LAUSR

Let $${\mathcal {C}}$$ C be a finite tensor category, and let $${\mathcal {M}}$$ M be an exact left $${\mathcal {C}}$$ C -module category. The action of $${\mathcal {C}}$$ C on $${\mathcal {M}}$$ M induces a functor $$\rho : {\mathcal {C}} \rightarrow \mathrm {Rex}({\mathcal {M}})$$ ρ : C → Rex ( M ) , where $$\mathrm {Rex}({\mathcal {M}})$$ Rex ( M ) is the category of k -linear right exact endofunctors on $${\mathcal {M}}$$ M . Our key observation is that $$\rho $$ ρ has a right adjoint $$\rho ^{\mathrm {ra}}$$ ρ ra given by the end $$\begin{aligned} \rho ^{\mathrm {ra}}(F) = \int _{M \in {\mathcal {M}}} \underline{\mathrm {Hom}}(M, F(M)) \quad (F \in \mathrm {Rex}({\mathcal {M}})). \end{aligned}$$ ρ ra ( F ) = ∫ M ∈ M Hom ̲ ( M , F ( M ) ) ( F ∈ Rex ( M ) ) . As an application, we establish the following results: (1) We give a description of the composition of the induction functor $${\mathcal {C}}_{{\mathcal {M}}}^* \rightarrow {\mathcal {Z}}({\mathcal {C}}_{{\mathcal {M}}}^*)$$ C M ∗ → Z ( C M ∗ ) and Schauenburg’s equivalence $${\mathcal {Z}}({\mathcal {C}}_{{\mathcal {M}}}^*) \approx {\mathcal {Z}}({\mathcal {C}})$$ Z ( C M ∗ ) ≈ Z ( C ) . (2) We introduce the space $$\mathrm {CF}({\mathcal {M}})$$ CF ( M ) of ‘class functions’ of $${\mathcal {M}}$$ M and initiate the character theory for pivotal module categories. (3) We introduce a filtration for $$\mathrm {CF}({\mathcal {M}})$$ CF ( M ) and discuss its relation with some ring-theoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that $$\mathrm {Ext}_{{\mathcal {C}}}^{\bullet }(1, \rho ^{\mathrm {ra}}(\mathrm {id}_{{\mathcal {M}}}))$$ Ext C ∙ ( 1 , ρ ra ( id M ) ) is isomorphic to the Hochschild cohomology of $${\mathcal {M}}$$ M . As an application, we show that the modular group acts projectively on the Hochschild cohomology of a modular tensor category.