If $X$ is an affine scheme over the field $k$ than algebraic invariants of the coordinate ring $k[X]$ usually have a geometric interpretation in terms of $X$ (and vice versa). As an example, the minimal primes of $k[X]$ correspond to the irreducible components of $X$.

Now suppose $G$ is a finite group scheme over $k$. Thus $k[G]$ is a finite dimensional Hopf algebra and by a well-known theorem of Larson-Sweedler, $k[G]$ has a non-zero integral, i.e. an element $a_0 \in k[G], a_0 \neq 0$ such that $a\cdot a_0 = \epsilon(a)a_0$ for all $a \in k[G]$, where $\epsilon: k[G] \to k$ is the augementation induced by the identity $e: \operatorname{Spec}(k) \to G$.

Is there a colorful geometric interpretation of this integral ?