  Let $(R, \mathfrak m)$ denote an $n$-dimensional Gorenstein ring. For an
ideal $I \subset R$ of height $c$ we are interested in the endomorphism ring $B
= \Hom_R(H^c_I(R), H^c_I(R)).$ It turns out that $B$ is a commutative ring. In
the case of $(R,\mathfrak m)$ a regular local ring containing a field $B$ is a
Cohen-Macaulay ring. Its properties are related to the highest Lyubeznik number
$l = \dim_k \Ext_R^d(k,H^c_I(R)).$ In particular $R \simeq B$ if and only if $l
= 1.$ Moreover, we show that the natural homomorphism $\Ext_R^d(k, H^c_I(R))
\to k$ is non-zero.
