  Let $K$ be a field of characteristic zero and let $R=K[X_1, \ldots,X_n ]$,
with standard grading. Let $\mathfrak{m}= (X_1, \ldots, X_n)$ and let $E$ be
the $^*$injective hull of $R/\mathfrak{m}.$ Let $A_n(K)$ be the $n^{th}$ Weyl
algebra over $K$. Let $I, J$ be homogeneous ideals in $R$. Fix $i,j \geq 0$ 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}
  $H^l_\mathfrak{m}(\Tor^R_\nu(M, N)) \cong E(n)^{a_{l,\nu}}$ for some
$a_{l,\nu} \geq 0$.
  For all $\nu \geq 0$; the finite dimensional vector space $\Tor^{A_n(K)}_\nu(
M^\sharp, N)$ is concentrated in degree $-n$ (here $M^\sharp$ is the standard
right $A_n(K)$-module associated to $M$). \end{enumerate} We also conjecture
that for all $i \geq 0$ 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.
