Further results on the structure of (co)ends in finite tensor categories

Let $\mathcal{C}$ be a finite tensor category, and let $\mathcal{M}$ be an exact left $\mathcal{C}$-module category. The action of $\mathcal{C}$ on $\mathcal{M}$ induces a functor $\rho: $\mathcal{C}$ \to $\mathrm{R}\mathrm{e}\mathrm{x}$($\mathcal{M}$)$,where$$\mathrm{R}\mathrm{e}\mathrm{x}$($\mathcal{M}$)$ is the category of $k$-linear right exact endofunctors on $\mathcal{M}$. Our key observation is that $\rho$ has a right adjoint ${\rho }^{\mathrm{r}\mathrm{a}}$ given by the end $\rho^{$\mathrm{r}\mathrm{a}$}(F) = \int_{M \in $\mathcal{M}$} \underline{$\mathrm{H}\mathrm{o}\mathrm{m}$}(M, M)$. As an application, we establish the following results: (1) We give a description of the composition of the induction functor ${\mathcal{C}}_{\mathcal{M}}^{\ast }\to \mathcal{Z}({\mathcal{C}}_{\mathcal{M}}^{\ast })$ and Schauenburg’s equivalence $$\mathcal{Z}$($\mathcal{C}${$\mathcal{M}$}^*) \approx $\mathcal{Z}$($\mathcal{C}$)$\mathrm{.}(2)Weintroducethespace$$\mathrm{C}\mathrm{F}$($\mathcal{M}$)$of\mathrm{‘}classfunction{s}^{\mathrm{\prime }}of$$\mathcal{M}$$ and initiate the character theory for pivotal module categories. (3) We introduce a filtration for $\mathrm{C}\mathrm{F}(\mathcal{M})$ and discuss its relation with some ring-theoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that $$\mathrm{E}\mathrm{x}\mathrm{t}${$\mathcal{C}$}^{\bullet}(1, \rho^{$\mathrm{r}\mathrm{a}$}($\mathrm{i}\mathrm{d}$_{$\mathcal{M}$}))$ is isomorphic to the Hochschild cohomology of $\mathcal{M}$. As an application, we show that the modular group acts projectively on the Hochschild cohomology of a modular tensor category.