On derived functors of Graded local cohomology modules

Let $K$ be a field of characteristic zero and let $R=K[X_1,\dots ,X_n]$, with standard grading. Let $\mathfrak{m}=(X_1,\dots ,X_n)$ and let $E$ be the ${}^{\ast }$injective hull of $R\mathrm{/}\mathfrak{m}\mathrm{.}$ Let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. Let $I,J$ be homogeneous ideals in $R$. Fix $i,j\ge 0$ and set $M=H_I^i(R)$ and $N=H_J^j(R)$ considered as left $A_n(K)$-modules. We show the following two results for which no analogous result is known in charactersitc $p>0$. \begin{enumerate} $H_{\mathfrak{m}}^l({\text{Tor}}_{\nu }^R(M,N))\cong E(n{)}^{a_{l,\nu }}$ for some $a_{l,\nu }\ge 0$. For all $\nu \ge 0$; the finite dimensional vector space $\Tor^{A_n(K)}\nu( M^\sharp, N)$isconcentratedindegree$-n$(here$M^\sharp$ is the standard right $A_n(K)$-module associated to $M$). \end{enumerate} We also conjecture that for all $i\ge 0$ the finite dimensional vector space $\Ext^i{A_n(K)}(M, N)$ is concentrated in degree zero. We give a few examples which support this conjecture.