**Astérisque**

Volume: 359;
2014;
146 pp;
Softcover

MSC: Primary 20;
**Print ISBN: 978-2-85629-781-0
Product Code: AST/359**

List Price: $63.00

AMS Member Price: $50.40

# Split Spetses for Primitive Reflection Groups

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*Michel Broué; Gunter Malle; Jean Michel*

A publication of the Société Mathématique de France

Let \(W\) be an exceptional spetsial irreducible
reflection group acting on a complex vector space \(V\),
i.e., a group \(G{_n}\) for
\[n \in 4, 6, 8, 14, 23, 24, 25, 26, 27, 28,
29, 30, 32, 33, 34, 35, 36, 37\] in the Shephard-Todd notation.

The authors describe how to determine some data associated to the
corresponding (split) “spets” \(\mathbb{G} =(V,W)\), given
complete knowledge of the same data for all proper subspetses (the
method is thus inductive).

The data determined here are the set
\(\mathrm{Uch}(\mathbb{G})\) of “unipotent
characters” of \(\mathbb{G}\) and its repartition into
families, as well as the associated set of Frobenius eigenvalues. The
determination of the Fourier matrices linking unipotent characters and
“unipotent character sheaves” will be given in another
paper.

The approach works for all split reflection cosets
for primitive irreducible reflection groups. The result is that all the above
data exist and are unique (note that the cuspidal unipotent degrees are only
determined up to sign).

The authors keep track of the complete list of axioms used. In
order to do that, they explain in detail some general axioms of
“spetses”, generalizing (and sometimes correcting) along
the way.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in complex reflection groups, braid groups, Hecke algebras, finite reductive groups, and spetses.