Filtrations and positive characteristic Howe duality
Let \(k\) be a field and \(A\) a \(k\)-algebra, finite dimensional over \(k\). Index the simple left \(A\)-modules by a (necessarily finite) set \(\Omega\). For each \(\lambda\in\Omega\), let \(L(\lambda)\), \(P(\lambda)\), \(Q(\lambda)\) denote respectively the simple module corresponding to \(\lambda\), its projective cover, and its injective hull. Assume given a partial ordering \(\leq\) on \(\Omega\). For \(\lambda\in\Omega\), one defines the standard module, or Weyl module, \(\Delta(\lambda)\) as the quotient of \(P(\lambda)\) by the sum of all images \(P(\mu)\to P(\lambda)\) for all \(\mu\not\leq\lambda\) in \(\Omega\). Dually one defines the co-standard module \(\nabla(\lambda)\) as the intersection of all kernels of maps \(Q(\lambda)\to Q(\mu)\) for all \(\mu\not\leq\lambda\) in \(\Omega\). The data \((A,\Omega,\leq)\) is said to be quasi-hereditary provided that \(\leq\) is equivalent to a total ordering on \(\Omega\), the regular module \(_AA\) has a \(\Delta\)-filtration and for each \(\lambda\in\Omega\), \(L(\lambda)\) appears as a composition factor in \(\Delta(\lambda)\) precisely once. Quasi-hereditary algebras were originally introduced by Cline, Parshall and Scott in order to describe highest weight categories arising in the representation theory of complex Lie algebras and algebraic groups; their formulation of the definition was slightly different than, but equivalent to, the above one. The formulation here follows from \textit{C. M. Ringel} [Math. Z. 208, No. 2, 209-224 (1991; Zbl 0725.16011)] (see also [Lond. Math. Soc. Lect. Note Ser. 168, 200-224 (1992; Zbl 0793.16006)] by \textit{C. M. Ringel} and \textit{V. Dlab}). The category of finitely generated \(A\)-modules will be denoted by \(A\)-Mod. A module \(M\) in \(A\)-Mod is said to have a \(\Delta\)-filtration provided there is a sequence of submodules \(\{0\}=M_0\subset M_1\subset\cdots\subset M_r=M\) with \(M_i/M_{i-1}\simeq\Delta(\lambda_i)\) for some \(\lambda_i\in\Omega\) and for each \(i=1,2,\dots,r\). The notion of \(\nabla\)-filtration is analogous. The fact that \(A\) is quasi-hereditary is reflected in a number of ``homological'' properties. One has \[ \text{Ext}^i_A(\Delta(\lambda),\Delta(\mu))=\text{Ext}^i_A(\nabla(\lambda),\nabla(\mu))=0\quad\text{ for all \(i\geq 1\) if \(\lambda\not\leq\mu\).}\tag{\(*\)} \] Let \(\omega\) be the full subcategory of \(A\)-Mod consisting of modules having both a \(\Delta\)-filtration and a \(\nabla\)-filtration. Let \((A,\Omega,\leq)\) be a quasi-hereditary algebra. Then, \textit{C. M. Ringel} proved [in Math. Z., loc. cit.] that there exists a uniquely determined basic module \(T\) so that \(\omega\) is the full subcategory of \(A\)-Mod consisting of all direct sums of direct summands of \(T\). The modules in \(\omega\) are partial tilting modules. One has \(T\simeq\bigoplus_{\lambda\in\Omega}T(\lambda)\), for indecomposable non-isomorphic modules \(T(\lambda)\). Let \(R\) be a complete discrete valuation ring with maximal ideal \(\pi R\). Put \(k^{(i)}:=R/\pi^iR\) for \(i\geq 1\); in particular, \(k=k^{(1)}\) is the residue field. Let \(K\) be the field of fractions of \(R\). Let \(A_R\) be an admissible \(R\)-algebra (that is, one which is free of finite rank as an \(R\)-module). Put \(A_K:=A_R\otimes_RK\), \(A^{(i)}:=A_R\otimes_Rk^{(i)}\simeq A_R/\pi^iA_R\). An \(A_R\)-module \(M_R\) is said to be admissible in case it is a finite rank free \(R\)-module. Similarly, an \(A^{(i)}\)-module \(M^{(i)}\) is admissible if it is a finite rank free \(k^{(i)}\)-module. For \(j\geq 1\), one says that an admissible \(A_R\)-module \(X\) is a lift of the \(A^{(j)}\)-module \(X/\pi^j X\). For each \(\lambda\), there is a non-zero homomorphism \(c_\lambda\colon\Delta(\lambda)\to\nabla(\lambda)\) determined by the projection \(\Delta(\lambda)\to\text{head }\Delta(\lambda)=L(\lambda)\) and the inclusion \(L(\lambda)=\text{soc }\nabla(\lambda)\subset\nabla(\lambda)\). The author shows that \((*)\) implies that there are admissible lifts \(\Delta_R(\lambda)\), \(\nabla_R(\lambda)\), \(T_R(\lambda)\) of \(\Delta(\lambda)\), \(\nabla(\lambda)\), \(T(\lambda)\) respectively. Then he is able to define the Jantzen filtration of \(\Delta_R(\lambda)\) as \[ {\mathcal J}^i\Delta_R(\lambda):=\bigl\{x\in\Delta_R(\lambda):(x)c_\lambda\in\pi^i\nabla_R(\lambda)\bigr\}\quad\text{ for \(i\geq 0\).} \] The corresponding filtration of \(\Delta(\lambda)\) is obtained by taking the canonical image of each term of the filtration of \(\Delta_R(\lambda)\). The filtration \({\mathcal J}^i\) is independent of the choice of \(c_\lambda\); for each \(i\geq 1\), \({\mathcal J}^i\Delta_R(\lambda)\) is an \(A_R\)-submodule of \(\Delta_R(\lambda)\) and \({\mathcal J}^i\Delta(\lambda)\) is an \(A\)-submodule of \(\Delta(\lambda)\) and \(\Delta(\lambda)/{\mathcal J}^1\Delta(\lambda)\simeq L(\lambda)\). Let \(Q_R\) be an admissible \(R\)-lift of the partial tilting \(A\)-module \(Q\). Choose \(\lambda\in\Omega\), put \[ F_\lambda(Q_R):=\Hom_{A_R}(\Delta_R(\lambda),Q_R),\qquad E_\lambda(Q_R):=\Hom_{A_R}(Q_R,\nabla_R(\lambda)), \] and denote by \(F_\lambda(Q)\) (resp. \(E_\lambda(Q)\)) the reduction modulo \(\pi\) of the \(R\)-module \(F_\lambda(Q_R)\) (resp. \(E_\lambda(Q_R)\)). Ones defines a filtration of \(F_\lambda(Q_R)\) (the Andersen filtration) by \[ \mathcal A^jF_\lambda(Q_R):=\bigl\{\phi\in F_\lambda(Q_R):\phi\circ\psi\in c_\lambda\pi^jD_R^\lambda\;\forall\psi\in E_\lambda\bigr\}, \] where \(D_R^\lambda:=\text{End}_{A_R}(\Delta_R(\lambda))\). Assume now that \((A,\Omega,\leq,\iota)\) is a quasi-hereditary algebra with duality (then \(\iota\colon A\to A\) is an antiautomorphism, regarding it as an isomorphism \(\iota\colon A\to A^{\text{op}}\)), one gets the covariant equivalence \({\mathcal F}_\iota\colon A^{\text{op}}\text{-Mod}\to A\text{-Mod}\), on the other hand, one has a contravariant equivalence of categories \({\mathcal D}\colon A\text{-Mod}\to\text{Mod-}A=A^{\text{op}}\text{-Mod}\) given by \({\mathcal D}(A):=M^*\), and we set \(M^\vee:=({\mathcal F}_\iota\circ{\mathcal D})(M)\), and that \(Q\) is a self-dual full tilting \(A\)-module. For each \(\lambda\in\Omega\), evidently \(L(\lambda)^\vee\) is simple; it follows that there exists a function \(w\colon\Omega\to\Omega\) with the property that \(L(\lambda)^\vee\simeq L(w\lambda)\). Let \(A_R\) be an \(R\)-form of \(A\) and assume that \(\iota\) is defined over \(R\). Then the author proves the existence of an isomorphism \(F_\lambda(Q_R)\to\Delta_R'((w\lambda)')\) of \(A_R'\)-modules, where \(A_R':=\text{End}_{A_R}(Q_R)\). Then he applies his result to algebraic groups, and, in particular, to Howe dual pairs.
scientific article; zbMATH DE number 1559188