Let W be an exceptional spetsial irreducible reflection group acting

on a complex vector space V, i.e. , a group G_n for

n (...) {4, 6, 8, 14, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37}

in the Shephard-Todd notation. We describe how to determine

some data associated to the corresponding (split) "spets"

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 Uch(G) of "unipotent

characters" of 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).

We keep track of the complete list of axioms used. In order to

do that, we explain in detail some general axioms of "spetses", generalizing

(and sometimes correcting) our paper "Toward Spetses",

Transformation groups 4 (1999), along the way.

Note that to make the induction work, we must consider a class

of reflection cosets slightly more general than the split irreducible

ones: the reflection cosets with split semi-simple part, i.e. , cosets

( V, W Phi ) such that V = V ₁ (...) V ₂ with W (...) GL ( V ₁) and Phi| V ₁ = Id.

We need also to consider some non-exceptional cosets, those associated

to imprimitive complex reflection groups which appear as

parabolic subgroups of the exceptional ones.