Geometric interpretation of integrals of coordinate rings 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 ? 
 A: Just a quick answer: I have maybe slightly different Hopf algebras in mind as you, but in my applications the integral often behaves like the fundamental class of a manifold. 
[Added as answer:] The example I have in mind is as follows: Take $H=\mathcal{U}_q(\mathfrak{g})$, for $q$ a root of unity the finite-dimensional quotient (Frobenius-Lusztig kernel) $\bar{H}$ and $\mathfrak{B}(M)$ the Borel part, the so-called Nichols algebra (generated be all $E_i$'s). The integral of $\mathfrak{B}(M)$ corresponds to the longest element in the Weyl group. I have read people connect that to Schubert cells but I can not repeat these arguments.
You can use it to prove e.g. Poincare Duality of $\mathfrak{B}(M)$. The Hilbert series hence exhibit the nice palindrom symmetry.
I'm very sure you can argue similarly for non-truncations, but then it seems to me you have to do topological completion for that. Maybe the regular functions on a Lie group give you a more accessible example? Generally it should be a good idea to think of the integral in terms of a Frobenius algebra!
