&-\, (4\delta _1+1)(\delta _1-1)(\delta _1+1)(\delta _1+2)(4\delta _1^2+\delta _1-6),\\ -4\tilde{q}_{2,0}&= (4\delta _2+1)^2\gamma ^2(1-\gamma )^2 +2(4\delta _2+1)((4\delta _2+1)\delta _1^2\\&-\,2(4\delta _2+1)(\delta _2+1)\delta _1+4\delta _2^3+5\delta _2^2+2\delta _2+4)\gamma (1-\gamma ) +(4\delta _2+1)^2\delta _1^4\\&\quad -\,4(4\delta _2+1)^2(\delta _2+1)\delta _1^3 +(32\delta _2^4+112\delta _2^3+142\delta _2^2+52\delta _2-13)\delta _1^2 \\&-\, (64\delta _2^5+32\delta _2^4-92\delta _2^3+68\delta _2^2+82\delta _2-4)\delta _1 \\&\quad -\, (4\delta _2+1)(\delta _2-1)(\delta _2+1)(\delta _2+2)(4\delta _2^2+\delta _2-6),\\ \frac{1}{2}\tilde{q}_{1,1}&= (28\delta _1^2\delta _2^2-4(\delta _1+\delta _2)\delta _1\delta _2+(\delta _1^2+\delta _2^2)-20\delta _1\delta _2-(\delta _1+\delta _2))\gamma (1-\gamma ) \\&\quad +\,4(7(\delta _1^2+\delta _2^2)-10\delta _1\delta _2)(\delta _1\delta _2)^2 -4(\delta _1^3+\delta _2^3)\delta _1\delta _2 \\&\quad +\,(\delta _1^4+\delta _2^4-12(\delta _1^2+\delta _2^2)\delta _1\delta _2-6(\delta _1\delta _2)^2)\\&\quad +\,2(\delta _1^2+\delta _1\delta _2+\delta _2^2)(\delta _1+\delta _2) -(\delta _1^2+\delta _2^2-14\delta _1\delta _2)-2(\delta _1+\delta _2). \end{aligned}$$