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 $\rho: \mathcal{C} \to \mathrm{Rex}(\mathcal{M})$, where $\mathrm{Rex}(\mathcal{M})$ is the category of $k$-linear right exact endofunctors on $\mathcal{M}$. Our key observation is that $\rho$ has a right adjoint $\rho^{\mathrm{ra}}$ given by the end $\rho^{\mathrm{ra}}(F) = \int_{M \in \mathcal{M}} \underline{\mathrm{Hom}}(M, M)$. As an application, we establish the following results: (1) We give a description of the composition of the induction functor $\mathcal{C}_{\mathcal{M}}^* \to \mathcal{Z}(\mathcal{C}_{\mathcal{M}}^*)$ and Schauenburg's equivalence $\mathcal{Z}(\mathcal{C}_{\mathcal{M}}^*) \approx \mathcal{Z}(\mathcal{C})$. (2) We introduce the space $\mathrm{CF}(\mathcal{M})$ of `class functions' of $\mathcal{M}$ and initiate the character theory for pivotal module categories. (3) We introduce a filtration for $\mathrm{CF}(\mathcal{M})$ and discuss its relation with some ring-theoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that $\mathrm{Ext}_{\mathcal{C}}^{\bullet}(1, \rho^{\mathrm{ra}}(\mathrm{id}_{\mathcal{M}}))$ 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.