Heisenberg idempotents on unipotent groups. (Q2275705)

Let \(k\) be an algebraically closed field of positive characteristic. Let \(G\) be a smooth algebraic group of finite type over \(k\) such that its neutral component \(H:=G^0\) is unipotent. Following \textit{M. Boyarchenko}, [Sel. Math., New Ser. 16, No. 4, 857-933 (2010; Zbl 1223.20038)], the author introduces \(\mathcal D(G)\), the bounded derived category of \(\overline{\mathbb Q}_\ell\)-complexes on \(G\), as well as their equivariant versions \(\mathcal D_G(G)\), \(\mathcal D_H(G)\) relative to the actions of \(G\) and \(H\), respectively. In op. cit., Boyarchenko introduced a closed idempotent \(e\in\mathcal D_G(G)\) by fixing an admissible pair \((N,\mathcal L)\). These idempotents are called Heisenberg idempotents and play a vital role in the study of irreducible characters of unipotent groups over finite fields and their relation with character sheaves. Let \(\mathcal M_e^{\mathrm{perv}}\) be the full subcategory of \(e\mathcal D_H(G)\) consisting of perverse sheaves and \(\mathcal M_e:=\mathcal M_e^{\mathrm{perv}}[\dim N]\). In the article under review, the authors proves three theorems on the structure of \(\mathcal M_e^{\mathrm{perv}}\) and \(\mathcal M_e\) (Theorem 1.1, 1.2 and 1.3). In particular, he shows that \(\mathcal M_e\) is a fusion category under the convolution product. Set \(\Gamma:=G/H\), it is also shown that the \(\Gamma\)-equivariantization \(\mathcal M^\Gamma_e\) is a braided fusion category. This verifies some conjectures of Drinfeld.