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Mackey formula in type \(A\). (English) Zbl 1037.20051
Proc. Lond. Math. Soc., III. Ser. 80, No. 3, 545-574 (2000); Corrigenda 86, No. 2, 435-442 (2003).
Let \(G\) be a connected, reductive algebraic group defined over a finite field with \(q\) elements. \(G\) is of type \(A\) if all the irreducible components of its root system are of type \(A\). In 2000 the author proved that the Mackey formula holds in groups of type \(A\).
The author noticed two errors in that paper. They concern Theorem 4.1.1 and Formulas (5.1.7) and (5.1.8). The sign “\(+\)” in these formulas must be changed to “\(-\)”. This has no consequence concerning the other results. Theorem 4.1.1 is false. However, its Corollary 4.1.2 is still correct (see Theorem 6 below) and only this corollary is used in the rest of that paper.
Theorem 6. If the Mackey formula holds in \(G\), then \(Cus_{uni}(G^F)\) and \({\mathcal CUS}_{uni}(G^F)\) are isomorphic as \(\overline\mathbb{Q}_l\text{Out}(G,f)\)-modules.
Note that in Corollary \(4.1.2(b)\) the term “cuspidal function” must be replaced by “absolutely cuspidal function”.

20G40 Linear algebraic groups over finite fields
20C15 Ordinary representations and characters
20G05 Representation theory for linear algebraic groups
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