Character sheaves and characters of unipotent groups over finite fields. (Q2851614)

Let \(k\) be a perfect field of characteristic \(p>0\). If \(X\) is a perfect variety, let \(\mathcal D(X)=D_c^b(X,\overline{\mathbb Q}_\ell)\) denote the bounded derived category of constructible complexes of \(\overline{\mathbb Q}_\ell\)-sheaves on \(X\). If \(G\) is a perfect unipotent group over \(k\) acting on \(X\), let \(\mathcal D_G(X)\) be the \(G\)-equivariant derived category defined by the author in his previous works. In particular, one may consider the derived category \(\mathcal D_G(G)\) for the conjugation action of \(G\) on itself.NEWLINENEWLINE The categories \(\mathcal D(G)\) and \(\mathcal D_G(G)\) are monoidal with respect to the bifunctor of convolution with compact supports: NEWLINE\[NEWLINE(M,N)\mapsto M*N:=\mu_!((p_1^* M)\otimes (p_2^*N)),NEWLINE\]NEWLINE where \(\mu\colon G\times G\to G\) is the multiplication morphism and \(p_1,p_2\colon G\times G\to G\) are the first and second projections. The unit object in each of these categories is the delta-sheaf at the identity element of \(G\), which will be denoted by \(\mathbf 1\).NEWLINENEWLINE An object \(e\) of \(\mathcal D_G(G)\) is called a \textit{closed idempotent} if there exists an arrow \(\mathbf 1\to e\) that becomes an isomorphism after convolution with \(e\). It is further said to be a \textit{minimal} closed idempotent if \(e\neq 0\) and for every closed idempotent \(e'\in\mathcal D_G(G)\), we have either \(e*e'\simeq e\), or \(e* e'=0\).NEWLINENEWLINE Assume from now on that \(k\) is algebraically closed. Let \(G\) be a perfect unipotent group over \(k\), and let \(e\) be a closed idempotent. The author defines the \textit{Hecke subcategory} defined by \(e\) to be the full subcategory \(e\mathcal D_G(G)\) of \(\mathcal D_G(G)\) consisting of objects \(M\) such that \(e*M\simeq M\). In a previous article joint with Drinfeld the author proved that the category \(e\mathcal D_G(G)\) is closed under \(*\) and is a monoidal category with unit object \(e\).NEWLINENEWLINE If \(e\) is minimal, let \(\mathcal M_e^{\text{perv}}\) denote the full subcategory of \(e\mathcal D_G(G)\) consisting of those objects for which the underlying \(\ell\)-adic complex is a perverse sheaf on \(G\). The category \(\mathcal M_e^{\text{perv}}\) is a semisimple Abelian category with finitely many simple objects. Define the \textit{Lusztig packet of character sheaves} on \(G\) (or \textit{\(\mathbb L\)-packet} for short) to be the set of (isomorphism classes of) indecomposable objects of the category \(\mathcal M_e^{\text{perv}}\). An object of \(\mathcal D_G(G)\) is called a \textit{character sheaf} if it lies in the Lusztig packet of characters sheaves defined by some minimal closed idempotent in \(\mathcal D_G(G)\).NEWLINENEWLINE Fix an algebraic closure \(\mathbb F\) of a field \(\mathbb F_p\) with \(p\) elements. Let \(q\) be a finite power of \(p\). If \(X_0\) is a perfect variety over \(\mathbb F_q\) and \(M_0\in\mathcal D(X_0)\), let \(t_{M_0}\colon X_0(\mathbb F_q)\to\overline{\mathbb Q}_\ell\) denote the function associated to \(M_0\) via the functions-sheaves dictionary.NEWLINENEWLINE Let \(G_0\) be a perfect unipotent group over \(\mathbb F_q\), and write \(G:=G_0\otimes_{{\mathbb F_q}}\mathbb F\). Given a class \(\alpha\in H^1(\mathbb F_q,G_0)\) in Galois cohomology, one can define the corresponding pure inner form \(G_0^\alpha\); it is a perfect group over \(\mathbb F_q\) which becomes isomorphic to \(G_0\) over \(\mathbb F\). Moreover, there is a natural monoidal equivalence \(\mathcal D_{G_0}(G_0)\simeq\mathcal D_{G_0^\alpha}(G_0^\alpha)\), which will be denoted \(M_0\mapsto M_0^\alpha\) and called the transport of equivariant complexes.NEWLINENEWLINE An object \(e_0\) of \(\mathcal D_{G_0}(G_0)\) is called a \textit{geometrically minimal idempotent} if \(e_0*e_0\simeq e_0\) and the corresponding object \(e\) in \(\mathcal D_G(G)\) is a minimal idempotent.NEWLINENEWLINE Given \(\alpha\in H^1(\mathbb F_q,G_0)\), we have the corresponding pure inner form \(G_0^\alpha\) and the geometrically minimal idempotent \(e_0^\alpha\in\mathcal D_{G_0^\alpha}(G_0^\alpha)\) obtained by \(e_0\) via transport of equivariant complexes; in particular, the function NEWLINE\[NEWLINEt_{e_0^\alpha}\colon G_0^\alpha(\mathbb F_q)\to\overline{\mathbb Q}_\ellNEWLINE\]NEWLINE is a central idempotent with respect to convolution.NEWLINENEWLINE The disjoint union, for \(\alpha\in H^1(\mathbb F_q,G_0)\), of the sets of (isomorphism classes of) irreducible representations of \(G_0^\alpha(\mathbb F_q)\) over \(\overline{\mathbb Q}_\ell\) on which the idempotent \( t_{e_0^\alpha}\) acts via identity is called the \textit{\(\mathbb L\)-packet of irreducible representations of \(G_0\) defined by \(e_0\)}. The set of characters of the representations belonging to an \(\mathbb L\) will be called an \textit{\(\mathbb L\)-packet of irreducible characters of \(G_0\)}.NEWLINENEWLINE The article studies the relationship between irreducible characters of \(G_0(\mathbb F_q)\) and character sheaves on \(G\) that are invariant under the Frobenius endomorphism \(\text{Fr}_q\colon G\to G\). The relationship is easier to formulate when \(G_0\) is connected. In this case the number of \(\text{Fr}_q^*\)-invariant character sheaves \(M\) on \(G\) equals the number of irreducible characters of \(G_0({\mathbb{F}_{q}})\), and every such \(M\) comes from an irreducible perverse sheaf \(M_0\) on \(G_0\) such that \(M_0\) is pure of weight \(0\) and the function \(t_{M_0}\) takes values in \(\mathbb Q^{\text{ab}}\), the Abelian closure of \(\mathbb Q\). With this choice of \(M_0\), the author proves (see Theorem 1.8) that as \(M\) ranges over the set of \(\text{Fr}_q^*\)-invariant character sheaves on \(G\), the functions \(t_{M_0}\) form a basis of the space of class functions from \(G_0(\mathbb F_q)\) to \(\mathbb Q^{\text{ab}}\), which are orthonormal with respect to the unnormalized standard inner product \(\langle f_1,f_2\rangle:=\sum_{g\in G_0(\mathbb F_q)}f_1(g)\overline{f_2(g)}\). The matrix which relates this basis to the basis formed by the irreducible characters of \(G_0(\mathbb F_q)\) is block-diagonal, with blocks labelled by the \(\mathbb L\)-packets (see Theorem~2.17).NEWLINENEWLINE The results of the paper are formulated and proven for \(G_0\) an arbitrary unipotent group over \(\mathbb F_q\). In contrast to the case when \(G_0\) is connected, in general there may be more \(\text{Fr}_q^*\)-invariant character sheaves on \(G\) than there are irreducible characters of \(G_0(\mathbb F_q)\). It is worth to note that even if ultimately one is only interested in the connected case, studying arbitrary unipotent groups seems to be necessary. For instance, the study of \(\mathbb L\)-packets of irreducible characters of \(G_0(\mathbb F_q)\) leads to consider characters of subgroups \(G_0'(\mathbb F_q)\) of \(G_0(\mathbb F_q)\) for certain closed subgroups \(G'_0\) of \(G_0\) that may be disconnected.