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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”.

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