Harish-Chandra series of modules and modular Howe correspondence
Let \(G\) be a reductive algebraic group defined over the finite field \(\mathbb{F}_q\) and let \(K\) be a local field of characteristic zero with residue field \(k\) of characteristic \(\ell\), where \((\ell,q)=1\) and \(q\) is odd. The author begins by defining induction and restriction functors \(R^G_L\), \(^*R^G_L\) for any 1-split Levi component \(L\) (a Levi component \(L\) is 1-split if it is a rational Levi subgroup of a rational parabolic subgroup \(P\) of \(G\)). Then she defines ``Harish-Chandra series'' of simple \(kG\)-modules; this definition is consistent with the work of \textit{G. Hiss} [J. Lond. Math. Soc., II. Ser. 48, No. 2, 219-228 (1993; Zbl 0814.20004)]. The Harish-Chandra series are parametrized by simple cuspidal \(kL\)-modules. Related work has been done by \textit{R. Dipper} and \textit{J. Du} [J. Reine Angew. Math. 437, 101-130 (1993; Zbl 0768.20004)]. The second part of the paper gives an analogue for the functors \(R^G_L\), \(^*R^G_L\) of the results of \textit{D. Benson} and \textit{R. A. Parker} [J. Algebra 87, 290-331 (1984; Zbl 0539.20009)]. It shows that the Green ring of \(G\) is a direct sum of \(\text{image}(R^G_L)\) and \(\text{ker}(^*R^G_L)\). It requires a new method of proof, since \(R^G_L\) and \(^*R^G_L\) do not satisfy Frobenius reciprocity. If \(G\), \(G'\) are an irreducible reductive dual pair defined over \(\mathbb{F}_q\), it is possible to define the Weil representation. The third part of the article uses it to establish a ``modular Howe correspondence'' from \(kG\)-modules to \(kG'\)-modules. Under certain conditions, this correspondence is shown to preserve cuspidal modules.
scientific article; zbMATH DE number 589961