centertableaux

# Rational Catalan Numbers for Complex Reflection Groups

Weston Miller  University of Texas at Dallas<EMAIL_ADDRESS>
###### Abstract.

Assuming standard conjectures, we show that the canonical symmetrizing trace
evaluated at powers of a Coxeter element produces rational Catalan numbers for
irreducible spetsial complex reflection groups. This extends a technique used
by Galashin, Lam, Trinh, and Williams to uniformly prove the enumeration of
their noncrossing Catalan objects for finite Coxeter groups.

##  1\. Introduction

###  1.1. Catalan combinatorics

††This work was supported in part by the National Science Foundation under
grant DMS-2246877

The prototypical noncrossing Coxeter-Catalan objects are the _noncrossing
partitions_. In type $A$ with the usual Coxeter element $c=(1,2,\ldots,n)$,
these correspond to partitions $\\{B_{1},\dots,B_{k}\\}$ of the set
$\\{1,2,\dots,n\\}$ such that there do not exist $a<b<c<d$ such that $a,c\in
B_{i}$ and $b,d\in B_{j}$ with $i\neq j$. These type $A$ noncrossing
partitions are counted by
$\operatorname{Cat}_{n}:=\frac{1}{n+1}\binom{2n}{n}$.

More generally, let $W$ be a finite Coxeter group acting in the reflection
representation $V$, and let $c$ be a Coxeter element of $W$. The _$c$
-noncrossing partition lattice_ $\mathrm{NC}(W,c)$ is the interval $[e,c]_{T}$
in the absolute order, and the number of noncrossing partitions is counted by
the _Coxeter-Catalan number_

$\operatorname{Cat}(W):=\prod_{i=1}^{n}\frac{h+1+e_{i}}{d_{i}},$

where $h=d_{n}$ is the Coxeter number of $W$, $d_{1}\leq\cdots\leq d_{n}$ are
the degrees of a set of algebraically independent homogeneous polynomials
which generate the algebra of invariants $\operatorname{Sym}(V^{*})^{W}$, and
$e_{i}=d_{i}-1$.

In type $A$, the _rational Catalan numbers_
$\operatorname{Cat}_{a,b}:=\frac{1}{a+b}\binom{a+b}{a}$ count _rational Dyck
paths_ —lattice paths in an $a\times b$ rectangle (for $a$ and $b$ coprime)
that stay above the diagonal. For $a=n$ and $b=n+1$, this recovers the usual
Catalan numbers. In other types, these rational Dyck paths are generalized to
certain _nonnesting objects_ , but such objects are only defined for Weyl
groups.

Figure 1. The rational Dyck paths counted by $\operatorname{Cat}_{3,5}$

In [GLTW22], the authors define rational noncrossing Coxeter-Catalan objects
called _maximal $\mathbf{c}^{p}$-Deograms_, counted by the _rational Coxeter-
Catalan numbers_

$\operatorname{Cat}_{p}(W):=\prod_{i=1}^{n}\frac{p+(pe_{i}\operatorname{mod}h)}{d_{i}},$

where $p$ is coprime to $h$. These objects are defined for finite Coxeter
groups. As part of their type-uniform proof of this enumeration, they use
Hecke algebra traces to compute the point count of braid Richardson varieties
over a finite field, producing $q$-deformed rational Catalan numbers:

$\operatorname{Cat}_{p}(W;q):=\prod_{i=1}^{n}\frac{[p+(pe_{i}\operatorname{mod}h)]_{q}}{[d_{i}]_{q}}.$

Many of the objects used in their proof can also be defined for the well-
generated complex reflection groups, so it is natural to try to compute these
traces in the complex case. It turns out that the well-generated condition is
too weak for certain representation-theoretic techniques to work—the necessary
condition is for the group to be spetsial. These spetsial complex reflection
groups (see Definition 3.14) are well-generated complex reflection groups that
behave as if they were the Weyl group for some connected reductive algebraic
group. In the Shephard-Todd classification, the _spetsial_ groups are

$W(A_{n-1}),\quad G(m,1,n),\quad G(m,m,n),\quad G_{i}\text{ where
}i\in\\{4,6,8,14,23,\dots,30,32,\dots,37\\}.$

Analogs of unipotent characters, Harish-Chandra theory, and Lusztig’s Fourier
transform can be defined combinatorially for these groups, which allows
techniques from the representation theory of finite groups of Lie type to be
extended to spetsial complex reflection groups.

As the main result of this paper, Theorem 7.1, we show that for irreducible
spetsial complex reflection groups the trace of a power of a Coxeter element
still produces a rational Catalan number even though there are not braid
Richardson varieties in this context. Precisely, we prove the following
result:

######  Theorem 1.1.

Let $W$ be an irreducible spetsial complex reflection group with Coxeter
number $h$, and let $c$ be a $\zeta_{h}$-regular element of $W$. Let
$\mathbf{c}\in B(W)$ be a lift of $c$ such that $\mathbf{c}^{h}=\bm{\pi}$.
Then

$\tau_{q}(T_{\mathbf{c}}^{-p})=q^{-np}(1-q)^{n}\operatorname{Cat}_{p}(W;q).$

###  1.2. Parking combinatorics

The _noncrossing parking functions_ in type $A$, called 2-partitions in
[Ede80], are sets of tuples $\\{(B_{1},L_{1}),\dots,(B_{k},L_{k})\\}$, where
$B_{i},L_{i}\subseteq\\{1,\dots,n\\}$ and

  * •

$\\{B_{1},\dots,B_{k}\\}$ is a noncrossing partition of $\\{1,\dots,n\\}$,

  * •

$\\{L_{1},\dots,L_{k}\\}$ is a set partition of $\\{1,\dots,n\\}$, and

  * •

$|B_{i}|=|L_{i}|$ for $i=1,\dots,k$.

The number of these noncrossing parking functions is $(n+1)^{n-1}$.

More generally, for $W$ a finite Coxeter group, let $\mathcal{L}$ be the
lattice of _flats_. That is, the elements of $\mathcal{L}$ are the
intersections of collections of reflection hyperplanes, and they are ordered
by reverse inclusion. Then by [BW02] there is an embedding
$\mathrm{NC}(W,c)\hookrightarrow\mathcal{L}$. Define an equivalence relation
on $W\times\mathcal{L}$ by setting $(w,X)\sim(w^{\prime},X^{\prime})$ if
$X=X^{\prime}$ and $wW_{X}=w^{\prime}W_{X}$, where $W_{X}$ is the pointwise
stabilizer of $X$. Denote by $[w,X]$ the equivalence class of $(w,X)$. Then
the _$W$ -noncrossing parking functions_ defined in [ARR15] are

$\operatorname{Park}^{\mathrm{NC}}_{W}:=\\{[w,X]:w\in W\text{ and
}X\in\mathrm{NC}(W,c)\\},$

and they are counted by $(h+1)^{n}$.

In type $A$, the _rational nonnesting parking functions_ for $a$ and $b$
coprime are rational Dyck paths in an $a\times b$ rectangle with a labelling
of the north steps with the numbers $\\{1,\dots,a\\}$, such that the labels
increase in each column going north. These parking functions are counted by
$b^{a-1}$ [ALW16, Corollary 4]. As with rational Catalan objects, nonnesting
parking objects again are only defined for Weyl groups.

24153 Figure 2. A $(5,7)$-parking function

In [GLTW22], the authors uniformly defined rational noncrossing parking
objects for finite Coxeter groups. These parking objects correspond to certain
walks in the Hasse digram of the weak Bruhat order and are counted by $p^{n}$.
As a corollary of our main result, we prove the following (see Corollary 7.4):

######  Corollary 1.2.

For $W$ an irreducible spetsial complex reflection group, let $\mathcal{B}$ be
a basis of the spetsial Hecke algebra $\mathcal{H}_{q}(W)$ (adapted to the
Wedderburn decomposition), and let $\mathbf{c}$ be a lift of a
$\zeta_{h}$-regular element such that $\mathbf{c}^{h}=\mathbf{\pi}$. Then

$\sum_{b\in\mathcal{B}}\tau_{q}(b^{\vee}T_{\mathbf{c}^{p}}b)=(q-1)^{n}[p]_{q}^{n}.$

In the real case, this is a key algebraic step in the proof of the enumeration
of rational noncrossing parking functions [GLTW22, Corollary 6.15]. Finding a
combinatorial interpretation of the left-hand-side of this equation, e.g.
rational noncrossing parking functions for spetsial complex reflection groups,
is an open problem.

###  1.3. Outline of the paper

In Section 2 and Section 3 we cover the necessary background on complex
reflection groups, braid groups, and Hecke algebras. A key result in these
sections is Tits’ deformation theorem (Theorem 3.11) which links the
representation theory of a complex reflection group with that of its Hecke
algebra. We also show in Theorem 3.17 that the exterior powers of Galois
twists of the reflection representation are precisely where the generic
degrees fail to vanish when evaluated at a root of unity. Some data associated
to these representations is presented throughout the paper as a running
example.

Then, in Section 4 we give some brief background on the representation theory
of Lie type to motivate the constructions in Section 5 and Section 6. In
particular, we introduce the uniform almost characters, principal series
characters, and families of unipotent characters.

The construction of unipotent characters and generic degrees for the infinite
families are then described in Section 5. We also highlight the role of
Clifford theory in describing the representation theory of $G(m,m,n)$. There
is a focus throughout this section on computing data for the exterior powers
of Galois twists. In particular, the proof of Theorem 3.17 for the infinite
families appears in this section.

The key lemma for our main result is the focus of Section 6. We describe
Lusztig’s original construction of a Fourier transform for Weyl groups, then
discuss generalizations to finite Coxeter groups and spetsial complex
reflection groups. Finally, in Section 7 we present our main result and a
corollary related to parking combinatorics.

For the exceptional groups, some of our results are checked by computer
[MW23].

### Acknowledgments

The author would like to thank Nathan Williams for his guidance, support, and
many helpful discussions.

##  2\. Complex reflection groups

Most of the background material in this section can be found in [LT09]. Let
$V$ be a finite-dimensional complex vector space. A linear transformation
$g\in\mathrm{GL}(V)$ is a _reflection_ if the order of $g$ is finite and the
subspace $\mathrm{Fix}(g):=\\{v\in V:gv=v\\}$ has codimension 1. In this case,
$\mathrm{Fix}(g)$ will be called the _reflection hyperplane_ of $g$.

######  Definition 2.1.

A _(finite) complex reflection group_ is a finite subgroup of $\mathrm{GL}(V)$
that is generated by reflections.

For a complex reflection group $W$, we will use $\mathcal{R}$ to denote the
set of reflections in $W$, and $\mathcal{A}$ will denote the corresponding set
of reflecting hyperplanes (for Coxeter groups, $|\mathcal{R}|=|\mathcal{A}|$).

For any finite subgroup $G$ of $\mathrm{GL}(V)$, there exists a $G$-invariant
inner product on $V$. So we may assume that a complex reflection group
$G\subset\mathrm{GL}(V)$ is a subgroup of the unitary group $\mathrm{U}(V)$
for an appropriate inner product. Moreover, one can show that finite subgroups
of $\mathrm{U}(V)$ are conjugate in $\mathrm{U}(V)$ if and only if they are
conjugate in $\mathrm{GL}(V)$.

######  Definition 2.2.

A complex reflection group $W\subset\mathrm{U}(V)$ is an _irreducible complex
reflection group_ if $V$ is an irreducible $W$-module.

For an orbit of hyperplanes $\mathcal{C}\in\mathcal{A}/W$, we will let
$e_{\mathcal{C}}$ denote the order of the pointwise stabilizer $W_{H}=\\{w\in
W:wh=h,\,\,\forall h\in H\\}$ for any $H\in\mathcal{C}$ (the order does not
depend on the choice of $H$). For any $H\in\mathcal{C}$, the group $W_{H}$ is
cyclic with order $e_{\mathcal{C}}$, and there is a reflection $s_{H}\in W$
with reflecting hyperplane $H$ and determinant
$\zeta_{e_{\mathcal{C}}}=\exp(2\pi i/e_{\mathcal{C}})$. Such reflections are
called _distinguished reflections_.

The _field of definition_ $k_{W}$ of a complex reflection group $W$ is the
field generated by the traces of the elements of $W$ on the reflection
representation. The field of definition is a subfield of $\mathbb{R}$ when $W$
is a finite Coxeter group and equals $\mathbb{Q}$ when $W$ is a Weyl group. It
is a theorem of [Ben76] and [Bes97] that every representation of $W$ is
definable over its field of definition.

###  2.1. Classification of complex reflection groups

Let $\mu_{m}$ denote the cyclic subgroup of $\mathbb{C}^{\times}$ consisting
of the $m$th roots of unity. If $G$ is a group acting on the set
$\\{1,2,\dots,n\\}$ and $V$ is an $n$-dimensional complex inner product space
with orthonormal basis $e_{1},\dots,e_{n}$, the _standard monomial
representation_ of $\mu_{m}\wr G$ is given by

$(h,g)e_{i}=h_{g(i)}e_{g(i)},$

where $h=(h_{1},\dots,h_{n})\in\mu_{m}^{n}$.

For $p$ a divisor of $m$, let
$A(m,p,n)=\\{(h_{1},\dots,h_{n})\in\mu_{m}^{n}:(h_{1}\cdots h_{n})^{m/p}=1$}.
Let $G(m,p,n)=A(m,p,n)\rtimes S_{n}$. The group $G(m,p,n)$ is a subgroup of
$\mu_{m}\wr S_{n}$, so it may be represented as a group of linear
transformations in the standard monomial representation.

######  Definition 2.3.

A complex reflection group $W\subset U(V)$ is _imprimitive_ if for some $m>1$,
there is a direct sum $V=V_{1}\oplus V_{2}\oplus\cdots\oplus V_{m}$ of non-
zero subspaces $V_{i}$ such that the action of $W$ on $V$ permutes the
subspaces $V_{1}$, $V_{2}$, …, $V_{m}$. If $W$ is not imprimitive, then it is
_primitive_.

######  Proposition 2.4 ([LT09], Proposition 2.10).

The groups $G(m,p,n)$ with $n>1$ are imprimitive complex reflection groups.
$G(m,p,n)$ is irreducible except for the groups $G(1,1,n)$ and $G(2,2,2)$

######  Proposition 2.5 ([LT09], Proposition 2.14).

If $V$ is an $n$-dimensional complex inner product space and $G$ is an
irreducible imprimitive subgroup of $U(V)$, which is generated by reflections,
then $n>1$ and $G$ is conjugate to $G(m,p,n)$ for some $m>1$ and some divisor
$p$ of $m$.

The group $G(m,p,1)$ is the cyclic group of order $m/p$ and is an irreducible
primitive complex reflection group. The group $G(1,1,n)$ is the symmetric
group $S_{n}$. Let $W(A_{n-1})$ be the complex reflection group gotten by
restricting the action of $G(1,1,n)$ to the hyperplane orthogonal to
$e_{1}+\cdots+e_{n}$. Then $W(A_{n-1})$ is irreducible, and it is primitive
when $n\geq 5$.

By the classification due to Shephard and Todd, an irreducible complex
reflection group is conjugate in $\mathrm{U}(V)$ to a group $G(m,p,n)$ or one
of 34 exceptional (primitive) groups (see [LT09, §8.7]), which are numbered
$G_{4},G_{5},\dots,G_{37}$.

###  2.2. Degrees and exponents

For a complex reflection group $W$ with reflection representation $V$ of
dimension $n$, let $S:=\text{Sym}(V^{*})\simeq\mathbb{C}[x_{1},\dots,x_{n}]$
be the symmetric algebra of $V^{*}$. Then $W$ acts on $S$ via $(g\cdot
p)(v)=p(g^{-1}v)$ for $g\in W$, $p\in S$, and $v\in V$. By the Shephard-Todd-
Chevalley theorem (see [Che55]) the _algebra of invariants_ $S^{W}$ is
generated by a collection of algebraically independent homogeneous
polynomials. The _degrees_ of $W$ are defined to be the degrees
$d_{1}\leq\cdots\leq d_{n}$ of these generators. The Poincaré polynomial
$P_{W}$ is defined by

$P_{W}:=\prod_{j=1}^{n}[d_{j}]_{q},$

where $[n]_{q}$ denotes the $q$-analog $[n]_{q}:=(q^{n}-1)/(q-1)$ for
$n\in\mathbb{Z}$.

Let $S^{W}_{+}$ be the ideal of $S$ generated by the positive degree elements
of $S^{W}$. Then the action of $W$ on the _coinvariant algebra_ $S/S^{W}_{+}$
is the regular representation, so $S/S^{W}_{+}$ contains exactly $r$ copies of
any irreducible representation $M$ of $W$ of dimension $r$ (see [LT09] Section
3.6). The _exponents_ of $M$ are defined to be the degrees
$e_{1}(M)\leq\cdots\leq e_{r}(M)$ of the homogeneous components of
$S/S^{W}_{+}$ containing a copy of $M$. If $\chi$ is the irreducible character
corresponding to $M$, we will also denote $e_{i}(\chi):=e_{i}(M)$.

######  Definition 2.6.

The degrees of $W$ satisfy $d_{i}=e_{i}+1$ for $i=1,\dots,n$, where
$e_{i}:=e_{i}(V)$. One can define the _codegrees_ $d_{1}^{*}\geq\cdots\geq
d_{n}^{*}$ of $W$ by $d_{i}^{*}=e_{n-i+1}(V^{*})-1$. An irreducible complex
reflection group $W$ is _well-generated_ if the degrees and codegrees satisfy
$d_{i}+d_{i}^{*}=d_{n}$. Equivalently, $W$ is well-generated if and only if it
can be generated by $n$ reflections.

It is a theorem of Shephard and Todd (see [LS99, Theorem 4.14]) that the the
degrees and exponents satisfy $|W|=d_{1}\cdots d_{n}$ and
$|\mathcal{R}|=e_{1}+\cdots+e_{n}$.

######  Definition 2.7.

Let $W$ be an irreducible complex reflection group. The _Coxeter number_ of
$W$ is

$h:=\frac{|\mathcal{R}|+|\mathcal{A}|}{n}.$

If $W$ is well-generated, then $h=d_{n}$.

More generally, following [GG12] and [Dou18], define the _generalized Coxeter
number_ $h_{\chi}$ associated to a character $\chi$ to be the normalized trace
of the central element $\sum_{r\in\mathcal{R}}(1-r)$. That is,

$h_{\chi}=\frac{1}{\chi(1)}\sum_{r\in\mathcal{R}}(\chi(1)-\chi(r))=|\mathcal{R}|-\frac{1}{\chi(1)}\sum_{r\in\mathcal{R}}\chi(r).$

Equivalently (see [Dou18, Remark 4.15]),

$h_{\chi}=\frac{N(\chi)+N(\chi^{*})}{\chi(1)},$

where $N(\chi)$ is the sum of the exponents of $\chi^{*}$ (see [BM97b, §4.B]).
These generalized Coxeter numbers are integers by [Dou18, Corollary 4.16], and
$h_{\phi}=h$ when $\phi$ is the character of the reflection representation. In
[Tri21] and [GLTW22], the authors define the _content_ $c(\chi)$ of $\chi$ as

$c(\chi):=\frac{1}{\chi(1)}\sum_{r\in\mathcal{R}}\chi(r)=|\mathcal{R}|-h_{\chi}.$

######  Remark 2.8.

For the symmetric groups, the content of a character $\chi$ is equal to the
content of the partition of the integer partition
$(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq 0)$ corresponding to $\chi$. That
is,

$c(\chi)=\sum_{i=1}^{\infty}\sum_{j=1}^{\lambda_{i}}(j-i).$

∎

######  Example 2.9 (Galois twists).

Let $W$ be a well-generated irreducible complex reflection group with Coxeter
number $h$ and reflection representation $V$ of dimension $n$. For well-
generated groups, we have the containment
$k_{W}\subseteq\mathbb{Q}(\zeta_{h})$. Every element
$\sigma\in\operatorname{Gal}(\mathbb{Q}(\zeta_{h})/\mathbb{Q})$ acts as
$\sigma_{p}:\zeta_{h}\mapsto\zeta_{h}^{p}$ for some $p$ coprime to $h$. The
_Galois twist_ $V^{\sigma_{p}}$ of $V$ is the irreducible representation of
$W$ obtained by applying $\sigma_{p}$ to the matrices representing the
elements $w\in W$ as linear operators on $V$. These Galois twists may not be
distinct—different choices of $p$ may result in the same representation.

If $r\in\mathcal{R}$, then it has a unique eigenvalue $\zeta\neq 1$ (which is
a root of unity) and all other eigenvalues are 1. One can check case-by-case
that the order of $\zeta$ divides $h$ in the exceptional groups. In the
infinite family $G(m,p,n)$, either $\zeta$ is an $(m/p)$th root of unity (in
which case the order of $\zeta$ divides $h=\max\\{(n-1)m,mn/p\\}$) or
$\zeta=-1$ (see [LS99, Lemma 2.8]). Let $\chi_{k,p}$ denote the character of
the irreducible representation $\Lambda^{k}V^{\sigma_{p}}$ for $p$ coprime to
$h$ and $k\in\\{0,1,\dots,n\\}$. Then

$\chi_{k,p}(r)=e_{k}(\bar{\zeta},1,\dots,1)=\binom{n-1}{k-1}\bar{\zeta}+\binom{n-1}{k},$

where

$e_{k}(x_{1},\dots,x_{n})=\sum_{i_{1}<\cdots<i_{k}}x_{i_{1}}\cdots x_{i_{k}}$

is the $k$th elementary symmetric polynomial on $n$ variables and

$\bar{\zeta}=\sigma_{p}(\zeta)=\begin{cases}\zeta^{p}&\text{if
}\zeta^{h}=1,\\\ -1&\text{if }\zeta=-1.\end{cases}$

Let $h_{k,p}$ be the generalized Coxeter number of $\chi_{k,p}$. Hence

$\displaystyle h_{k,p}$
$\displaystyle=|\mathcal{R}|-\frac{1}{\binom{n}{k}}\sum_{\mathcal{C}\in\mathcal{A}/W}|\mathcal{C}|\sum_{j=1}^{e_{\mathcal{C}}-1}\left(\binom{n-1}{k-1}\bar{\zeta}_{e_{\mathcal{C}}}^{j}+\binom{n-1}{k}\right)$
$\displaystyle=|\mathcal{R}|-\frac{1}{\binom{n}{k}}\sum_{\mathcal{C}\in\mathcal{A}/W}|\mathcal{C}|\left(-\binom{n-1}{k-1}+(e_{\mathcal{C}}-1)\binom{n-1}{k}\right)$
$\displaystyle=|\mathcal{R}|-\frac{1}{\binom{n}{k}}\left(-\binom{n-1}{k-1}|\mathcal{A}|+\binom{n-1}{k}|\mathcal{R}|\right)$
$\displaystyle=|\mathcal{R}|+\frac{k}{n}|\mathcal{A}|-\left(1-\frac{k}{n}\right)|\mathcal{R}|$
$\displaystyle=kh.$

######  Definition 2.10.

A vector $v\in V$ is _regular_ if it is not contained in any reflection
hyperplane. Let $V^{\text{reg}}:=V\setminus\bigcup_{H\in\mathcal{A}}H$ denote
the set of regular vectors. An element $c\in W$ is _regular_ if it has a
regular eigenvector. Moreover, $c$ is _$\zeta$ -regular_ if this eigenvector
may be chosen to have eigenvalue $\zeta$. In this case, the multiplicative
order $d$ of $\zeta$ is a _regular number_ for $W$.

The following theorem was first proven case-by-case [LS99, Theorem C], but was
later proven uniformly [LM03, Theorem 3.1].

######  Theorem 2.11.

A positive integer $d$ is a regular number of $W$ if and only if it divides
the same number of degrees as codegrees.

If $W$ is well-generated, then the Coxeter number $h$ is a regular number
because it divides only $d_{n}$ and $d_{1}^{*}$. It follows that there exists
a $\zeta$-regular element for every $h$-th root of unity $\zeta$ (see [LT09,
Remark 11.23]). For $\zeta$ a primitive $h$-th root of unity, the
$\zeta$-regular elements of $W$ are called _Coxeter elements_.

######  Remark 2.12.

Some authors define the Coxeter elements to be only the $\zeta_{h}$-regular
elements. In several cases, we will find it useful to restrict our attention
to this subset of the Coxeter elements. It follows easily from the definition
that each Coxeter element is a power of some $\zeta_{h}$-regular element. ∎

######  Definition 2.13.

The _fake degree_ $\operatorname{Feg}_{\chi}(q)$ of an irreducible character
$\chi$ of $W$ is the graded multiplicity of the irreducible representation
with character $\chi$ in $S/S^{W}_{+}$:

$\operatorname{Feg}_{\chi}(q)=\sum_{i=1}^{r}q^{e_{i}(\chi)}.$

######  Remark 2.14.

The reader should be aware that some authors use the symmetric algebra
$\operatorname{Sym}(V)$ rather than $\operatorname{Sym}(V^{*})$ to define fake
degrees $\overline{\operatorname{Feg}}_{\chi}(q)$. In particular, this
alternative definition is used in GAP3. These two definitions are related by

$\operatorname{Feg}_{\chi}(q)=\overline{\operatorname{Feg}}_{\chi^{*}}(q).$

In the real case, the two definitions coincide. ∎

######  Example 2.15 (Galois twists).

Let $W$ be a well-generated irreducible complex reflection group with
reflection representation $V$ and Coxeter number $h$. By [OS80, Theorem 2.13,
Corollary 3.2], the fake degree of $\Lambda^{k}V^{\sigma_{p}}$ is given by

$\sum_{i_{1}<\cdots<i_{k}}q^{e_{i_{1}}(V^{\sigma_{p}})+\cdots+e_{i_{k}}(V^{\sigma_{p}})}.$

Moreover, the sets $\\{e_{1}(V^{\sigma_{p}}),\dots,e_{n}(V^{\sigma_{p}})\\}$
and $\\{pe_{1}\operatorname{mod}h,\dots,pe_{n}\operatorname{mod}h\\}$
coincide: It follows easily from [Spr74, Proposition 4.5] that the sets
$\\{e_{1}(V^{\sigma_{p}}),\dots,e_{n}(V^{\sigma_{p}})\\}$ and
$\\{pe_{1},\dots,pe_{n}\\}$ coincide mod $h$; this is [STW20, Proposition
8.1.2]. It then suffices to show that $e_{i}(V^{\sigma_{p}})<h$. For the
exceptional groups, this can be checked by computer [MW23]. We will prove it
for the families $G(m,1,n)$ and $G(m,m,n)$ later (see Example 5.6 and Example
5.15).

######  Question 2.16.

Is there a uniform proof that $e_{i}(V^{\sigma_{p}})<h$?

##  3\. Braid groups and Hecke algebras

###  3.1. Braid groups of complex reflection groups

By [Ste64, Corollary 1.6], the action of $W$ on $V$ is free precisely on
$V^{\text{reg}}$. It is then not hard to show that the quotient map
$\rho:V^{\text{reg}}\to V^{\text{reg}}/W$ is a Galois covering.

######  Definition 3.1.

The _pure braid group_ for a complex reflection group $W$ is
$P(W):=\pi_{1}(V^{\text{reg}})$. Its _braid group_ is
$B(W):=\pi_{1}(V^{\text{reg}}/W)$.

The quotient $\rho$ induces a surjection $\pi:B(W)\to W$, giving a short exact
sequence

$1\to P(W)\xrightarrow[]{\rho_{*}}B(W)\xrightarrow[]{\pi}W\to 1,$

where $W$ can be interpreted as the group of deck transformations of the
covering (see [Hat02, Proposition 1.40]).

The braid group $B(W)$ has a set of generators $\\{\mathbf{s}_{H,\gamma}\\}$
called _generators of the monodromy_ or _braid reflections_ (see [BMR98,
Section 2.B] or [Bro10, Section 4.2.5]), such that
$\pi(\mathbf{s}_{H,\gamma})=s_{H}$ is a distinguished reflection. Moreover,
the pure braid group $P(W)$ is generated by the
$\\{\mathbf{s}_{H,\gamma}^{e_{\mathcal{C}}}\\}$ (where $H\in\mathcal{C}$), and
so $W\cong B(W)/\langle\mathbf{s}_{H,\gamma}^{e_{\mathcal{C}}}\rangle$.

######  Definition 3.2.

Define the _full twist_ $\bm{\pi}\in P(W)$ by

$\displaystyle[0,1]$ $\displaystyle\to V^{\text{reg}}$ $\displaystyle t$
$\displaystyle\mapsto v\exp(2\pi it),$

where the basepoint $v\in V^{\text{reg}}$ is suppressed. The image
$\rho_{*}(\bm{\pi})\in B(W)$ is a central element of $B(W)$ and will also be
called the full twist and be denoted by $\bm{\pi}$.

######  Proposition 3.3 ([Bro10], Proposition 5.24).

If $g$ is a $\zeta$-regular element of $W$, where $\zeta=\exp(2\pi im/d)$ is a
primitive $d$th root of unity, then $g$ has a lift $\mathbf{g}\in B(W)$ such
that $\mathbf{g}^{d}=\bm{\pi}^{m}$.

###  3.2. Hecke algebras for complex reflection groups

Define variables
$\mathbf{u}:=(u_{\mathcal{C},j})_{(\mathcal{C}\in\mathcal{A}/W),(0\leq j\leq
e_{\mathcal{C}}-1)}$, and let $\mathbb{Z}[\mathbf{u}^{\pm 1}]$ be the ring of
Laurent polynomials in these variables.

######  Definition 3.4.

Let $J$ be the ideal of the group algebra $\mathbb{Z}[\mathbf{u}^{\pm}]B(W)$
generated by the elements

$(\mathbf{s}_{H}-u_{\mathcal{C},0})(\mathbf{s}_{H}-u_{\mathcal{C},1})\cdots(\mathbf{s}_{H}-u_{\mathcal{C},e_{\mathcal{C}}-1}),$

where $\mathcal{C}\in\mathcal{A}/W$, $H\in\mathcal{C}$, and $\mathbf{s}_{H}$
is a braid reflection (since generators $\mathbf{s}_{H,\gamma}$ and
$\mathbf{s}_{H,\gamma^{\prime}}$ of the monodromy around $H$ are conjugate in
$B(W)$ by [BMR98, Lemma 2.13], it suffices to use only one such braid
reflection for each $H$ in the above relations to generate $J$). The _generic
Hecke algebra_ $\mathcal{H}(W)$ is the quotient
$\mathbb{Z}[\mathbf{u}^{\pm}]B(W)/J$. For $\mathbf{g}\in B(W)$, we’ll denote
by $T_{\mathbf{g}}$ the corresponding element in the Hecke algebra.

######  Theorem 3.5 (The BMR freeness theorem).

The algebra $\mathcal{H}(W)$ is a free $\mathbb{Z}[\mathbf{u}^{\pm}]$-module
of rank $|W|$.

A ring homomorphism $\mathbb{Z}[\mathbf{u}^{\pm}]\to R$ induces a
specialization $\mathcal{H}(W)\otimes_{\mathbb{Z}[\mathbf{u}^{\pm}]}R$ of the
Hecke algebra. For example, the specialization
$u_{\mathcal{C},j}\xmapsto{\sigma}\zeta_{e_{\mathcal{C}}}^{j}$ gives the group
algebra of $W$ (so that the Hecke algebra may be thought of as a deformation
of the group algebra of $W$). A specialization of the Hecke algebra is
_admissible_ if it factors through this specialization.

######  Definition 3.6.

The _spetsial Hecke algebra_ $\mathcal{H}_{q}(W)$ is the admissible cyclotomic
Hecke algebra induced by the map

$\theta_{q}:u_{\mathcal{C},j}\mapsto\begin{cases}q&\text{if }j=0\\\
\zeta_{e_{\mathcal{C}}}^{j}&\text{if }j>0.\end{cases}$

This is a generalization of the 1-parameter Iwahori-Hecke algebra of Coxeter
groups.

###  3.3. The representation theory of Hecke algebras

Most of the material in this section is from the more general representation
theory of symmetric algebras and can be found in [GP00, Section 7].

If $H$ is an associative $R$-algebra (where $R$ is a commutative ring), then a
_trace function_ on $H$ is an $R$-linear map $\tau:H\to R$ such that
$\tau(hh^{\prime})=\tau(h^{\prime}h)$ for all $h,h^{\prime}\in H$. A trace
function $\tau$ is a _symmetrizing trace_ if the bilinear form

$H\times H\to R\qquad(h,h^{\prime})\mapsto\tau(hh^{\prime})$

is non-degenerate. We say that $H$ is a _symmetric algebra_ if it possesses a
symmetrizing trace.

######  Example 3.7.

There is a canonical symmetrizing trace on the group algebra $\mathbb{C}W$
given by $\tau(w)=\delta_{1w}$. If $W$ is a real reflection group, then
$\mathcal{H}(W)$ has an analogous canonical symmetrizing trace given by
$\tau(T_{w})=\delta_{1w}$ (see [GP00, Proposition 8.1.1]).

For non-real complex reflection groups, there is not an obvious choice of
canonical lift $T_{w}$ for each $w\in W$, so there is not a simple
generalization to these groups of the construction in Example 3.7. It is
conjectured that $\mathcal{H}(W)$ has a canonical symmetrizing trace for all
complex reflection groups $W$. This is the BMM symmetrizing trace conjecture:

######  Assumption 3.8 ([BMM99], Assumption 2.1(1)).

There exists a $\mathbb{Z}[\mathbf{u}^{\pm}]$-linear map
$\tau:\mathcal{H}(W)\to\mathbb{Z}[\mathbf{u}^{\pm}]$ such that:

  1. (1)

$\tau$ is a symmetrizing trace.

  2. (2)

Through the specialization
$u_{\mathcal{C},j}\mapsto\zeta_{e_{\mathcal{C}}}^{j}$, the form $\tau$ becomes
the canonical symmetrizing trace on the group algebra.

  3. (3)

For all $\mathbf{b}\in B(W)$,

$\tau(T_{\mathbf{b}^{-1}})^{\vee}=\frac{\tau(T_{\mathbf{b}\bm{\pi}})}{\tau(T_{\bm{\pi}})},$

where $\alpha\mapsto\alpha^{\vee}$ is the automorphism on
$\mathbb{Z}[\mathbf{u}^{\pm}]$ consisting of simultaneous inversion of the
indeterminates.

By [BMM99, Proposition 2.2], if such a symmetrizing trace exists, it is
unique. We will call $\tau$ the canonical symmetrizing trace on
$\mathcal{H}(W)$. The BMM symmetrizing trace conjecture has been proved for
the infinite family $G(m,p,n)$ (see [BM97a] and [MM98]), but remains open for
most of the exceptional groups.

The next result is essentially [GP00, Theorem 7.2.6], but the precise
formulation presented here follows [BMM99, Proposition 7.12] so as to be more
easily applied to Hecke algebras. Let $R$ be an integrally closed commutative
noetherian domain with field of fractions $F$, and let $H$ be an $R$-algebra
which is a free $R$-module of finite rank. Assume that $H$ is endowed with a
symmetrizing trace $\tau$. For each (absolutely) irreducible character $\chi$
of $FH:=F\otimes_{R}H$, let $\omega_{\chi}:Z(H)\to R$ denote the central
character associated with $V_{\chi}$ (i.e., the restriction of the natural map
$FH\to\operatorname{End}_{F}(V_{\chi})$), where $V_{\chi}$ is the $FH$-module
corresponding to $\chi$. The _Schur element of $\chi$_ is the element of $R$
defined by $S_{\chi}:=\omega_{\chi}(\chi^{\vee})$, where $\chi^{\vee}$ is the
unique element of $H$ satisfying $\tau(\chi^{\vee}h)=\chi(h)$ for all $h\in
H$.

######  Theorem 3.9.

If $FH$ is split semisimple. Then

$\tau=\sum_{\chi\in\operatorname{Irr}(FH)}\frac{1}{S_{\chi}}\chi.$

Before applying this result to Hecke algebras and their admissible
specializations, we first discuss Tits’ deformation theorem [GP00, Theorem
7.4.6], which gives a bijection between the irreducible representations of an
admissible specialization of the Hecke algebra and those of the group algebra.

######  Theorem 3.10 ([Mal99], Theorem 5.2).

Let $\mu(k_{W})$ be the group of roots of unity in $k_{W}$, and define
variables
$\mathbf{v}=(v_{\mathcal{C},j})_{(\mathcal{C}\in\mathcal{A}/W),(0\leq j\leq
e_{\mathcal{C}}-1)}$ that satisfy

$v_{\mathcal{C},j}^{|\mu(k_{W})|}=\zeta_{e_{\mathcal{C}}}^{-j}u_{\mathcal{C},j}.$

Then the field $K_{W}:=k_{W}(\mathbf{v})$ is a splitting field for
$\mathcal{H}(W)$. We will define
$\mathcal{H}_{K_{W}}(W):=\mathcal{H}(W)\otimes_{\mathbb{Z}[\mathbf{u}^{\pm}]}K_{W}$.

We may extend the specialization $\sigma$ of $\mathcal{H}(W)$ (giving the
group algebra) by $v_{\mathcal{C},j}\xmapsto{\sigma}1$. This induces a map
$\tilde{\sigma}:\mathcal{H}_{K_{W}}(W)\to k_{W}[W]$ which agrees with the
surjection $\pi:B(W)\to W$. We are now in the proper setting to apply Tits’
deformation theorem (the precise statement here follows [Dou18, Theorem 4.6]).

######  Theorem 3.11 (Tits’ deformation theorem).

The algebra $\mathcal{H}_{K_{W}}(W)$ is semisimple and the specialization
$\sigma$ induces a bijection

$d_{\sigma}:\operatorname{Irr}(\mathcal{H}_{K_{W}}(W))\to\operatorname{Irr}(k_{W}[W])$

that respects the spectra of elements. That is, for any irreducible
$\mathcal{H}_{K_{W}}(W)$-module $U$ the following diagram commutes:

${\mathcal{H}_{K_{W}}(W)}$${k_{W}[\mathbf{v}^{\pm}][x]}$${k_{W}[W]}$${k_{W}[x]}$$\scriptstyle{p_{U}}$$\scriptstyle{\tilde{\sigma}}$$\scriptstyle{t_{\sigma}}$$\scriptstyle{p_{d_{\sigma}(U)}}$

where the horizontal maps send an element to its characteristic polynomial,
and the vertical maps are naturally induced by $\sigma$. Let
$\chi_{\mathbf{v}}$ and $\chi$ be the characters associated to $U$ and
$d_{\sigma}(U)$, respectively. Then we have

$\chi(g)=\sigma(\chi_{\mathbf{v}}(T_{\mathbf{g}})),$

for all $\mathbf{g}\in B(W)$ where $g=\pi(\mathbf{g})$.

Tits’ deformation theorem can be applied to any admissible specialization of
$\mathcal{H}(W)$ by first moving to the splitting field determined by Theorem
3.10. In particular, for the spetsial Hecke algebra $\mathcal{H}_{q}(W)$, the
splitting field is $k_{W}(y)$ with $y^{|\mu(k_{W})|}=q$. We will denote by
$\chi_{q}$ the irreducible character of $\mathcal{H}_{q}(W)$ corresponding to
the irreducible character $\chi$ of $W$.

The next result is a slight reformulation of [BM97b, Proposition 4.18]
following [Dou18, Proposition 4.11]

######  Proposition 3.12.

Suppose that $W$ is an irreducible complex reflection group. If $\mathbf{w}\in
B(W)$ is a lift of $w\in W$ such that $\mathbf{w}^{d}=\bm{\pi}^{\ell}$, then

$\chi_{q}(T_{\mathbf{w}})=\chi(w)\cdot q^{(nh-h_{\chi})\ell/d},$

for $\chi$ an irreducible character of $W$.

######  Corollary 3.13.

Suppose $W$ is an irreducible well-generated complex reflection group with
Coxeter number $h$, and $c$ is a $\zeta_{h}$-regular element of $W$. Let
$\mathbf{c}\in B(W)$ be a lift of $c$ such that $\mathbf{c}^{h}=\bm{\pi}$
(which exists by Proposition 3.3). Then

$\chi_{q}(T_{\mathbf{c}}^{-p})=q^{(h_{\chi}-nh)p/h}\operatorname{Feg}_{\chi}(e^{2\pi
ip/h}),$

for $\chi$ an irreducible character of $W$ and $p$ an integer.

###### Proof.

This follows from the previous result and [Spr74, Proposition 4.5]. ∎

Returning to Schur elements, by [BMM99, Proposition 2.5], there exist
$S_{\chi}(\mathbf{v})\in\mathbb{Z}_{k_{W}}[\mathbf{v}^{\pm}]$ for
$\chi\in\operatorname{Irr}(W)$, such that

$\tau=\sum_{\chi\in\operatorname{Irr}(W)}\frac{1}{S_{\chi}(\mathbf{v})}\chi_{\mathbf{v}},$

where $\tau$ is the canonical symmetrizing trace on $\mathcal{H}(W)$ and
$\mathbb{Z}_{k_{W}}$ is the ring of integers of $k_{W}$. These Schur elements
have been computed even in the cases for which Assumption 3.8 is still open
(see [Mal00] for instance). The specialization of the canonical symmetrizing
trace on $\mathcal{H}(W)$ is a symmetrizing trace $\tau_{q}$ on
$\mathcal{H}_{q}(W)$ whose Schur elements, which will be denoted
$S_{\chi}(q)$, are the images of the Schur elements of $\mathcal{H}(W)$ via
the specialization $\theta_{q}$.

We now define the class of spetisal complex reflection groups.

######  Definition 3.14.

An complex reflection group is called _spetsial_ if all of its irreducible
components $W$ satisfy any of the following equivalent conditions (equivalence
of the conditions is shown in [Mal00, Proposition 8.1]):

  1. (1)

$S_{\bm{1}}(q)=P_{W}$, where $\bm{1}$ denotes the trivial representation of
$W$

  2. (2)

$P_{W}/S_{\chi}(q)\in k_{W}(q)$ for all $\chi\in\operatorname{Irr}(W)$

  3. (3)

$P_{W}/S_{\chi}(q)\in k_{W}[y]$ for all $\chi\in\operatorname{Irr}(W)$, where
$y^{|\mu(k_{W})|}=q$

  4. (4)

$W$ is one of the following groups:

$W(A_{n-1}),\quad G(m,1,n),\quad G(m,m,n),\quad G_{i}\text{ where
}i\in\\{4,6,8,14,23,\dots,30,32,\dots,37\\}.$

######  Definition 3.15.

For $W$ an irreducible spetsial complex reflection group, define the _generic
degree_ of an irreducible character $\chi$ by

$\operatorname{Deg}_{\chi}(q):=P_{W}/S_{\chi}(q)\in k_{W}(q).$

Let $a_{\chi}$ be the largest non-negative integer such that $q^{a_{\chi}}$
divides $\operatorname{Deg}_{\chi}(q)$, and let $A_{\chi}$ be the degree of
$q$ in $\operatorname{Deg}_{\chi}(q)$. Then $h_{\chi}=a_{\chi}+A_{\chi}$
[BMM99, Corollary 6.9].

Similarly, define $b_{\chi}$ to be the smallest integer $i\geq 0$ such that
$\chi$ occurs in the character of the $i$th symmetric power of the reflection
representation of $W$. That is, $b_{\chi}$ is the largest non-negative integer
such that $q^{b_{\chi}}$ divides the fake degree of $\chi^{*}$. Also, let
$B_{\chi}$ be the degree in $q$ of the fake degree of $\chi^{*}$.

A representation $\psi\in\operatorname{Irr}(W)$ is _special_ if
$a(\psi)=b(\psi)$. If $W$ is a spetsial group, then for all
$\chi\in\operatorname{Irr}(W)$, we have $a(\chi)\leq b(\chi)$ and there exists
a special $\psi\in\operatorname{Irr}(W)$ with $a(\chi)=a(\psi)$ [Mal00,
Proposition 8.1].

######  Proposition 3.16.

Suppose $W$ is an irreducible spetsial complex reflection group with Coxeter
number $h$, and let $c$ be a $\zeta_{h}$-regular element of $W$. Let
$\mathbf{c}\in B(W)$ be a lift of $c$ such that $\mathbf{c}^{h}=\bm{\pi}$.
Then

$\tau_{q}(T_{\mathbf{c}}^{-p})=\frac{1}{P_{W}}\sum_{\chi\in\operatorname{Irr}(W)}q^{(h_{\chi}-nh)p/h}\operatorname{Feg}_{\chi}(e^{2\pi
ip/h})\operatorname{Deg}_{\chi}(q),$

for $\chi$ an irreducible character of $W$.

###### Proof.

It is straightforward to check case-by-case that an irreducible spetsial
complex reflection group is well-generated. The result then follows from
Corollary 3.13. ∎

######  Theorem 3.17.

Suppose $W$ is an irreducible well-generated complex reflection group with
Coxeter number $h$. Then for $p$ relatively prime to $h$,

$[S_{\bm{1}}/S_{\chi}](e^{2\pi ip/h})=\begin{cases}(-1)^{k}&\text{if
}\chi=\chi_{k,p},\\\ 0&\text{otherwise},\end{cases}$

for $\chi$ an irreducible character of $W$ and $\chi_{k,p}$ the character of
$\Lambda^{k}V^{\sigma_{p}}$. In particular, if $W$ is spetsial, then

$\operatorname{Deg}_{\chi}(e^{2\pi ip/h})=\begin{cases}(-1)^{k}&\text{if
}\chi=\chi_{k,p},\\\ 0&\text{otherwise}.\end{cases}$

###### Proof.

For the symmetric groups (in fact, all real groups), the proof is sketched in
[GLTW22, remark 6.10]. For the exceptional irreducible well-generated groups,
this is checked by computer [MW23]. The proof for the groups $G(m,1,n)$ and
$G(m,m,n)$ follows the proof of the “untwisted case” in [Mic22, Proposition 4]
and will be given in the next section. ∎

##  4\. Finite groups of Lie type

This section serves primarily to motivate the constructions in the subsequent
sections by explaining what is going on for Weyl groups. Most of the material
in this section can be found in [Car85], [DM20], and [GM20]. For background on
(linear) algebraic groups, the standard references are [Bor12], [Hum12], and
[Spr98]. See also [Gec13] for a slightly more elementary introduction. Our
conventions and notation will usually follow [GM20].

We fix the following objects for this section: Let $W$ be an irreducible Weyl
group with generating set of simple reflections $S$, and let $q$ be a power of
a prime $p$. Let $\mathbf{G}$ be a simple connected reductive group over
$\overline{\mathbb{F}}_{p}$ with connected center which has Weyl group $W$,
and let $F:\mathbf{G}\to\mathbf{G}$ be a Frobenius map with respect to some
$\mathbb{F}_{q}$-rational structure which acts trivially on $W$. We denote by
$\mathbf{G}^{F}$ the corresponding finite group of Lie type and fix a
maximally split torus $\mathbf{T}_{0}$.

###  4.1. Uniform almost characters and principal series characters

A character $\rho\in\operatorname{Irr}(\mathbf{G}^{F})$ is called a _unipotent
character_ if $\langle
R_{\mathbf{T}}^{\mathbf{G}}(1_{\mathbf{T}}),\rho\rangle\neq 0$ for some
$F$-stable maximal torus $\mathbf{T}\subseteq\mathbf{G}$. Here
$1_{\mathbf{T}}$ is the trivial character of $\mathbf{T}^{F}$ and
$R_{\mathbf{T}}^{\mathbf{G}}(1_{\mathbf{T}})$ is the induced Deligne-Lusztig
character of $\mathbf{G}^{F}$. We denote by
$\operatorname{Uch}(\mathbf{G}^{F})$ the set of all unipotent characters of
$\mathbf{G}^{F}$. We now discuss two important subsets of the unipotent
characters of $\mathbf{G}^{F}$ whose degrees are equal to the fake degrees and
generic degrees, respectively, of corresponding characters of $W$ evaluated at
$q$.

Any maximal torus of $\mathbf{G}$ has the form
${}^{g}\mathbf{T}_{0}:=g\mathbf{T}_{0}g^{-1}$ for some $g\in\mathbf{G}$.
Moreover, ${}^{g}\mathbf{T}_{0}$ is $F$-stable if and only if $g^{-1}F(g)\in
N_{\mathbf{G}}(\mathbf{T}_{0})$, and the map
${}^{g}\mathbf{T}_{0}\mapsto\pi(g^{-1}F(g))$ determines a bijection between
the $\mathbf{G}^{F}$-classes of $F$-stable maximal tori of $G$ and the
conjugacy classes of $W$. Here $\pi:N_{\mathbf{G}}(\mathbf{T}_{0})\to W$ is
the quotient map. If $w\in W$ is conjugate to $\pi(g^{-1}F(g))$, we say that
$\mathbf{T}_{w}:={}^{g}\mathbf{T}_{0}$ is obtained from $\mathbf{T}_{0}$ by
twisting with $w$. We denote by $R_{w}$ the Deligne-Lusztig character
$R_{\mathbf{T}_{w}}^{\mathbf{G}}(1_{\mathbf{T}_{w}^{F}})$.

For each $\chi\in\operatorname{Irr}(W)$, we then define the _unipotent uniform
almost character_

$R_{\chi}:=\frac{1}{|W|}\sum_{w\in W}\chi(w)R_{w},$

which satisfies $R_{\chi}(1)=\operatorname{Feg}_{\chi}(q)$.

The elements of the Harish-Chandra series
$\mathcal{E}(\mathbf{G}^{F},(\mathbf{T}_{0}^{F},1))$ are called the _unipotent
principal series characters_ of $\mathbf{G}^{F}$. A choice of a square root of
$q$ determines a canonical bijection (see [GM20, Example 3.2.6])

$\operatorname{Irr}(W)\to\mathcal{E}(\mathbf{G}^{F},(\mathbf{T}_{0}^{F},1))\qquad\chi\mapsto\rho_{\chi}.$

######  Proposition 4.1.

Every $\rho_{\chi}\in\mathcal{E}(\mathbf{G}^{F},(\mathbf{T}_{0}^{F},1))$
satisfies $\rho_{\chi}(1)=\operatorname{Deg}_{\chi}(q)$.

###### Proof.

We will include an outline of the proof following [DM20, §6.1-6.3] and [GJ11,
§4.2-4.3], but see also [GM20, Theorem 3.2.18].

As described in [GM20, Example 3.2.6], the spetsial Hecke algebra
$\mathcal{H}_{q}(W)$ “evaluated” at $q$ is isomorphic to
$\operatorname{End}_{\mathbf{G}^{F}}\left(R_{\mathbf{T}_{0}}^{\mathbf{G}}(1)\right)^{\operatorname{opp}}$.
The Hom functor

$\mathbb{C}\mathbf{G}^{F}\text{-mod}\to\mathcal{H}_{q}(W)\text{-mod},\qquad
Y\mapsto\operatorname{Hom}_{\mathbf{G^{F}}}\left(R_{\mathbf{T}_{0}}^{\mathbf{G}}(1),Y\right)$

induces (see [GM20, Theorem 3.1.18]) a bijection
$\mathcal{E}(\mathbf{G}^{F},(\mathbf{T}_{0}^{F},1))\to\operatorname{Irr}(\mathcal{H}_{q}(W))$.
Composing this with the bijection from Tits’ deformation theorem gives the
correspondence $\chi\mapsto\rho_{\chi}$. As a
$\mathcal{H}_{q}(W)\times\mathbb{C}\mathbf{G}^{F}$-module,
$R_{\mathbf{T}_{0}}^{\mathbf{G}}(1)$ decomposes as

$R_{\mathbf{T}_{0}}^{\mathbf{G}}(1)=\bigoplus_{\chi\in\operatorname{Irr}(W)}\chi_{q}\otimes\rho_{\chi}.$

In particular, the multiplicity of $\chi_{q}$ is equal to $\rho_{\chi}(1)$
[DM20, Theorem 6.1.1]. Also, note that as a character of $\mathbf{G}^{F}$, we
have
$R_{\mathbf{T}_{0}}^{\mathbf{G}}(1)=\sum_{\chi\in\operatorname{Irr}(W)}\chi(1)\rho_{\chi}$.

Let $\mathbf{B}_{0}$ be an $F$-stable Borel containing $\mathbf{T}_{0}$. Then
$R_{\mathbf{T}_{0}}^{\mathbf{G}}(1)\cong\mathbb{C}[\mathbf{G}^{F}/\mathbf{B}_{0}^{F}]$
is the permutation representation of $\mathbf{G}^{F}$ on the cosets of
$\mathbf{B}_{0}^{F}$. Fix a set of representatives $\\{\bar{w}\in
N_{\mathbf{G}}(\mathbf{T}_{0})^{F}:w\in W\\}$. The basis $\\{T_{w}:w\in W\\}$
of $\mathcal{H}_{q}(W)$ can be given explicitly using the Bruhat decomposition
by defining
$T_{w}:\mathbb{C}[\mathbf{G}^{F}/\mathbf{B}_{0}^{F}]\to\mathbb{C}[\mathbf{G}^{F}/\mathbf{B}_{0}^{F}]$
by

$T_{w}\left(x\mathbf{B}_{0}^{F}\right)=\text{ sum of all cosets
$y\mathbf{B}_{0}^{F}\in\mathbb{C}[\mathbf{G}^{F}/\mathbf{B}_{0}^{F}]$ such
that $x^{-1}y\in\mathbf{B}_{0}^{F}\bar{w}\mathbf{B}_{0}^{F}$}.$

From this, one can show that
$\psi:T_{w}\mapsto\operatorname{Trace}\left(T_{w}\,\,|\,\,R_{\mathbf{T}_{0}}^{\mathbf{G}}(1)\right)$
is a multiple of the symmetrizing trace $\tau_{q}$ with
$\psi(T_{1})=\operatorname{dim}R_{\mathbf{T}_{0}}^{\mathbf{G}}(1)=[\mathbf{G}^{F}:\mathbf{B}^{F}]$
and $\psi(T_{w})=0$ for $w\neq 1$ (see the proof of [GJ11, Lemma 4.3.2]). Then
by [DM20, Corollary 6.3.10] for each $\chi\in\operatorname{Irr}(W)$ the
multiplicity of $\chi_{q}$ in $R_{\mathbf{T}_{0}}^{\mathbf{G}}(1)$ is
$[\mathbf{G}^{F}:\mathbf{B}^{F}]/S_{\chi}(q)$. Hence
$\rho_{\chi}(1)=[\mathbf{G}^{F}:\mathbf{B}^{F}]/S_{\chi}(q)$. Since
$\rho_{\bm{1}}(1)=1$, it follows that
$\operatorname{Deg}_{\chi}(q)=S_{\bm{1}}(q)/S_{\chi}(q)=\rho_{\chi}(1)$. ∎

###  4.2. Families of unipotent characters

Denote by $\varepsilon$ the sign character of $W$. Define a relation $\preceq$
on $\operatorname{Irr}(W)$ inductively as follows. For $W=\\{1\\}$, the
trivial character is related to itself. Now assume that $W\neq\\{1\\}$ and
that $\preceq$ has been defined for all proper parabolic subgroups of $W$. For
$\phi,\phi^{\prime}\in\operatorname{Irr}(W)$, we write
$\phi\preceq\phi^{\prime}$ if there is a sequence
$\phi=\phi_{0},\phi_{1},\dots,\phi_{m}=\phi^{\prime}$ such that for each
$i\in\\{1,\dots,m\\}$ there exists a subset $I_{i}\subsetneq S$ and
$\psi_{i},\psi_{i}^{\prime}\in\operatorname{Irr}(W_{I_{i}})$, where
$\psi_{i}\preceq\psi_{i}^{\prime}$, such that either

$\langle\operatorname{Ind}_{I_{i}}^{S}(\psi_{i}),\phi_{i-1}\rangle\neq
0,\quad\langle\operatorname{Ind}_{I_{i}}^{S}(\psi_{i}^{\prime}),\phi_{i}\rangle\neq
0,\quad\text{and}\quad a_{\psi_{i}^{\prime}}=a_{\phi_{i}},$

or

$\langle\operatorname{Ind}_{I_{i}}^{S}(\psi_{i}),\varepsilon\phi_{i}\rangle\neq
0,\quad\langle\operatorname{Ind}_{I_{i}}^{S}(\psi_{i}^{\prime}),\varepsilon\phi_{i-1}\rangle\neq
0,\quad\text{and}\quad a_{\psi_{i}^{\prime}}=a_{\varepsilon\phi_{i-1}}.$

######  Definition 4.2.

We say that $\phi,\phi^{\prime}$ belong to the same _family_ in
$\operatorname{Irr}(W)$ if $\phi\preceq\phi^{\prime}$ and
$\phi^{\prime}\preceq\phi$.

One can show [GM20, Proposition 4.1.19] that the functions $\phi\mapsto
a_{\phi}$ and $\phi\mapsto A_{\phi}$ are constant on the families of
$\operatorname{Irr}(W)$. Moreover, in each family
$\mathscr{F}\subseteq\operatorname{Irr}(W)$, there is a unique special
character $\phi\in\mathscr{F}$ with $a_{\phi}=b_{\phi}$ [GM20, Proposition
4.1.20].

Define a graph on the set of vertices $\operatorname{Uch}(\mathbf{G}^{F})$ as
follows: two unipotent characters
$\rho_{1},\rho_{2}\in\operatorname{Uch}(\mathbf{G}^{F})$ are joined if and
only if there is an irreducible character $\chi\in\operatorname{Irr}(W)$ such
that $\langle R_{\chi},\rho_{i}\rangle\neq 0$ for $i=1,2$. The sets of
vertices corresponding to the connected components of the graph are called the
_families_ in $\operatorname{Uch}(\mathbf{G}^{F})$. There is another
perspective on these families via their unipotent support [GM00].

There is an induced partition of $\operatorname{Irr}(W)$ wherein two
characters $\chi,\chi^{\prime}\in\operatorname{Irr}(W)$ are equivalent if and
only if $R_{\chi}$ and $R_{\chi^{\prime}}$ have unipotent constituents lying
in the same family of $\operatorname{Uch}(\mathbf{G}^{F})$. This recovers the
families of $\operatorname{Irr}(W)$.

Alternatively [Car85, §12], the families of $\operatorname{Irr}(W)$ can be
recovered from the correspondence $\chi\to\rho_{\chi}$ identifying the
irreducible characters of $\operatorname{Irr}(W)$ with the unipotent principle
series characters. That is, $\chi,\chi^{\prime}\in\operatorname{Irr}(W)$ are
in the same family of $\operatorname{Irr}(W)$ if and only if
$\rho_{\chi},\rho_{\chi^{\prime}}$ are in the same family of
$\operatorname{Uch}(\mathbf{G}^{F})$. Moreover, the family of
$\operatorname{Uch}(\mathbf{G}^{F})$ corresponding to $R_{\chi}$ is the same
as the family corresponding to $\rho_{\chi}$. That is, $\rho_{\chi}$ is in the
family of $\operatorname{Uch}(\mathbf{G}^{F})$ containing the components of
$R_{\chi}$.

Each $\rho\in\operatorname{Uch}(\mathbf{G}^{F})$ has a _degree polynomial_
$\mathbb{D}_{\rho}\in\mathbb{R}[x]$, such that $\rho(1)=\mathbb{D}_{\rho}(q)$.
See [GM20, definition 2.3.25] for a precise definition. For the unipotent
uniform almost characters, the degree polynomial is the corresponding fake
degree. For the unipotent principal series characters, the degree polynomial
is the corresponding generic degree. We then define

$\displaystyle A_{\rho}$ $\displaystyle:=\text{degree of $x$ in
}\mathbb{D}_{\rho}$ $\displaystyle a_{\rho}$ $\displaystyle:=\text{largest
non-negative integer such that $x^{a_{\rho}}$ divides }\mathbb{D}_{\rho}.$

######  Proposition 4.3 ([GM20], Proposition 4.2.7).

The $a$ and $A$ values are constant on each family of
$\operatorname{Uch}(\mathbf{G}^{F})$ and agree with the $a$ and $A$ values of
the corresponding family of $\operatorname{Irr}(W)$.

##  5\. Spetses

###  5.1. Unipotent characters and Generic Degrees for $G(m,1,n)$

The irreducible representations of $G(m,1,n)$ can be parametrized by
$m$-partitions of $n$, that is, tuples
$\bm{\lambda}=(\lambda^{(0)},\lambda^{(1)},\dots,\lambda^{(m-1)})$ of
partitions such that $|\lambda^{(0)}|+\cdots+|\lambda^{(m-1)}|=n$. We will
generally write the parts of a partition in decreasing order:
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots)$.

The group $G(m,1,n)$ is generated by the $n$ elements
$\\{t,s_{1},s_{2},\dots,s_{n-1}\\}$, where $t$ is given by the matrix
$\operatorname{Diag}(\zeta_{m},1,\dots,1)$ and $s_{i}$ is given by the
permutation matrix corresponding to the transposition $(i,i+1)$ (see [LT09,
§2.7]). The parametrization of irreducible characters of $G(m,1,n)$ is as
follows:

$\bm{\lambda}\leftrightarrow\chi_{\bm{\lambda}}=\operatorname{Ind}_{G(m,1,|\lambda^{(0)}|)\times\cdots\times
G(m,1,|\lambda^{(m-1)}|)}^{G(m,1,n)}\left((\chi_{0}\otimes\gamma_{0})\boxtimes\cdots\boxtimes(\chi_{m-1}\otimes\gamma_{m-1})\right),$

where

  * •

$\gamma_{k}$ is the linear character of $G(m,1,|\lambda^{(k)}|)$ defined by
$t\mapsto\zeta_{m}^{k}$ and $s_{i}\mapsto 1$ for
$i=1,\dots,|\lambda^{(k)}|-1$.

  * •

$\chi_{k}$ is the character of the symmetric group $S_{|\lambda^{(k)}|}$
corresponding to $\lambda^{(k)}$ (see [Sag13] for instance) considered as a
character of $G(m,1,|\lambda^{(k)}|)$ via the surjection
$G(m,1,|\lambda^{(k)}|)\to S_{|\lambda^{(k)}|}$.

Every irreducible character of $G(m,1,n)$ is equal to $\chi_{\bm{\lambda}}$
for some $\bm{\lambda}$, and $\chi_{\bm{\alpha}}=\chi_{\bm{\beta}}$ if and
only if $\bm{\alpha}=\bm{\beta}$. We will describe explicit models for the
irreducible representations in terms of the $m$-partitions of $n$ as given in
[AK94] (although the precise notation will follow [MM10]).

Fix $\bm{\lambda}$, an $m$-partition of $n$. Denote by
$\mathcal{T}(\bm{\lambda})$ the set of standard Young tableaux with shape
$\bm{\lambda}$. That is, the elements of $\mathcal{T}(\bm{\lambda})$ are
tuples $\mathbf{T}=(T_{0},\dots,t_{m-1})$ such that $T_{i}$ is a filling of
the Young diagram of $\lambda^{i}$ with numbers in $\\{1,\dots,n\\}$ such that
the numbers increase across the rows and columns of each $T_{i}$, and each
number appears exactly once in one of the filled diagrams. We will let
$\mathbf{T}(k)=i$ if $k$ is placed in $T_{i}$. Let $V(\bm{\lambda})$ be the
$\mathbb{C}$-vector space with basis $\mathcal{T}(\bm{\lambda})$, and define
the representation $\rho_{\bm{\lambda}}:G(m,1,n)\to V(\bm{\lambda})$ by

  * •

$\rho_{\bm{\lambda}}(t)\mathbf{T}=\zeta_{m}^{\mathbf{T}(1)}\mathbf{T}$.

  * •

Let $\mathbf{T}_{i\leftrightarrow i+1}$ be the tuple obtained by exchanging
the numbers $i$ and $i+1$ if this results in a tuple of standard Young
tableaux, and 0 otherwise. If $i$ and $i+1$ occur in the same tableau, say $i$
is in row $\ell_{1}$ column $m_{1}$ and $i+1$ is in row $\ell_{2}$ column
$m_{2}$, define the axial distance $a(i,i+1)$ between them to be
$a(i,i+1)=(m_{2}-\ell_{2})-(m_{1}-\ell_{1})$. If $i$ and $i+1$ do not occur in
the same tableau, then set $a(i,i+1)=\infty$ so that $1/a(i,i+1)=0$. Then

$\rho_{\bm{\lambda}}(s_{i})\mathbf{T}=\frac{1}{a(i,i+1)}\mathbf{T}+\left(1+\frac{1}{a(i,i+1)}\right)\mathbf{T}_{i\leftrightarrow
i+1}.$

It is easy to see from this construction that the reflection representation is
given by the $m$-partition $(n-1,1,\varnothing,\dots,\varnothing)$.

######  Example 5.1 (Exterior powers).

Let $V$ be the reflection representation of $G(m,1,n)$. We will show that the
$m$-partition of $n$ corresponding to $\Lambda^{k}V$ is
$(n-k,1^{k},\varnothing,\dots,\varnothing)$.

Let $\bm{\lambda}(k)$ be the partition
$(n-k,1^{k},\varnothing,\dots,\varnothing)$. Consider the model
$V(\bm{\lambda}(1))$ given above for the reflection representation with basis
$\mathcal{T}(\bm{\lambda}(1))$ consisting of standard Young tableaux
$\mathbf{T}=(T_{0},T_{1},\dots,T_{m})$ of shape $\bm{\lambda}(1)$. Let
$\mathbf{T}^{a_{1},a_{2},\dots,a_{k}}$ be the standard Young tableaux of shape
$\bm{\lambda}(k)$, with

$T^{a_{1},a_{2},\dots,a_{k}}_{1}=\ytableau a_{1}\\\ a_{2}\\\ \vdots\\\ a_{k}$

for $1\leq a_{1}<a_{2}<\cdots<a_{k}\leq n$. So
$\mathcal{T}(\bm{\lambda}(k))=\\{\mathbf{T}^{a_{1},\dots,a_{k}}:1\leq
a_{1}<\cdots<a_{k}\leq n\\}$. Define a linear map
$\varphi:\Lambda^{k}V(\bm{\lambda}(1))\to V(\bm{\lambda}(k))$ by

$\mathbf{T}^{a_{1}}\wedge\mathbf{T}^{a_{2}}\wedge\cdots\wedge\mathbf{T}^{a_{k}}\mapsto\mathbf{T}^{a_{1},\dots,a_{k}},$

for $1\leq a_{1}<\cdots<a_{k}\leq n$. We need to show that $\varphi$ gives an
isomorphism of representations. Now

$\displaystyle\varphi(t\cdot(\mathbf{T}^{a_{1}}\wedge\cdots\wedge\mathbf{T}^{a_{k}}))$
$\displaystyle=\varphi\left(\zeta_{m}^{\mathbf{T}^{a_{1}}(1)+\cdots+\mathbf{T}^{a_{k}}(1)}(\mathbf{T}^{a_{1}}\wedge\cdots\wedge\mathbf{T}^{a_{k}})\right)$
$\displaystyle=\zeta_{m}^{\mathbf{T}^{a_{1},\dots,a_{k}}(1)}\mathbf{T}^{a_{1},\dots,a_{k}}$
$\displaystyle=t\cdot\mathbf{T}^{a_{1},\dots,a_{k}}.$

For each $i=1,\dots,n-1$, there are three cases to consider:

  1. (1)

$\mathbf{T}^{a_{1},\dots,a_{k}}(i)=\mathbf{T}^{a_{1},\dots,a_{k}}(i+1)=0$. In
this case,

$\displaystyle\varphi(s_{i}\cdot(\mathbf{T}^{a_{1}}\wedge\cdots\wedge\mathbf{T}^{a_{k}}))$
$\displaystyle=\varphi(\mathbf{T}^{a_{1}}\wedge\cdots\wedge\mathbf{T}^{a_{k}})$
$\displaystyle=\mathbf{T}^{a_{1},\dots,a_{k}}=s_{i}\cdot\mathbf{T}^{a_{1},\dots,a_{k}}.$

  2. (2)

$\mathbf{T}^{a_{1},\dots,a_{k}}(i)=\mathbf{T}^{a_{1},\dots,a_{k}}(i+1)=1$. In
this case,

$\displaystyle\varphi(s_{i}\cdot(\mathbf{T}^{a_{1}}\wedge\cdots\mathbf{T}^{i}\wedge\mathbf{T}^{i+1}\wedge\cdots\wedge\mathbf{T}^{a_{k}}))$
$\displaystyle=\varphi(\mathbf{T}^{a_{1}}\wedge\cdots\mathbf{T}^{i+1}\wedge\mathbf{T}^{i}\wedge\cdots\wedge\mathbf{T}^{a_{k}})$
$\displaystyle=-\mathbf{T}^{a_{1},\dots,a_{k}}=s_{i}\cdot\mathbf{T}^{a_{1},\dots,a_{k}}.$

  3. (3)

$\mathbf{T}^{a_{1},\dots,a_{k}}(i)\neq\mathbf{T}^{a_{1},\dots,a_{k}}(i+1)$.
That is, exactly one of $i$, $i+1$ is in $\\{a_{1},\dots,a_{k}\\}$, say
$a_{j}=i$ or $i+1$. In this case,

$\displaystyle\varphi(s_{i}\cdot(\mathbf{T}^{a_{1}}\wedge\cdots\wedge\mathbf{T}^{a_{j}}\wedge\cdots\wedge\mathbf{T}^{a_{k}}))$
$\displaystyle=\varphi(\mathbf{T}^{a_{1}}\wedge\cdots\wedge\mathbf{T}^{a_{j}}_{i\leftrightarrow
i+1}\wedge\cdots\wedge\mathbf{T}^{a_{k}})$
$\displaystyle=\mathbf{T}^{a_{1},\dots,a_{k}}_{i\leftrightarrow
i+1}=s_{i}\cdot\mathbf{T}^{a_{1},\dots,a_{k}}.$

Thus, the $m$-partition of of $n$ corresponding to $\Lambda^{k}V$ is
$(n-k,1^{k},\varnothing,\dots,\varnothing)$.

######  Example 5.2 (Galois twists).

One can easily show, using Schur orthogonality relations, that the elements of
$\operatorname{Gal}(\mathbb{Q}(\zeta_{h})/\mathbb{Q})$ act as permutations on
$\operatorname{Irr}(W)$ for $W$ a well-generated irreducible complex
reflection group with Coxeter number $h$. For the groups $G(m,1,n)$, we will
characterize this action in terms of the $m$-partitions of $n$.

The following is [MM10, Lemma 5.2] (which contains a small typo). For
$\bm{\lambda}=(\lambda^{(0)},\lambda^{(1)},\dots,\lambda^{(m-1)})$, let
$\sigma_{p}(\bm{\lambda})=(\lambda^{(0)},\lambda^{(p^{-1})},\dots,\lambda^{((m-1)p^{-1})})$,
where the indices are taken mod $m$ and $p^{-1}$ is the multiplicative inverse
of $p$ mod $mn$. That is, $[\sigma_{p}(\bm{\lambda})]^{(pk)}=\lambda^{(k)}$
(indices taken mod $m$). Then $V(\bm{\lambda})^{\sigma_{p}}\cong
V(\sigma_{p}(\bm{\lambda}))$.

To see this, define $\varphi:V(\bm{\lambda})^{\sigma_{p}}\to
V(\sigma_{p}(\bm{\lambda}))$ by

$\mathbf{T}\mapsto\mathbf{T}^{\sigma_{p}},\quad\text{where}\quad
T^{\sigma_{p}}_{pk}=T_{k}.$

This is an isomorphism of representations because

$\varphi(t\cdot\mathbf{T})=\varphi(\zeta_{m}^{p\mathbf{T}(1)}\mathbf{T})=\zeta_{m}^{p\mathbf{T}(1)}\mathbf{T}^{\sigma_{p}}=t\cdot\mathbf{T}^{\sigma_{p}}$

and

$\varphi(s_{i}\cdot\mathbf{T})=\frac{1}{a(i,i+1)}\mathbf{T}^{\sigma_{p}}+\left(1+\frac{1}{a(i,i+1)}\right)\mathbf{T}^{\sigma_{p}}_{i\leftrightarrow
i+1}=s_{i}\cdot\mathbf{T}^{\sigma_{p}}.$

One can see that $\Lambda^{k}V^{\sigma_{p}}\cong(\Lambda^{k}V)^{\sigma_{p}}$
as representations of $W$ by noticing that they induce the same character. So
the $m$-partition of $n$ corresponding to $\Lambda^{k}V^{\sigma_{p}}$ is
$(n-k,\varnothing,\dots,\varnothing,1^{k},\varnothing,\dots,\varnothing)$,
where the $1^{k}$ is in the $p$th slot (mod $m$; indexing starts with 0).

In [Mal95], Malle gives a combinatorial construction of unipotent characters
and generic degrees for the groups $G(m,1,n)$ and $G(m,m,n)$ which enjoy many
of the same properties as the corresponding objects for Weyl groups. We will
now describe the construction for the groups $G(m,1,n)$. The construction for
the groups $G(m,m,n)$ will be given later.

######  Definition 5.3.

An _$m$ -symbol_ is a tuple $S=(S_{0},\dots,S_{m-1})$ of $m$ finite sequences
$S_{i}=(0\leq S_{i,1}<S_{i,2}<\cdots<S_{i,M_{i}})$ of strictly increasing non-
negative integers. An $m$-symbol will be presented as

$S=\begin{pmatrix}S_{0,1}&\cdots&S_{0,M_{0}}\\\ \vdots&&\vdots\\\
S_{m-1,1}&\cdots&S_{m-1,M_{m-1}}\end{pmatrix}.$

The _content_ of an $m$-symbol $S$ is $I(S)=M_{0}+\cdots+M_{m-1}$. Its _rank_
is

$\operatorname{rk}(S)=\sum_{i,j}\lambda_{i,j}-\left\lfloor\frac{(I(S)-1)(I(S)-m+1)}{2m}\right\rfloor.$

For an $m$-symbol $S$ with $I(S)\equiv 1\operatorname{mod}m$, define its
_defect_ by

$\operatorname{def}(S)=\left(\frac{(m-1)(I(S)-1)}{2}-\sum_{i=0}^{m-1}iM_{i}\right)\operatorname{mod}m.$

Such a symbol will be called _reduced_ if $\operatorname{def}(S)=0$ and not
all of the $\lambda_{i,1}$ are zero.

######  Definition 5.4.

The _unipotent characters_ of $G(m,1,n)$ are

$\operatorname{Uch}(G(m,1,n)):=\\{\text{reduced $m$-symbols
}S:\operatorname{rk}(S)=m,\,\,I(S)\equiv 1\operatorname{mod}m\\}.$

We’ll say that two unipotent characters $S$, $S^{\prime}$ belong to the same
family if the entries of $S$ and $S^{\prime}$ coincide as multisets.

The _generic degree_ of a unipotent character is defined by

$\operatorname{Deg}_{S}(q):=\frac{(-1)^{\binom{m}{2}\binom{\ell}{2}}\left(\prod_{i=1}^{n}(q^{mi}-1)\right)\cdot\prod_{i=0}^{m-1}\prod_{j=i}^{m-1}\prod_{\begin{subarray}{c}(\lambda,\mu)\in
S_{i}\times S_{j}\\\ \mu<\lambda\text{ if
}i=j\end{subarray}}(q^{\lambda}\zeta_{m}^{i}-q^{\mu}\zeta_{m}^{j})}{\tau(m)^{\ell}q^{\binom{m(\ell-1)+1}{2}+\binom{m(\ell-2)+1}{2}+\cdots}\prod_{i=0}^{m-1}\Theta(S_{i},q^{m})}$

where

$\ell=\frac{I(S)-1}{m},\qquad\tau(m)=\prod_{i=0}^{m-1}\prod_{j=i+1}^{m-1}(\zeta^{i}-\zeta^{j})=\sqrt{-1}^{\binom{m-1}{2}}\sqrt{m}^{m},$

and for any finite $A\subset\mathbb{Z}$,

$\Theta(A,q)=\prod_{\begin{subarray}{c}\lambda\in A\\\ \lambda\geq
1\end{subarray}}\prod_{j=1}^{\lambda}(q^{j}-1).$

There is an inclusion
$\operatorname{Irr}(G(m,1,n))\subset\operatorname{Uch}(G(m,1,n))$. Let
$\bm{\lambda}=(\lambda^{(0)},\dots,\lambda^{(m-1)})$ be an $m$-partition of
$n$. Let $\alpha_{0}=(0\leq\alpha_{0,1}\leq\cdots\leq\alpha_{0,M+1})$ be the
parts of $\lambda^{(0)}$ written in increasing order with 0’s added if
necessary. Let $\alpha_{i}=(0\leq\alpha_{i,1}\leq\cdots\leq\alpha_{i,M})$ be
the parts of $\lambda^{(i)}$ written in increasing order for $i>0$ with 0’s
added if necessary. Choose $M$ as small as possible. Define an $m$-symbol
$S_{\bm{\lambda}}=(S_{0},\dots,S_{m-1})$ by $S_{i,j}=\alpha_{i,j}+j-1$. Then
$S_{\bm{\lambda}}$ is a reduced symbol with $I(S_{\bm{\lambda}})=mM+1$. The
inclusion is defined by

$\operatorname{Irr}(G(m,1,n))\to\operatorname{Uch}(G(m,1,n))\qquad\chi_{\bm{\lambda}}\mapsto
S_{\bm{\lambda}}.$

Under this inclusion, the definition of the generic degree of an irreducible
character given in Definition 3.15 coincides with the definition for unipotent
characters. That is,
$\operatorname{Deg}_{\chi_{\bm{\lambda}}}(q)=\operatorname{Deg}_{S_{\bm{\lambda}}}(q)$.

Malle also gives a formula for the fake degree of an irreducible character in
terms of the corresponding $m$-symbol. This formula has to be slightly
adjusted (following [Las12, Proposition 5.3.11]) because Malle uses a
different definition of fake degrees (see Remark 2.14).

######  Proposition 5.5 ([Mal95], Proposition 2.10).

The fake degree of $\chi_{\bm{\lambda}}$, with corresponding $m$-symbol
$S_{\bm{\lambda}}=(S_{0},\dots,S_{m-1})$, is equal to

$\left(\prod_{i=1}^{n}(q^{mi}-1)\right)\cdot\left(\prod_{i=0}^{m-1}\frac{\Delta(S_{i},q^{m})}{\Theta(S_{i},q^{m})}\right)\cdot\frac{\prod_{i=1}^{m-1}\prod_{\lambda\in
S_{i}}q^{(m-i)\lambda}}{q^{\binom{m(\ell-1)+1}{2}+\binom{m(\ell-2)+1}{2}+\cdots}},$

where for any finite $A\subset\mathbb{Z}$,

$\Delta(A,q)=\prod_{\begin{subarray}{c}\lambda,\lambda^{\prime}\in A\\\
\lambda^{\prime}<\lambda\end{subarray}}(q^{\lambda}-q^{\lambda^{\prime}}),$

and $\ell$ and $\Theta$ are as in Definition 5.4.

######  Example 5.6.

The $m$-symbol corresponding to $V^{\sigma_{p}}$ is given by

$S_{i}=\begin{cases}(0,n)&\text{if }i=0,\\\ (1)&\text{if
}i=p\operatorname{mod}m,\\\ (0)&\text{otherwise}.\end{cases}$

So

$\Delta(S_{i},q^{m})=\begin{cases}q^{mn}-1&\text{if }i=0,\\\
1&\text{otherwise},\end{cases}\qquad\text{and}\qquad\Theta(S_{i},q^{m})=\begin{cases}\prod_{j=1}^{n}(q^{mj}-1)&\text{if
}i=0,\\\ (q^{m}-1)&\text{if }i=p\operatorname{mod}m,\\\
1&\text{otherwise}.\end{cases}$

Letting $\chi_{1,p}$ denote the character of $V^{\sigma_{p}}$, we then have

$\displaystyle\operatorname{Feg}_{\chi_{1,p}}(q)$
$\displaystyle=\left(\prod_{i=1}^{n}(q^{mi}-1)\right)\cdot\frac{q^{m-(p\operatorname{mod}m)}}{(q^{m}-1)\prod_{j=1}^{n-1}(q^{mj}-1)}$
$\displaystyle=q^{m-(p\operatorname{mod}m)}+q^{2m-(p\operatorname{mod}m)}+\cdots+q^{mn-(p\operatorname{mod}m)}.$

Since $p$ is relatively prime to $mn$, we can rewrite this as

$\operatorname{Feg}_{\chi_{1,p}}(q)=\sum_{j=1}^{n}q^{(mj-p)\operatorname{mod}mn}=\sum_{j=1}^{n}q^{p(mj-1)\operatorname{mod}mn}=\sum_{j=1}^{n}q^{pe_{j}\operatorname{mod}h},$

which demonstrates Example 2.15.

The Schur elements of irreducible representations of $G(m,1,n)$ can be
expressed in terms of the corresponding $m$-partition using the next result.

######  Theorem 5.7 ([Mat04], Corollary 5.16).

For partitions $\lambda,\mu$ (whose parts are written in decreasing order),
define

$\displaystyle[\lambda]$ $\displaystyle:=\\{(i,j):i\geq 1,\,\,1\leq
j\leq\lambda_{i}\\},$ $\displaystyle\lambda^{\prime}_{k}$
$\displaystyle:=\\#\\{i:i\geq 1\text{ such that }\lambda_{i}\geq k\\},$
$\displaystyle n(\lambda)$ $\displaystyle:=\sum_{i\geq 1}(i-1)\lambda_{i},$
$\displaystyle h_{i,j}^{\lambda,\mu}$
$\displaystyle:=\lambda_{i}-i+\mu^{\prime}_{j}-j+1\quad\text{for
}(i,j)\in[\lambda].$

Let $\bm{\lambda}=(\lambda^{(0)},\lambda^{(1)},\dots,\lambda^{(m-1)})$ be the
$m$-partition of $n$ corresponding to the irreducible representation $\chi$.
Then

$S_{\chi}(q)=(-1)^{mn}q^{-\alpha(\bm{\lambda})-n}\prod_{s=0}^{m-1}\prod_{(i,j)\in[\lambda^{(s)}]}Q_{s}\left[h_{i,j}^{\lambda^{(s)},\lambda^{(s)}}\right]_{q}\prod_{t=s+1}^{m-1}X_{s,t}^{\bm{\lambda}},$

where

$X_{s,t}^{\bm{\lambda}}=\left(\prod_{(i,j)\in[\lambda^{(t)}]}(q^{j-i}Q_{t}-Q_{s})\right)\cdot\left(\prod_{(i,j)\in[\lambda^{(s)}]}\left(q^{j-i}Q_{s}-q^{\lambda_{1}^{(t)}}Q_{t}\right)\prod_{k=1}^{\lambda_{1}^{(t)}}\frac{q^{j-i}Q_{s}-q^{k-1-\lambda_{k}^{(t)^{\prime}}}Q_{t}}{q^{j-i}Q_{s}-q^{k-\lambda_{k}^{(t)^{\prime}}}Q_{t}}\right),$

and

$Q_{s}=\begin{cases}q&\text{if }s=0\\\
\zeta_{m}^{s}&\text{otherwise}.\end{cases}\qquad\text{ and
}\qquad\alpha(\bm{\lambda})=\sum_{0\leq s\leq m-1}n(\lambda^{(s)}).$

We will want to be able to evaluate the Schur elements at the values
$\zeta_{mn}^{p}$ for $p$ relatively prime to $mn$. For this, Mathas’s formula
will be difficult to work with, so we will prefer the following
simplification:

######  Theorem 5.8 ([CJ12], Theorem 3.2).

Let $\bm{\lambda}=(\lambda^{(0)},\lambda^{(1)},\dots,\lambda^{(m-1)})$ be the
$m$-partition of $n$ corresponding to the irreducible representation $\chi$,
and let $\bar{\bm{\lambda}}$ be the (decreasing) sequence consisting of all of
the parts of $\lambda^{(0)},\lambda^{(1)},\dots,\lambda^{(m-1)}$ combined.
Then

$S_{\chi}(q)=(-1)^{n(m-1)}q^{-n(\bar{\bm{\lambda}})}(q-1)^{-n}\prod_{s=0}^{m-1}\prod_{(i,j)\in[\lambda^{(s)}]}\prod_{t=0}^{m-1}\left(q^{h_{i,j}^{\lambda^{(s)},\lambda^{(t)}}}Q_{s}Q_{t}^{-1}-1\right),$

where

$Q_{s}=\begin{cases}q&\text{if }s=0\\\
\zeta_{m}^{s}&\text{otherwise}.\end{cases}$

We are now ready to prove Theorem 3.17 for the spetsial groups $G(m,1,n)$.

###### Proof of Theorem 3.17.

We will first compute the generic degrees for the exterior powers of Galois
twists of the reflection representation. We saw in Example 5.2 that the
$m$-partition of $n$ corresponding to $\Lambda^{k}V^{\sigma_{p}}$ is
$\bm{\lambda}=(n-k,\varnothing,\dots,\varnothing,1^{k},\varnothing,\dots,\varnothing)$.
Now

$h_{1,j}^{\lambda^{(0)},\lambda^{(t)}}=\begin{cases}n-k-j+1&\text{if }t=0\\\
n-1&\text{if }t=p\operatorname{mod}m\text{ and }j=1\\\
n-k-j&\text{otherwise},\end{cases}\quad\text{ and }\quad
h_{i,1}^{\lambda^{(p)},\lambda^{(t)}}=\begin{cases}2-i&\text{if }t=0\\\
k-i+1&\text{if }t=p\operatorname{mod}m\\\ 1-i&\text{otherwise}.\end{cases}$

So, letting $\chi$ be the character corresponding to
$\Lambda^{k}V^{\sigma_{p}}$,

$\displaystyle S_{\chi}(q)$
$\displaystyle=(-1)^{n(m-1)}q^{-k(k+1)/2}(q-1)^{-n}\left(\prod_{j=1}^{n-k}\left(q^{n-k-j+1}-1\right)\right)\left(\frac{q^{n}\zeta_{m}^{-p}-1}{q^{n-k}\zeta_{m}^{-p}-1}\right)$
$\displaystyle\qquad\left(\prod_{t=1}^{m-1}\prod_{j=1}^{n-k}\left(q^{n-k-j+1}\zeta_{m}^{-t}-1\right)\right)\left(\prod_{i=1}^{k}\left(q^{k-i+1}-1\right)\right)\left(\prod_{\begin{subarray}{c}t=0\\\
t\neq
p\operatorname{mod}m\end{subarray}}^{m-1}\prod_{i=1}^{k}\left(q^{1-i}\zeta_{m}^{p-t}-1\right)\right)$
$\displaystyle=\frac{(-1)^{n(m-1)}(q^{n}-\zeta_{m}^{p})}{q^{k(k+1)/2}(q-1)^{n}(q^{n-k}-\zeta_{m}^{p})}\left(\prod_{j=1}^{n-k}\left(q^{j}-1\right)\right)\left(\prod_{j=1}^{n-k}(-1)^{m-1}\frac{q^{mj}-1}{q^{j}-1}\right)$
$\displaystyle\qquad\left(\prod_{i=1}^{k}\left(q^{i}-1\right)\right)\left((-1)^{m-1}\prod_{\begin{subarray}{c}t=0\\\
t\neq
p\operatorname{mod}m\end{subarray}}^{m-1}\left(1-\zeta_{m}^{p-t}\right)\right)\left(\prod_{i=1}^{k-1}(-1)^{m-1}q^{-i(m-1)}\frac{q^{mi}-1}{q^{i}-1}\right)$
$\displaystyle=\frac{m(q^{k}-1)(q^{n}-\zeta_{m}^{p})\prod_{j=1}^{n-k}(q^{mj}-1)\prod_{i=1}^{k-1}(q^{mi}-1)}{q^{k+m\binom{k}{2}}(q-1)^{n}(q^{n-k}-\zeta_{m}^{p})}.$

The Poincare polynomial for $G(m,1,n)$ is
$\prod_{i=1}^{n}\frac{q^{mi}-1}{q-1}$, so

(5.1.1) $\displaystyle\operatorname{Deg}_{\chi}(q)$
$\displaystyle=\frac{q^{k+m\binom{k}{2}}(q^{n-k}-\zeta_{m}^{p})\prod_{i=k}^{n}(q^{mi}-1)}{m(q^{k}-1)(q^{n}-\zeta_{m}^{p})\prod_{j=1}^{n-k}(q^{mj}-1)}$
$\displaystyle=\frac{q^{k+m\binom{k}{2}}(q^{n-k}-\zeta_{m}^{p})\prod_{i=k}^{n-1}(q^{mi}-1)}{m(q^{k}-1)\prod_{j=1}^{n-k}(q^{mj}-1)}\prod_{\begin{subarray}{c}j=0\\\
j\neq p\operatorname{mod}m\end{subarray}}^{m-1}(q^{n}-\zeta_{m}^{j}).$

Hence

$\displaystyle\operatorname{Deg}_{\chi}(\zeta_{mn}^{p})$
$\displaystyle=\frac{\zeta_{mn}^{pk}\zeta_{n}^{p\binom{k}{2}}(\zeta_{mn}^{-pk}\zeta_{m}^{p}-\zeta_{m}^{p})\prod_{i=k}^{n-1}(\zeta_{n}^{pi}-1)}{m(\zeta_{mn}^{pk}-1)\prod_{j=1}^{n-k}(\zeta_{n}^{pj}-1)}\prod_{\begin{subarray}{c}j=0\\\
j\neq p\operatorname{mod}m\end{subarray}}^{m-1}(\zeta_{m}^{p}-\zeta_{m}^{j})$
$\displaystyle=\frac{\zeta_{n}^{p\binom{k}{2}}\zeta_{m}^{p}(1-\zeta_{mn}^{pk})\prod_{i=k}^{n-1}(\zeta_{n}^{pi}-1)}{m(\zeta_{mn}^{pk}-1)\prod_{j=1}^{n-k}(-\zeta_{n}^{pj})(\zeta_{n}^{p(n-j)}-1)}(\zeta_{m}^{-p})\prod_{j=1}^{m-1}(1-\zeta_{m}^{j})$
$\displaystyle=\frac{-\zeta_{n}^{p\binom{k}{2}}}{(-1)^{n-k}\zeta_{n}^{p\binom{n-k+1}{2}}}\frac{\prod_{i=k}^{n-1}(\zeta_{n}^{pi}-1)}{\prod_{j=1}^{n-k}(\zeta_{n}^{p(n-j)}-1)}\frac{\prod_{j=1}^{m-1}(1-\zeta_{m}^{j})}{m}$
$\displaystyle=(-1)^{n-k+1}\zeta_{n}^{p\left(kn-\frac{n(n+1)}{2}\right)}=(-1)^{k}.$

Now let $\bm{\lambda}$ be any $m$-partition of $n$. Notice that the Poincare
polynomial evaluated at $\zeta_{mn}^{p}$ is always 0. So if the generic degree
of the representation corresponding to $\lambda$ evaluated at $\zeta_{mn}^{p}$
is nonzero, then the Schur element evaluated at $\zeta_{mn}^{p}$ must be 0.
Looking at the formula in Theorem 5.8, notice that the Schur element vanishes
at $\zeta_{mn}^{p}$ if and only if one of the factors
$\left(q^{h_{i,j}^{\lambda^{(s)},\lambda^{(t)}}}Q_{s}Q_{t}^{-1}-1\right)$
vanishes at $\zeta_{mn}^{p}$. This happens if and only if at least one of the
following is a multiple of $mn$:

  1. (1)

$ph_{i,j}^{\lambda^{(0)},\lambda^{(0)}}$

  2. (2)

$p(h_{i,j}^{\lambda^{(0)},\lambda^{(t)}}+1)-nt$

  3. (3)

$p(h_{i,j}^{\lambda^{(s)},\lambda^{(0)}}-1)+ns$

  4. (4)

$ph_{i,j}^{\lambda^{(s)},\lambda^{(t)}}+n(s-t)$,

where $s,t\geq 1$.

Expression (1) is a multiple of $mn$ if and only if
$h_{i,j}^{\lambda^{(0)},\lambda^{(0)}}$ is a multiple of $mn$, but this is not
possible because $1\leq h_{i,j}^{\lambda^{(0)},\lambda^{(0)}}\leq n$.

If expression (2) is a multiple of $mn$, then
$(h_{i,j}^{\lambda^{(0)},\lambda^{(t)}}+1)$ is a multiple of $n$. Now
$-(n-1)\leq h_{i,j}^{\lambda^{(0)},\lambda^{(t)}}\leq n-1$, so either
$h_{i,j}^{\lambda^{(0)},\lambda^{(t)}}=-1$ or $n-1$. If it equals $-1$, then
$-nt$ is a multiple of $mn$, which is not possible because $1\leq t\leq m-1$.
So $h_{i,j}^{\lambda^{(0)},\lambda^{(t)}}=n-1$, which can only happen if
$i=j=1$ and $\lambda^{(0)}_{1}+\lambda_{1}^{(t)^{\prime}}=n$. So for some
$1\leq k\leq n-1$ we have that $\lambda^{(0)}=n-k$ and $\lambda^{(t)}=1^{k}$.
Moreover, we must have $t=p\operatorname{mod}m$ because $p-t$ needs to be a
multiple of $m$. It follows that $\bm{\lambda}$ corresponds to the $k$th
exterior power of the $p$th Galois twist.

If expression (3) is a multiple of $mn$, then
$(h_{i,j}^{\lambda^{(s)},\lambda^{(0)}}-1)$ is a multiple of $n$, so
$h_{i,j}^{\lambda^{(s)},\lambda^{(0)}}=1$ or $-(n-1)$. If it equals $1$, then
$s$ must be a multiple of $m$ which is not possible. So
$h_{i,j}^{\lambda^{(s)},\lambda^{(0)}}=-(n-1)$, which can only happen if
$i=n$, $j=1$, $\lambda^{(s)}_{i}=1$, and $\lambda_{j}^{(0)^{\prime}}=0$. So we
have $\lambda^{(s)}=1^{n}$. Moreover, $s=p\operatorname{mod}m$, because $-p+s$
needs to be a multiple of $m$. It follows that $\bm{\lambda}$ corresponds to
the representation $\Lambda^{n}V^{\sigma_{p}}$.

If expression (4) is a multiple of $mn$, then
$h_{i,j}^{\lambda^{(s)},\lambda^{(t)}}$ is a multiple of $n$, which can only
happen if $h_{i,j}^{\lambda^{(s)},\lambda^{(t)}}=0$. So $s-t$ needs to be a
multiple of $m$, which implies that $s=t$. But if $s=t$, then
$h_{i,j}^{\lambda^{(s)},\lambda^{(t)}}\geq 1$, so we have a contradiction.

Thus we can conclude that $\operatorname{Deg}_{\chi}(\zeta_{mn}^{p})=0$ if
$\chi$ is not an exterior power of $V^{\sigma_{p}}$. ∎

###  5.2. Clifford Theory

######  Theorem 5.9 (Clifford’s Theorem, [Cli37]).

Let $k$ be a field, $G$ a group, and $N$ a normal subgroup of finite index in
$G$. Suppose that $V$ is a simple $kG$-module. Fix a simple summand $W$ of
$\operatorname{Res}^{G}_{N}(V)$, and define the inertia subgroup, $T(W)$, to
be the set of all $g\in G$ such that the $kN$-module $W^{g}=\\{g\cdot w:w\in
W\\}$ is isomorphic to $W$. If $X$ is a transversal to $T(W)$ in $G$, then
there is a positive integer $e$ (called the ramification index of $V$ in $N$)
such that

$\operatorname{Res}^{G}_{N}(V)\cong\bigoplus_{x\in X}(W^{x})^{\oplus e}.$

Clifford’s theorem has led to an area of representation called Clifford theory
that relates representations of $N$ with representations of $G$. See [Isa06]
or [Cra19, §7] for more on the general theory.

We will now describe the irreducible characters of the groups $G(m,m,n)$
following [Las12, §3.4.2] to emphasize the role of Clifford theory.

The character version of Clifford’s theorem is as follows:

######  Theorem 5.10 (Clifford’s Theorem).

(We will assume that $G$ is finite) Let $H$ be a normal subgroup of $G$,
$\phi$ an irreducible character of $G$, and $\chi$ an irreducible character of
$H$ such that $\langle\phi_{H},\chi\rangle_{H}\neq 0$. Then

$\phi_{H}=\langle\phi_{H},\chi\rangle_{H}\sum_{g\in G/I_{G}(\chi)}\chi^{g},$

where $\chi^{g}(h)=\chi(g^{-1}hg)$ and $I_{G}(\chi)=\\{g\in
G:\chi^{g}(h)=\chi(h)\text{ for all }h\in H\\}$.

We will also need:

######  Theorem 5.11 ([Isa06], Theorem 6.11).

Let $H\trianglelefteq G$, $\chi\in\operatorname{Irr}H$, and $T=I_{G}(\chi)$.
Define

$A=\\{\psi\in\operatorname{Irr}T:\langle\psi_{H},\chi\rangle_{H}\neq
0\\},\qquad
B=\\{\phi\in\operatorname{Irr}G:\langle\phi_{H},\chi\rangle_{H}\neq 0\\}.$

  1. (1)

If $\psi\in A$ then $\operatorname{Ind}_{T}^{G}\psi$ is irreducible.

  2. (2)

The map $\psi\mapsto\operatorname{Ind}_{T}^{G}\psi$ is a bijection of $A$ onto
$B$.

  3. (3)

If $\operatorname{Ind}_{T}^{G}\psi=\phi$ with $\psi\in A$, then $\psi$ is the
unique irreducible constituent of $\phi_{T}$ which lies in $A$, and
$\langle\psi_{H},\chi\rangle_{H}=\langle\phi_{H},\chi\rangle_{H}$.

Suppose that $G=H\rtimes\langle t\rangle$, with $t^{m}=1$. Now for
$\chi\in\operatorname{Irr}H$, the inertia group is
$T=I_{G}(\chi)=H\rtimes\langle t^{d}\rangle$ for $d$ a divisor of $m$. To see
this, note that if $t^{k}\in T$, then

$\chi((nt^{k})^{-1}h(nt^{k}))=\chi(t^{-k}n^{-1}hnt^{k})=\chi(n^{-1}hn)=\chi(h)\implies
nt^{k}\in T,$

for all $n,h\in H$. If $V$ is a $\mathbb{C}$-vector space affording the
irreducible representation corresponding to $\chi$, then there exists an
isomorphism $\varphi_{0}:V\to V$ such that
$t^{-d}ht^{d}\varphi_{0}(v)=\varphi_{0}(hv)$ for any $h\in H$ and $v\in V$
(since $t^{d}\in T$). By Schur’s lemma, there exists $\lambda\in\mathbb{C}$
such that $\varphi:=\lambda\varphi_{0}$ satisfies
$\varphi^{m/d}=\text{Id}_{V}$ (the exponent here signifies repeated
composition). Now define an action of $t^{d}$ on $V$ by $t^{d}\cdot
v=f^{-1}(v)$. This gives an action of $T$ on $V$ because

$h_{1}f^{-k_{1}}(h_{2}f^{-k_{2}}(v))=h_{1}(t^{dk_{1}}h_{2}t^{-dk_{1}})f^{-(k_{1}+k_{2})}(v).$

Hence there exists $\psi\in\operatorname{Irr}T$ such that $\psi_{H}=\chi$.

Let $\bar{\chi}$ be any irreducible character of $T$ such that
$\bar{\chi}_{H}=\chi$ (we have just shown that there is at least one such
character). Now,
$\operatorname{Ind}_{H}^{T}\chi=\sum_{j=1}^{m/d}\bar{\chi}\otimes\xi^{j}$,
where $\xi\in\operatorname{Irr}T$ is the linear character with $\xi(H)=1$ and
$\xi(t^{d})=\exp(2d\pi i/m)$ (this can be easily checked using Frobenius
reciprocity and comparing degrees). So the
$\sum_{j=1}^{m/d}\bar{\chi}\otimes\xi^{j}$ are precisely the irreducible
characters of $T$ which restrict to $\chi$ on $H$, and
$\operatorname{Ind}_{H}^{G}\chi=\sum_{j=1}^{m/d}\operatorname{Ind}_{T}^{G}(\bar{\chi}\otimes\xi^{j})$,
where each $\operatorname{Ind}_{T}^{G}(\bar{\chi}\otimes\xi^{j})$ is
irreducible by Theorem 5.11. Now
$\operatorname{Ind}_{T}^{G}(\bar{\chi}\otimes\xi^{j})=\operatorname{Ind}_{T}^{G}(\bar{\chi})\otimes\epsilon^{j}$,
where $\epsilon$ is the linear character with $\epsilon(H)=1$ and
$\epsilon(t)=\exp(2\pi i/m)$. So
$\operatorname{Ind}_{H}^{G}\chi=\sum_{j=1}^{m/d}\operatorname{Ind}_{T}^{G}(\bar{\chi})\otimes\epsilon^{j}$,
and $m/d$ can be characterized as the smallest positive integer $k$ such that
$\operatorname{Ind}_{T}^{G}(\bar{\chi})\otimes\epsilon^{k}=\operatorname{Ind}_{T}^{G}(\bar{\chi})$.

Now suppose that we have some $\phi\in\operatorname{Irr}G$ such that
$\langle\phi_{H},\chi\rangle_{H}\neq 0$. By Frobenius reciprocity, it then
follows that $\phi=\operatorname{Ind}_{T}^{G}(\bar{\chi})\otimes\epsilon^{j}$
for some $j\in\\{1,\dots,m/d\\}$. Since all of the $\bar{\chi}\otimes\xi^{j}$
satisfy $(\bar{\chi}\otimes\xi^{j})_{H}=\chi$, we can assume that
$\phi=\operatorname{Ind}_{T}^{G}(\bar{\chi})$. So $m/d$ is the smallest
positive integer $k$ such that $\phi\otimes\epsilon^{k}=\phi$. Now
$\langle\phi_{H},\chi\rangle_{H}=\langle\bar{\chi}_{H},\chi\rangle_{H}=1$, so
by Clifford’s theorem we have

$\phi_{H}=\sum_{j=0}^{d-1}\chi^{t^{j}}.$

Recall (see [LT09, §2.7]) that $G(m,m,n)$ is the subgroup of $G(m,1,n)$
generated by the reflections $\\{t^{-1}s_{1}t,s_{1},s_{2},\dots,s_{n-1}\\}$.
Moreover, $G(m,m,n)$ is a normal subgroup of $G(m,1,n)$, and the quotient
$G(m,1,n)/G(m,m,n)$ is cyclic, generated by the image of $t$. Hence
$G(m,1,n)\cong G(m,m,n)\rtimes\langle t\rangle$, so we can apply the above
result to these groups.

For an $m$-partition $\bm{\lambda}=(\lambda^{(0)},\dots,\lambda^{(m-1)})$ of
$n$, denote by $\pi(\bm{\lambda})$ the cyclic permutation
$(\lambda^{(m-1)},\lambda^{(0)},\dots,\lambda^{(m-2)})$. Let
$\langle\pi\rangle$ denote the cyclic group of order $m$, and let
$s(\bm{\lambda})$ be the size of the subgroup of $\langle\pi\rangle$ fixing
$\bm{\lambda}$. Notice that
$\chi_{\bm{\lambda}}\otimes\epsilon=\chi_{\pi(\bm{\lambda})}$, so the smallest
positive integer $k$ satisfying
$\chi_{\bm{\lambda}}=\chi_{\bm{\lambda}}\otimes\epsilon^{k}=\chi_{\pi^{k}(\bm{\lambda})}$
is $m/s(\bm{\lambda})$. It follows that the restriction of
$\chi_{\bm{\lambda}}$ to $G(m,m,n)$ is the sum of $s(\bm{\lambda})$ distinct
(conjugate) irreducible characters. Moreover, each irreducible character of
$G(m,m,n)$ is in the restriction of exactly $m/s(\bm{\lambda})$ irreducible
characters of $G(m,1,n)$, which are equivalent under the action of
$\langle\pi\rangle$ and have the same decomposition when restricted to
$G(m,m,n)$. That is, the correspondence is as follows:

$\displaystyle\\{\chi_{\bm{\lambda}},\chi_{\pi(\bm{\lambda})},\dots,\chi_{\pi^{m/s(\bm{\lambda})-1}(\bm{\lambda})}\\}\in\operatorname{Irr}(G(m,1,n))$
$\displaystyle\leftrightarrow\left\\{\psi,\psi^{t},\dots,\psi^{t^{s(\bm{\lambda})-1}}\right\\}\in\operatorname{Irr}(G(m,m,n))$
$\displaystyle(\chi_{\pi^{j}(\bm{\lambda})})_{G(m,m,n)}$
$\displaystyle=\psi+\psi^{t}+\cdots+\psi^{t^{s(\bm{\lambda})-1}}$
$\displaystyle\chi_{\bm{\lambda}}+\chi_{\pi(\bm{\lambda})}+\cdots+\chi_{\pi^{m/s(\bm{\lambda})-1}(\bm{\lambda})}$
$\displaystyle=\operatorname{Ind}_{G(m,m,n)}^{G(m,1,n)}\psi^{t^{j}}.$

There is a similar description of the irreducible characters of $G(m,p,n)$,
and there are are explicit models for the irreducible representations of
$G(m,p,n)$ related to those shown above in the case $G(m,1,n)$. See [Ari95]
and [MM10, §2.4].

######  Example 5.12 (Exterior powers and Galois twists).

Let $V$ be the reflection representation of $G(m,1,n)$. Then the reflection
representation $\tilde{V}$ of $G(m,m,n)$ is the restriction of the action of
$G(m,1,n)$ on $V$. Now consider the $G(m,m,n)$ representation
$\Lambda^{k}\tilde{V}^{\sigma_{p}}$, where $p$ is coprime to
$h=\max\\{n,(n-1)m\\}$ (the Coxeter number of $G(m,m,n)$) and
$\sigma_{p}\in\operatorname{Gal}(\mathbb{Q}(\zeta_{h})/\mathbb{Q})$. Assuming
that $\Lambda^{k}V^{\sigma_{p}}$ is well-defined as a representation of
$G(m,1,n)$, it is easy to see (by looking at the characters, for example) that
the restriction of $\Lambda^{k}V^{\sigma_{p}}$ to $G(m,m,n)$ is precisely
$\Lambda^{k}\tilde{V}^{\sigma_{p}}$.

For the groups $G(m,1,n)$, the field of definition is $\mathbb{Q}(\zeta_{m})$.
If $n>(n-1)m$, there are only two possibilities:

  * •

If $n=1$, then $G(m,1,1)$ is a cyclic group and $G(m,m,1)$ is the trivial
group. In this case $\operatorname{Gal}(\mathbb{Q}(\zeta_{h})/\mathbb{Q})$ is
trivial, so $\Lambda^{k}V^{\sigma_{p}}$ is clearly well-defined.

  * •

If $n\geq 2$ and $m=1$, then $G(m,1,n)=G(1,1,n)$ is not irreducible.

Otherwise $(n-1)m>n$, and $\Lambda^{k}V^{\sigma_{p}}$ is well defined because
$\mathbb{Q}(\zeta_{m})\subseteq\mathbb{Q}(\zeta_{h})$.

So the $m$-partition (up to cyclic permutation) corresponding to
$\Lambda^{k}\tilde{V}^{\sigma_{p}}$ is
$\bm{\lambda}_{k,p}=(n-k,\varnothing,\dots,\varnothing,1^{k},\varnothing,\dots,\varnothing)$,
where the $1^{k}$ is in the $p$th slot (mod $m$; indexing starts at 0). If
$\bm{\lambda}_{k,p}$ has cyclic symmetry, then $n=2$, $m$ is even, and
$p=m/2$. In this case $(n-1)m>n$, so since $p$ must be relatively prime to
$m$, we have $m=2$. But the group $G(2,2,2)$ is not irreducible. So we have
$s(\bm{\lambda}_{k,p})=1$ which agrees with the observation that the
restriction of $\Lambda^{k}V^{\sigma_{p}}$ to $G(m,1,n)$ is precisely
$\Lambda^{k}\tilde{V}^{\sigma_{p}}$.

Notice that $\Lambda^{n}\tilde{V}^{\sigma_{p}}$ does not depend on $p$ because
$\bm{\lambda}_{n,p}=\pi^{p}(\bm{\lambda}_{n,1})$.

###  5.3. Unipotent characters and Generic Degrees for $G(m,m,n)$

We now give Malle’s construction [Mal95] [Las12] of the unipotent characters
and generic degrees for the groups $G(m,m,n)$.

Define an equivalence relation on $m$-symbols as the symmetric transitive
closure of the two operations: cyclic permutation of the $S_{i}$ in $S$, and
_simultaneous shift_ of all of the $S_{i}$ given by

$(\lambda_{i,1},\dots,\lambda_{i,M_{i}})\mapsto(0,\lambda_{i,1}+1,\dots,\lambda_{i,M_{i}}+1).$

Denote by $[S]$ the equivalence class of an $m$-symbol $S$.

For $S$ an $m$-symbol with $I(S)\equiv 0\operatorname{mod}m$, define its
_defect_ by

$\operatorname{def}(S)=\left(\frac{m-1}{2}I(S)-\sum_{i=0}^{m-1}iM_{i}\right)\operatorname{mod}m.$

Define $s(S)$ as the cardinality of the set $\\{0\leq i\leq
e-1:\pi^{i}(S)=S\\}$, where $\pi$ cyclically permutes the $S_{i}$ in $S$. In
the case $I(S)=mk$, the formula for rank takes the form

$\operatorname{rk}(S)=\sum_{i=0}^{m-1}\sum_{\lambda\in
S_{i}}\lambda-m\binom{k}{2}.$

######  Definition 5.13.

The _unipotent characters_ of $G(m,m,n)$ are the elements of the multiset

$\operatorname{Uch}(G(m,m,n)):=\\{\text{equivalence classes
$[S]$}:\operatorname{rk}(S)=n,\,\,I(S)\equiv\operatorname{def}(S)\equiv
0\operatorname{mod}m\\},$

where the multiplicity of $[S]$ in $\operatorname{Uch}(G(m,m,n))$ is $s(S)$
(none of these conditions/values depend on the choice of representative of
$[S]$).

Families of unipotent characters will be defined as follows: If $s(S)=m$,
we’ll say that each copy of $[S]$ belongs to its own family consisting of only
one element. If $[S]$, $[S^{\prime}]$ have representatives $S_{1}\in[S]$ and
$S_{1}^{\prime}\in[S^{\prime}]$ such that the entries of $S_{1}$ and
$S_{1}^{\prime}$ coincide as multisets and $s(S)<m$, then $s(S^{\prime})<m$,
and we’ll say that all copies of $[S]$ and $[S^{\prime}]$ belong to the same
family.

Let $S$ be any representative of $[S]$, then the _generic degree_ of $[S]$ is

$\operatorname{Deg}_{[S]}(q):=\frac{m}{s(S)}\cdot\frac{(-1)^{\binom{m}{2}\binom{\ell}{2}+\gamma(S)}(q^{n}-1)\left(\prod_{i=1}^{n-1}(q^{mi}-1)\right)\cdot\prod_{i=0}^{m-1}\prod_{j=i}^{m-1}\prod_{\begin{subarray}{c}(\lambda,\mu)\in
S_{i}\times S_{j}\\\ \mu<\lambda\text{ if
}i=j\end{subarray}}(q^{\lambda}\zeta_{m}^{i}-q^{\mu}\zeta_{m}^{j})}{\tau(m)^{\ell}q^{\binom{m(\ell-1)}{2}+\binom{m(\ell-2)}{2}+\cdots}\prod_{i=0}^{m-1}\Theta(S_{i},q^{m})}$

where

$\ell=\frac{I(S)}{m},\qquad\gamma(S)=\frac{\operatorname{def}(S)}{m}(m\ell-1),$

and $\tau(m)$ and $\Theta$ are as in Definition 5.4. Malle’s definition in
[Mal95] does not include the $(-1)^{\gamma(S)}$ term, in which case the
generic degree is defined only up to sign. We have followed [Las12] in this
definition so that the generic degree does not depend on the choice of
representative.

Again there is an inclusion
$\operatorname{Irr}(G(m,m,n))\subset\operatorname{Uch}(G(m,m,n))$. Let
$\bm{\lambda}=(\lambda^{(0)},\dots,\lambda^{(m-1)})$ be an $m$-partition of
$n$. Let $\alpha_{i}=(0\leq\alpha_{i,1}\leq\cdots\leq\alpha_{i,M})$ be the
parts of $\lambda^{(i)}$ written in increasing order for $i\geq 0$ with 0’s
added if necessary. Define an $m$-symbol
$\tilde{S}_{\bm{\lambda}}=(\tilde{S}_{0},\dots,\tilde{S}_{m-1})$ by
$\tilde{S}_{i,j}=\alpha_{i,j}+j-1$. Then $\tilde{S}_{\bm{\lambda}}$ is a
symbol of rank $n$ with $I(\tilde{S}_{\bm{\lambda}})=mM$ and
$\operatorname{def}(\tilde{S}_{\bm{\lambda}})=0$. The inclusion is then
defined by mapping the $s(\bm{\lambda})$ components of the restriction
$(\chi_{\bm{\lambda}})_{G(m,m,n)}$ to the
$s(\tilde{S}_{\bm{\lambda}})=s(\bm{\lambda})$ copies of
$[\tilde{S}_{\bm{\lambda}}]$ in $\operatorname{Uch}(G(m,m,n))$.

Under this inclusion, the definition of the generic degree of an irreducible
character given in Definition 3.15 coincides with the definition for unipotent
characters.

######  Proposition 5.14 ([Mal95], Proposition 5.5).

The fake degree of a component of the restriction
$(\chi_{\bm{\lambda}})_{G(m,m,n)}$, with corresponding $m$-symbol
$\tilde{S}_{\bm{\lambda}}=(\tilde{S}_{0},\dots,\tilde{S}_{m-1})$, is equal to

$\frac{q^{n}-1}{s(\tilde{S}_{\bm{\lambda}})}\cdot\left(\prod_{i=1}^{n-1}(q^{mi}-1)\right)\cdot\left(\prod_{i=0}^{m-1}\frac{\Delta(\tilde{S}_{i},q^{m})}{\Theta(\tilde{S}_{i},q^{m})}\right)\cdot\frac{\sum_{j=0}^{m-1}\prod_{i=1}^{m-1}\prod_{\lambda\in\tilde{S}_{(i+j)\operatorname{mod}m}}q^{(m-i)\lambda}}{q^{\binom{m(\ell-1)}{2}+\binom{m(\ell-2)}{2}+\cdots}},$

where $\Delta$ is as in Proposition 5.5 and $\ell$ and $\Theta$ are as in
Definition 5.13. We have again followed [Las12, Proposition 5.4.13] so that
this formula agrees with the usual definition of fake degrees.

######  Example 5.15.

The $m$-symbol corresponding to $\tilde{V}^{\sigma_{p}}$ is given by

$\tilde{S}_{i}=\begin{cases}(n-1)&\text{if }i=0,\\\ (1)&\text{if
}i=p\operatorname{mod}m,\\\ (0)&\text{otherwise}.\end{cases}$

So

$\Delta(\tilde{S}_{i},q^{m})=1,\qquad\text{and}\qquad\Theta(\tilde{S}_{i},q^{m})=\begin{cases}\prod_{j=1}^{n-1}(q^{mj}-1)&\text{if
}i=0,\\\ (q^{m}-1)&\text{if }i=p\operatorname{mod}m,\\\
1&\text{otherwise}.\end{cases}$

Letting $\chi_{1,p}$ denote the character of $\tilde{V}^{\sigma_{p}}$, we then
have

$\displaystyle\operatorname{Feg}_{\chi_{1,p}}(q)$
$\displaystyle=\frac{q^{n}-1}{q^{m}-1}\sum_{j=0}^{m-1}q^{j(n-1)+((j-p)\operatorname{mod}m)}$
$\displaystyle=\frac{q^{n}-1}{q^{m}-1}\left(\sum_{j=0}^{(p\operatorname{mod}m)-1}q^{m+jn-(p\operatorname{mod}m))}+\sum_{j=p\operatorname{mod}m}^{m-1}q^{jn-(p\operatorname{mod}m)}\right)$
$\displaystyle=\frac{q^{n}-1}{q^{m}-1}\left((q^{m}-1)\sum_{j=0}^{(p\operatorname{mod}m)-1}q^{jn-(p\operatorname{mod}m)}+\sum_{j=0}^{m-1}q^{jn-(p\operatorname{mod}m)}\right)$
$\displaystyle=(q^{n}-1)\sum_{j=0}^{(p\operatorname{mod}m)-1}q^{jn-(p\operatorname{mod}m)}+q^{-(p\operatorname{mod}m)}\cdot\frac{q^{n}-1}{q^{m}-1}\cdot\frac{q^{mn}-1}{q^{n}-1}$
$\displaystyle=q^{-(p\operatorname{mod}m)}\left(q^{n(p\operatorname{mod}m)}-1\right)+\sum_{j=0}^{n-1}q^{jm-(p\operatorname{mod}m)}$
$\displaystyle=q^{(n-1)(p\operatorname{mod}m)}+\sum_{j=1}^{n-1}q^{jm-(p\operatorname{mod}m)}=q^{p(n-1)\operatorname{mod}m(n-1)}+\sum_{j=1}^{n-1}q^{p(jm-1)\operatorname{mod}m(n-1)}$
$\displaystyle=\sum_{j=1}^{n}q^{pe_{i}\operatorname{mod}h},$

which demonstrates Example 2.15.

###### Proof of Theorem 3.17.

Let $\bm{\lambda}$ be the $m$-partition of $n$ corresponding to
$\Lambda^{k}V^{\sigma_{p}}$, where $V$ is the reflection representation of
$G(m,1,n)$ and $p$ is coprime to $(n-1)m$. The result is obvious for $k=0$, so
suppose that $k\geq 1$. Then $S_{\bm{\lambda}}=(S_{0},\dots,S_{m-1})$, where

$S_{i}=\begin{cases}(0,1,\dots,k-1,n)&\text{if }i=0,\\\ (1,2,\dots,k)&\text{if
}i=p\operatorname{mod}m,\\\ (0,1,\dots,k-1)&\text{otherwise},\end{cases}$

and $\tilde{S}_{\bm{\lambda}}=(\tilde{S}_{0},\dots,\tilde{S}_{m-1})$, where

$\tilde{S}_{i}=\begin{cases}(0,1,\dots,k-2,n-1)&\text{if }i=0,\\\
(1,2,\dots,k-1,k)&\text{if }i=p\operatorname{mod}m,\\\
(0,1,\dots,k-2,k-1)&\text{otherwise}.\end{cases}$

When $k=1$, $\tilde{S}_{0}=(n-1)$. Comparing Definition 5.4 and Definition
5.13, one sees that

$\displaystyle\frac{\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)}{\operatorname{Deg}_{S_{\bm{\lambda}}}(q)}$
$\displaystyle=\frac{mq^{m\binom{k}{2}}(q^{n}-1)}{(q^{mn}-1)}\cdot\left((q^{mn}-1)\prod_{i=1}^{k-1}(q^{mi}-1)\right)\cdot\frac{\left(\prod_{i=0}^{k-2}(q^{n-1}-q^{i})\right)}{\left(\prod_{i=0}^{k-2}(q^{k-1}-q^{i})\right)\left(\prod_{i=0}^{k-1}(q^{n}-q^{i})\right)}$
$\displaystyle\qquad\left(\prod_{j=1}^{m-1}\prod_{\mu=0}^{k-1}\frac{(q^{n-1}-q^{\mu}\zeta_{m}^{j})}{(q^{k-1}-q^{\mu}\zeta_{m}^{j})(q^{n}-q^{\mu}\zeta_{m}^{j})}\right)\cdot\frac{(q^{n-1}-q^{k}\zeta_{m}^{p})(q^{k-1}-\zeta_{m}^{p})(q^{n}-\zeta_{m}^{p})}{(q^{k-1}-q^{k}\zeta_{m}^{p})(q^{n}-q^{k}\zeta_{m}^{p})(q^{n-1}-\zeta_{m}^{p})}$
$\displaystyle=mq^{m\binom{k}{2}}(q^{n}-1)\left(\prod_{i=1}^{k-1}(q^{mi}-1)\right)\frac{1}{q^{(k-1)+\binom{k-1}{2}}(q^{n}-1)\prod_{i=1}^{k-1}(q^{i}-1)}$
$\displaystyle\qquad\left(\prod_{\mu=0}^{k-2}\frac{(q^{(n-\mu-1)m}-1)(q^{k-\mu-1}-1)(q^{n-\mu}-1)}{q^{\mu(m-1)}(q^{(k-\mu-1)m}-1)(q^{(n-\mu)m}-1)(q^{n-\mu-1}-1)}\right)$
$\displaystyle\qquad\left(\frac{(q^{n-k+1}-1)}{mq^{(k-1)(m-1)}(q^{(n-k+1)m}-1)}\prod_{j=1}^{m-1}(q^{n-k}-\zeta_{m}^{j})\right)\cdot\frac{(q^{n-k-1}-\zeta_{m}^{p})(q^{k-1}-\zeta_{m}^{p})(q^{n}-\zeta_{m}^{p})}{(q^{k-1}-q^{k}\zeta_{m}^{p})(q^{n-k}-\zeta_{m}^{p})(q^{n-1}-\zeta_{m}^{p})}$
$\displaystyle=\left(\prod_{i=1}^{k-1}(q^{mi}-1)\right)\frac{q^{m\binom{k}{2}}}{q^{(k-1)+\binom{k-1}{2}}\prod_{i=1}^{k-1}(q^{i}-1)}$
$\displaystyle\qquad\left(\frac{1}{q^{(m-1)\binom{k-1}{2}}}\cdot\frac{\left(\prod_{\mu=n-k+1}^{n-1}(q^{m\mu}-1)\right)\left(\prod_{\mu=1}^{k-1}(q^{\mu}-1)\right)\left(\prod_{\mu=n-k+2}^{n}(q^{\mu}-1)\right)}{\left(\prod_{\mu=1}^{k-1}(q^{m\mu}-1)\right)\left(\prod_{\mu=n-k+2}^{n}(q^{m\mu}-1)\right)\left(\prod_{\mu=n-k+1}^{n-1}(q^{\mu}-1)\right)}\right)$
$\displaystyle\qquad\left(\frac{(q^{n-k+1}-1)}{q^{(k-1)(m-1)}(q^{(n-k+1)m}-1)}\prod_{j=1}^{m-1}(q^{n-k}-\zeta_{m}^{j})\right)\cdot\frac{(q^{n-k-1}-\zeta_{m}^{p})(q^{k-1}-\zeta_{m}^{p})(q^{n}-\zeta_{m}^{p})}{(q^{k-1}-q^{k}\zeta_{m}^{p})(q^{n-k}-\zeta_{m}^{p})(q^{n-1}-\zeta_{m}^{p})}$
$\displaystyle=\frac{q^{m\binom{k}{2}}\prod_{j=1}^{m-1}(q^{n-k}-\zeta_{m}^{j})}{q^{m(k-1)+m\binom{k-1}{2}}}\left(\frac{(q^{m(n-k+1)}-1)(q^{n}-1)}{(q^{mn}-1)(q^{n-k+1}-1)}\right)\left(\frac{(q^{n-k+1}-1)}{(q^{(n-k+1)m}-1)}\right)$
$\displaystyle\qquad\frac{(q^{n-k-1}-\zeta_{m}^{p})(q^{k-1}-\zeta_{m}^{p})(q^{n}-\zeta_{m}^{p})}{(q^{k-1}-q^{k}\zeta_{m}^{p})(q^{n-k}-\zeta_{m}^{p})(q^{n-1}-\zeta_{m}^{p})}$
$\displaystyle=\frac{\alpha(q^{n}-1)(q^{n-k-1}-\zeta_{m}^{p})(q^{k-1}-\zeta_{m}^{p})(q^{n}-\zeta_{m}^{p})}{(q^{mn}-1)(q^{k-1}-q^{k}\zeta_{m}^{p})(q^{n-k}-\zeta_{m}^{p})(q^{n-1}-\zeta_{m}^{p})},$

where

$\alpha=\prod_{j=1}^{m-1}(q^{n-k}-\zeta_{m}^{j})=\begin{cases}m&\text{if
}k=n,\\\ \frac{q^{m(n-k)}-1}{q^{n-k}-1}&\text{otherwise}.\end{cases}$

We will first handle the case $k=n$. By Equation 5.1.1,

$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)=\frac{q^{n+m\binom{n}{2}}}{q^{n}}=q^{m\binom{n}{2}}\implies\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta^{p}_{m(n-1)})=(-1)^{np}.$

If $n$ is odd, then $p$ must be odd because it is coprime to $n-1$, so then
$np$ is odd. If $n$ is even, then certainly $np$ is even. Thus
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta^{p}_{m(n-1)})=(-1)^{n}$.

Now for $0<k<n$, by Equation 5.1.1,

$\displaystyle\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)$
$\displaystyle=\frac{q^{k+m\binom{k}{2}}(q^{n}-1)(q^{n-k-1}-\zeta_{m}^{p})(q^{k-1}-\zeta_{m}^{p})\prod_{i=k}^{n-1}(q^{mi}-1)}{m(q^{n-k}-1)(q^{k-1}-q^{k}\zeta_{m}^{p})(q^{n-1}-\zeta_{m}^{p})(q^{k}-1)\prod_{j=1}^{n-k-1}(q^{mj}-1)}$
$\displaystyle=\frac{q^{k+m\binom{k}{2}}(q^{n}-1)(q^{n-k-1}-\zeta_{m}^{p})(q^{k-1}-\zeta_{m}^{p})\prod_{i=k}^{n-2}(q^{mi}-1)}{m(q^{n-k}-1)(q^{k-1}-q^{k}\zeta_{m}^{p})(q^{k}-1)\prod_{j=1}^{n-k-1}(q^{mj}-1)}\prod_{\begin{subarray}{c}j=0\\\
j\neq p\operatorname{mod}m\end{subarray}}^{m-1}(q^{n-1}-\zeta_{m}^{j}).$

Hence,

$\displaystyle\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta_{m(n-1)}^{p})$
$\displaystyle=\frac{\zeta_{m(n-1)}^{pk}\zeta_{n-1}^{p\binom{k}{2}}(\zeta_{m(n-1)}^{pn}-1)(\zeta_{m}^{p}\zeta_{m(n-1)}^{-kp}-\zeta_{m}^{p})(\zeta_{m(n-1)}^{p(k-1)}-\zeta_{m}^{p})\prod_{i=k}^{n-2}(\zeta_{n-1}^{pi}-1)}{m(\zeta_{m(n-1)}^{p(n-k)}-1)(\zeta_{m(n-1)}^{p(k-1)}-\zeta_{m(n-1)}^{pk}\zeta_{m}^{p})(\zeta_{m(n-1)}^{pk}-1)\prod_{j=1}^{n-k-1}(\zeta_{n-1}^{pj}-1)}$
$\displaystyle\qquad\cdot\prod_{\begin{subarray}{c}j=0\\\ j\neq
p\operatorname{mod}m\end{subarray}}^{m-1}(\zeta_{m}^{p}-\zeta_{m}^{j})$
$\displaystyle=\frac{\zeta_{m(n-1)}^{pk}\zeta_{n-1}^{p\binom{k}{2}}(\zeta_{m(n-1)}^{pn}-1)(\zeta_{m}^{p})(\zeta_{m(n-1)}^{-pk}-1)(-\zeta_{m(n-1)}^{p(k-1)})(\zeta_{m(n-1)}^{p(n-k)}-1)}{m(\zeta_{m(n-1)}^{p(n-k)}-1)(-\zeta_{m(n-1)}^{p(k-1)})(\zeta_{m(n-1)}^{pn}-1)(\zeta_{m(n-1)}^{pk}-1)}$
$\displaystyle\qquad\frac{\prod_{i=k}^{n-2}(\zeta_{n-1}^{pi}-1)}{\prod_{j=1}^{n-k-1}(-\zeta_{n-1}^{pj})(\zeta_{n-1}^{p(n-j-1)}-1)}(\zeta_{m}^{-p}m)$
$\displaystyle=\frac{\zeta_{m(n-1)}^{pk}\zeta_{n-1}^{p\binom{k}{2}}(\zeta_{m(n-1)}^{-pk}-1)}{(-1)^{n-k-1}\zeta_{n-1}^{p\binom{n-k}{2}}(\zeta_{m(n-1)}^{pk}-1)}=(-1)^{p(2k-n)-(n-k)}=(-1)^{np}(-1)^{n}(-1)^{k}=(-1)^{k}.$

Now suppose that $\bm{\lambda}$ is any $m$-partition of $n$ such that
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta_{m(n-1)}^{p})\neq 0$.
Choose $\tilde{S}_{\bm{\lambda}}=(\tilde{S}_{0},\dots,\tilde{S}_{m-1})$ so
that $I(\tilde{S}_{\bm{\lambda}})$ is minimal. Let
$\tilde{k}:=I(\tilde{S}_{\bm{\lambda}})/m=\max\\{\text{number of parts in
}\lambda^{(i)}:i=0,\dots,n-1\\}$. Since
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)$ has the factor
$(q^{m(n-1)}-1)$ in the numerator, at least one of the
$\Theta(\tilde{S}_{i},\zeta_{n-1}^{p})$ is zero. This implies that there is at
least one $\mu\in\bigcup\tilde{S}_{i}$ with $\mu\geq n-1$. There are four
cases to consider:

  1. (1)

Suppose that there is some $j>0$ such that (up to cyclic permutation)

$\lambda^{(t)}=\begin{cases}n-\tilde{k}&\text{if }t=0\\\
1^{\tilde{k}}&\text{if }t=j\\\ \varnothing&\text{otherwise}.\end{cases}$

Then the $m$-symbol is

$\tilde{S}_{t}=\begin{cases}(0,1,\dots,\tilde{k}-2,n-1)&\text{if }t=0,\\\
(1,2,\dots,\tilde{k}-1,\tilde{k})&\text{if }t=j,\\\
(0,1,\dots,\tilde{k}-2,\tilde{k}-1)&\text{otherwise}.\end{cases}$

If $\tilde{k}=n$, then $\bm{\lambda}$ corresponds to
$\Lambda^{n}\tilde{V}^{\sigma_{p}}$. So suppose that $\tilde{k}\leq n-1$. If
$j=p\operatorname{mod}m$, then $\bm{\lambda}$ corresponds to
$\Lambda^{\tilde{k}}\tilde{V}^{\sigma_{p}}$. We will show that this is the
only possibility. Suppose for the sake of contradiction that $j\neq
p\operatorname{mod}m$. Then $m\geq 3$ and there are two cases to consider:

     * •

If $\tilde{k}=n-1$, then there are exactly two $\mu\in\bigcup\tilde{S}_{i}$
equal to $n-1$. Since $p$ is relatively prime to $m$, we have that $p\neq
2p\,\,(\operatorname{mod}m)$, so the numerator of
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)$ has the factor
$\pm(q^{n-1}\zeta_{m}^{p}-\zeta_{m}^{2p})$. Additionally, since we are
assuming $j\neq p$, the numerator of
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)$ has the factor
$(q^{n-1}-\zeta_{m}^{p})$. But it then follows that
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta_{m(n-1)}^{p})=0$, a
contradiction.

     * •

If $\tilde{k}<n-1$, then there is only one $\mu\in\bigcup\tilde{S}_{i}$ equal
to $n-1$. Since we are assuming $j\neq p$, the numerator of
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)$ has the factor
$(q^{n-1}-\zeta_{m}^{p})$. So
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta_{m(n-1)}^{p})=0$, a
contradiction.

  2. (2)

Suppose that $1\leq\tilde{k}\leq n-1$ and (up to cyclic permutation)

$\lambda^{(t)}=\begin{cases}(n-\tilde{k}+1,1^{\tilde{k}-1})&\text{if }t=0\\\
\varnothing&\text{otherwise}.\end{cases}$

Then the $m$-symbol is

$\tilde{S}_{\bm{\lambda}}=\begin{pmatrix}1&2&\cdots&\tilde{k}-1&n\\\
0&1&\cdots&\tilde{k}-2&\tilde{k}-1\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&1&\cdots&\tilde{k}-2&\tilde{k}-1\end{pmatrix}$

If $\tilde{k}=1$, then $\bm{\lambda}$ corresponds to
$\Lambda^{0}\tilde{V}^{\sigma_{p}}$. So suppose that $2\leq\tilde{k}\leq n-1$.
The numerator of $\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)$ has the
factor $(q^{n}-q\zeta_{m}^{p})$. So
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta_{m(n-1)}^{p})=0$, a
contradiction.

  3. (3)

Suppose that $2\leq\tilde{k}\leq n-2$ and (up to cyclic permutation)

$\lambda^{(t)}=\begin{cases}(n-\tilde{k},2,1^{\tilde{k}-2})&\text{if }t=0\\\
\varnothing&\text{otherwise}.\end{cases}$

Then the $m$-symbol is

$\tilde{S}_{\bm{\lambda}}=\begin{pmatrix}1&2&\cdots&\tilde{k}-2&\tilde{k}&n-1\\\
0&1&\cdots&\tilde{k}-3&\tilde{k}-2&\tilde{k}-1\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&1&\cdots&\tilde{k}-3&\tilde{k}-2&\tilde{k}-1\end{pmatrix}$

The numerator of $\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)$ has the
factor $(q^{n-1}-\zeta_{m}^{p})$. So
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta_{m(n-1)}^{p})=0$, a
contradiction.

  4. (4)

Suppose that $2\leq\tilde{k}\leq n-2$ and there is some $j>0$ such that (up to
cyclic permutation)

$\lambda^{(t)}=\begin{cases}(n-\tilde{k},1^{\tilde{k}-1})&\text{if }t=0\\\
1&\text{if }t=j\\\ \varnothing&\text{otherwise}.\end{cases}$

Then the $m$-symbol is

$\tilde{S}_{t}=\begin{cases}(1,2,\dots,\tilde{k}-1,n-1)&\text{if }t=0\\\
(0,1,\dots,\tilde{k}-2,\tilde{k})&\text{if }t=j\\\
(0,1,\dots,\tilde{k}-2,\tilde{k}-1)&\text{otherwise}.\end{cases}$

The numerator of $\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(q)$ has the
factor $(q^{n-1}-\zeta_{m}^{p})$. So
$\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta_{m(n-1)}^{p})=0$, a
contradiction.

To see that these are the only possible cases, choose $\ell$ such that the
number of parts in $\lambda^{(\ell)}$ is $\tilde{k}$. Suppose that there is
some $\mu\in\tilde{S}_{i}$ with $\mu\geq n-1$ and $i\neq\ell$. Then
$\mu\leq(n-\tilde{k})-\tilde{k}+1=n-1$ with equality only if $\bm{\lambda}$ is
in case (1).

Suppose now that there is some $\mu\in\tilde{S}_{\ell}$ with $\mu\geq n-1$.

  * •

If $\tilde{S}_{\ell,\tilde{k}}=n$, then $\lambda^{(\ell)}_{1}=n-\tilde{k}+1$.
This is case (1) if $\tilde{k}=n$, and it is case (2) otherwise.

  * •

If $\tilde{S}_{\ell,\tilde{k}}=n-1$, then $\lambda^{(\ell)}_{1}=n-\tilde{k}$.
This is case (1) if $\tilde{k}=1$ or $n-1$. Otherwise it is case (3) or (4).

Thus $\operatorname{Deg}_{[\tilde{S}_{\bm{\lambda}}]}(\zeta_{m(n-1)}^{p})=0$
if $\bm{\lambda}$ does not correspond to an exterior power of
$\tilde{V}^{\sigma_{p}}$. ∎

######  Question 5.16.

Let $W$ be a well-generated irreducible complex reflection group with Coxeter
number $h$. Is it true that
$S_{\sigma_{p}(\chi)}(\sigma_{p}(q))=\sigma_{p}(S_{\chi}(q))$ for all
$\sigma_{p}\in\operatorname{Gal}(\mathbb{Q}(\zeta_{h})/\mathbb{Q})$ and
$q\in\mathbb{Q}(\zeta_{h})$?

###  5.4. Families of unipotent characters in general

Let $\mathcal{O}$ be a commutative ring with a unit, and let $A$ be an
$\mathcal{O}$-algebra. An _idempotent_ in $A$ is an element satisfying
$e^{2}=e$. Idempotents $e_{1}$ and $e_{2}$ are _orthogonal_ if
$e_{1}e_{2}=e_{2}e_{1}=0$. An idempotent is _central_ if it is contained in
the center of $A$. An idempotent is _primitive_ if it is non-zero and is not
equal to the sum of two non-zero orthgonal idempotents.

Let $e$ be a central primitive idempotent. The two-sided ideal $Ae$ inherits
an algebra structure, and we call the algebra $Ae$ a _block_ of $A$.

Let $W$ be an irreducible spetsial complex reflection group. Define the
_Rouquier ring_ $\mathcal{R}_{W}(y)$ to be the $\mathbb{Z}_{k_{W}}$-subalgebra
of $k_{W}(y)$ given by

$\mathcal{R}_{W}(y):=\mathbb{Z}_{k_{W}}[y,y^{-1},(y^{n}-1)^{-1}_{n\geq 1}].$

Then the _Rouquier blocks_ of $\mathcal{H}_{q}(W)$ are the blocks of the
algebra $\mathcal{R}_{W}(y)\mathcal{H}_{q}(W)$, where $y^{|\mu(k_{W})|}=q$.
See [Chl09] for much more on these blocks.

To each $\chi\in\operatorname{Irr}(W)$ we can associate a central primitive
idempotent $e_{\chi}$ in $k_{W}(y)\mathcal{H}_{q}(W)$ given by

$e_{\chi}:=\frac{1}{S_{\chi}(q)}\sum_{b\in\mathcal{B}}\chi_{q}(b)b^{\vee},$

where $\mathcal{B}$ is a basis of $\mathcal{H}_{q}(W)$ adapted to the
Wedderburn decomposition, and the $b^{\vee}$ form the dual basis with respect
to $\tau_{q}$ (see [GP00, Proposition 7.2.7] and [Chl09, Proposition 2.2.12]).

By [Chl09, Theorem 2.1.6], there exists a unique partition
$\operatorname{RB(W)}$ of $\operatorname{Irr}(W)$ such that

  * •

for each $B\in\operatorname{RB}(W)$, the element $e_{B}:=\sum_{\chi\in
B}e_{\chi}$ is a central primitive idempotent in
$\mathcal{R}_{W}(y)\mathcal{H}_{q}(W)$,

  * •

$1=\sum_{B\in\operatorname{RB}(W)}e_{B}$ and for every central idempotent $e$
of $\mathcal{R}_{W}(y)\mathcal{H}_{q}(W)$ there exists a subset
$\operatorname{RB}(W,e)$ of $\operatorname{RB}(W)$ such that
$e=\sum_{B\in\operatorname{RB}(W,e)}e_{B}$.

We then say that two characters $\chi,\phi\in\operatorname{Irr}(W)$ belong to
the same Rouquier block of $\mathcal{H}_{q}(W)$ if they belong to the same
element of $\operatorname{RB}(W)$. We then have

######  Theorem 5.17.

If $\chi$ and $\phi$ belong to the same Rouquier block of the spetsial Hecke
algebra, then $a_{\chi}=a_{\phi}$ and $A_{\chi}=A_{\phi}$.

See the remark at the end of chapter 4 of [Chl09] for notes on the proof and a
more general statement. For Weyl groups, these Rouquier blocks are precisely
the families of $\operatorname{Irr}(W)$. For the spetsial groups $G(m,1,n)$
and $G(m,m,n)$, the families defined above for unipotent characters recover
the Rouquier blocks via the inclusion
$\operatorname{Irr}(W)\hookrightarrow\operatorname{Uch}(W)$.

For $W$ an exceptional irreducible spetsial complex reflection group, there is
a set $\operatorname{Uch}(\mathbb{G})$ defined in [BMM14] for the
corresponding split spets. There is also a principal series
$\operatorname{Uch}(\mathbb{G},1)$ with bijection
$\operatorname{Irr}(W)\to\operatorname{Uch}(\mathbb{G},1)$. Moreover, by
[BMM14, Axiom 5.20] there is a partition of $\operatorname{Uch}(\mathbb{G})$
into families which recovers the Rouquier blocks of $\mathcal{H}_{q}(W)$ when
restricted to the principal series.

##  6\. Lusztig’s Fourier transform

The following lemma is our motivation for this section.

######  Lemma 6.1.

Suppose that there exists a pairing
$\\{-,-\\}_{W}:\operatorname{Irr}(W)\times\operatorname{Irr}(W)\to\mathbb{C}$
satisfying

  1. (T1)

For all $\chi\in\operatorname{Irr}(W)$, we have

$\operatorname{Deg}_{\chi}(q)=\sum_{\phi\in\operatorname{Irr}(W)}\\{\chi,\phi\\}_{W}\operatorname{Feg}_{\phi}(q).$

  2. (T2)

For all $\chi,\phi\in\operatorname{Irr}(W)$, we have
$\\{\chi,\phi\\}_{W}=\\{\phi,\chi\\}_{W}$.

  3. (T3)

For all $\chi,\phi\in\operatorname{Irr}(W)$ with $\\{\chi,\phi\\}_{W}\neq 0$,
we have $h_{\chi}=h_{\phi}$.

Then

$\sum_{\chi\in\operatorname{Irr}(W)}q_{1}^{f(\chi)}\operatorname{Feg}_{\chi}(q_{2})\operatorname{Deg}_{\chi}(q_{3})=\sum_{\chi\in\operatorname{Irr}(W)}q_{1}^{f(\chi)}\operatorname{Feg}_{\chi}(q_{3})\operatorname{Deg}_{\chi}(q_{2}),$

if $f$ satisfies $h_{\chi}=h_{\phi}\implies f(\chi)=f(\phi)$.

###### Proof.

This follows from a double-summation argument. ∎

######  Conjecture 6.2.

For all irreducible spetsial complex reflection groups $W$, there exists a
pairing $\\{-,-\\}_{W}$ satisfying (T1), (T2), and (T3).

This conjecture has been proven for $W$ a finite Coxeter group and
$W=G(m,1,n)$.

###  6.1. A general construction

This construction is due to Lusztig, but we will follow [GM20, 4.2.9]. Let $G$
be a finite group and $\mathscr{M}(G)$ the set of pairs $(x,\sigma)\in
G\times\operatorname{Irr}(C_{G}(x))$ modulo the relation
$(x,\sigma)\sim({}^{g}\\!x,\sigma^{g})$, where
$\sigma^{g}({}^{g}\\!y):=\sigma(y)$ for $y\in C_{G}(x)$. Lusztig defined a
pairing
$\\{\cdot,\cdot\\}_{G}:\mathscr{M}(G)\times\mathscr{M}(G)\to\mathbb{C}$ by

$\\{(x,\sigma),(y,\tau)\\}:=\frac{1}{|C_{G}(x)||C_{G}(y)|}\sum_{\begin{subarray}{c}g\in
G\\\ xgyg^{-1}=gyg^{-1}x\end{subarray}}\sigma(gyg^{-1})\tau(g^{-1}x^{-1}g).$

Let $\mathbf{M}_{G}$ be the operator on the space of functions
$\mathscr{M}(G)\to\mathbb{C}$ defined by

$(\mathbf{M}_{G}f)(x,\sigma):=\sum_{(y,\tau)\in\mathcal{M}(G)}\\{(x,\sigma),(y,\tau)\\}f(y,\tau).$

This is called the _non-abelian Fourier transform_ associated to $G$.

###  6.2. The Fourier transform for Weyl groups

Suppose we are in the setting of Section 4. That is, $\mathbf{G}$ is a simple
connected reductive group over $\overline{\mathbb{F}}_{p}$ with connected
center which has Weyl group $W$, and $F:\mathbf{G}\to\mathbf{G}$ is a
Frobenius map with respect to some $\mathbb{F}_{q}$-rational structure which
acts trivially on $W$. Also fix some maximally split torus $\mathbf{T}_{0}$.

For each family $\mathscr{U}\subseteq\operatorname{Uch}(\mathbf{G}^{F})$ with
corresponding family $\mathscr{F}\subseteq\operatorname{Irr}(W)$, Lusztig
[Lus84, §4] defines the following data:

  * •

A finite group $\mathscr{G}_{\mathscr{U}}$

  * •

A bijection $\mathscr{U}\to\mathscr{M}(\mathscr{G}_{\mathscr{U}})$,
$\rho\mapsto x_{\rho}$

  * •

A map $\Delta:\mathscr{M}(\mathscr{G}_{\mathscr{U}})\to\\{\pm 1\\}$ such that

$\langle\rho,R_{\phi}\rangle=\Delta(x_{\rho})\\{x_{\rho},x_{\phi}\\}$

for all $\rho\in\mathscr{U}$ and $\phi\in\mathscr{F}$, where $x_{\phi}$ is
defined via the inclusion
$\operatorname{Irr}(W)\hookrightarrow\mathcal{E}(\mathbf{G}^{F},(\mathbf{T}_{0}^{F},1))$
and $\\{\cdot,\cdot\\}$ is the non-abelian Fourier transform associated to
$\mathscr{G}_{\mathscr{U}}$. In our context $\Delta(x_{\rho})=1$ except for
some non-principal series characters in types $E_{7}$ and $E_{8}$.

The class functions

$R_{\psi}:=\sum_{\rho\in\mathscr{U}}\Delta(x_{\rho})\\{x_{\rho},x_{\psi}\\}\rho$

for $x\in\mathscr{M}(\mathscr{G}_{\mathscr{U}})$ are called _unipotent almost
characters_ of $\mathbf{G}^{F}$. These have a geometric interpretation as the
characteristic functions of certain $F$-stable character sheaves on
$\mathbf{G}$ [Gec17, §7]. When $\psi\in\operatorname{Irr}(W)$, this agrees
with the unipotent uniform almost character $R_{\psi}$. If
$\psi\not\in\operatorname{Irr}(W)$, then $R_{\psi}(1)=0$. So the “fake degree”
of a non-principal series unipotent character $\psi$ is 0.

Denote by $\operatorname{Uch}(\mathscr{F}):=\mathscr{U}$ and
$\operatorname{Alm}(\mathscr{F}):=\\{R_{\psi}:\psi\in\operatorname{Uch}(\mathscr{F})\\}$.
Then the corresponding _non-abelian Fourier transform matrix_
$\mathbf{M}(\mathscr{F}):=(\\{x_{\rho},x_{\psi}\\})$ has columns indexed by
$\rho\in\operatorname{Uch}(\mathscr{F})$ and rows indexed by
$R_{\psi}\in\operatorname{Alm}(\mathscr{F})$. These matrices satisfy several
important properties which form an axiomatic framework for Fourier matrices.

###  6.3. The Fourier transform for finite Coxeter groups

This axiomatic framework is presented in [GM03, Theorem 6.9]; the authors also
describe Fourier matrices for non-Weyl Coxeter groups and show that they
satisfy these axioms.

Let $W$ be a finite Coxeter group, and let $\sigma:W\to W$ be an automorphism
which leaves a set of simple reflections invariant. For Weyl groups, $\sigma$
will be the action of $F$ on $W$. For
$\mathscr{F}\subseteq\operatorname{Irr}(W)$ a $\sigma$-invariant family, let
$\mathbf{M}(\mathscr{F})=(a_{\lambda,\mu})$ be the corresponding Fourier
matrix with columns indexed by $\mu\in\operatorname{Uch}(\mathscr{F})$ and
rows indexed by $\lambda\in\operatorname{Alm}(\mathscr{F})$. For Weyl groups
with $\sigma$ trivial, these objects are as we defined above; for Weyl groups
with $\sigma$ not trivial, the construction is similar [GM20, §4.2]. For non-
Weyl Coxeter groups, these objects are described in [GM03] along with an
inclusion $\mathscr{F}\hookrightarrow\operatorname{Uch}(\mathscr{F})$. The
unipotent degree of an element of $\operatorname{Uch}(\mathscr{F})$ is a
polynomial in $q$ which recovers the generic degree of characters of $W$ via
the inclusion.

A _twisting operator_ $t_{1}^{*}$ on the almost characters
$\operatorname{Alm}(\mathscr{F})$ is defined for non-Weyl Coxeter groups in
[GM03]. For Weyl groups, this operator is defined on all class functions on
$\mathbf{G}^{F}$ by

$t_{1}^{*}(f)(x):=f(yxy^{-1}),$

for $x\in\mathbf{G}^{F}$, where $y\in\mathbf{G}$ satisfies $x=y^{-1}F(y)$
[Asa84]. The almost characters are eigenvectors of $t_{1}^{*}$, and we denote
by $\mathbf{F}_{1}$ the diagonal matrix of the corresponding eigenvalues.

To each unipotent character $\rho$ can be associated a root of unity
$\omega_{\rho}$ called the _Frobenius eigenvalue_ of $\rho$. For Weyl groups,
this can be described as follows [GM20, 4.2.21]:

Fix an $F$-stable Borel $\mathbf{B}_{0}$ containing $\mathbf{T}_{0}$. For any
$w\in W$, define the corresponding _Deligne-Lusztig variety_

$\mathbf{X}_{w}:=\\{g\mathbf{B}_{0}\in
G/\mathbf{B}_{0}:g^{-1}F(g)\in\mathbf{B}_{0}w\mathbf{B}_{0}\\}.$

Let $\delta>0$ be the smallest positive integer such that $\sigma^{\delta}$
acts trivially on $W$. There exists some $w\in W$, $i\geq 0$, and
$\mu\in\overline{\mathbb{Q}}_{\ell}^{\times}$ such that $\rho$ is in the
character of the generalized $\mu$-eigenspace of $F^{\delta}$ on the
$\ell$-adic cohomology group
$H_{c}^{i}(\mathbf{X}_{w},\overline{\mathbb{Q}}_{\ell})$. Moreover, $\mu$ is
uniquely determined by $\rho$ (independently of $w$ and $i$) up to an integral
power of $q^{\delta}$. There is then a well-defined root of unity
$\omega_{\rho}\in\overline{\mathbb{Q}}_{\ell}$, element
$\lambda_{\rho}\in\\{1,q^{\delta/2}\\}$, and integer $s\geq 0$ such that
$\mu=\omega_{\rho}\lambda_{\rho}q^{s\delta}$.

Denote by $\mathbf{F}_{2}$ the diagonal matrix of eigenvalues of Frobenius.
When $\sigma$ is trivial, $\mathbf{F}_{1}=\mathbf{F}_{2}$, which will just be
denoted by $\mathbf{F}$. Let $\Delta$ be the permutation matrix describing the
complex conjugation of the almost characters.

######  Theorem 6.3 ([GM03], Theorem 6.9).

For $W$ a finite Coxeter group, the following axioms are satisfied for each
$\mathbf{M}=\mathbf{M}(\mathscr{F})$.

  1. (F1)

$\mathbf{M}$ transforms the vector of unipotent degrees to the vector of fake
degrees (extended by zeros).

  2. (F2)

All entries of $\mathbf{M}$ are real.

  3. (F3)

$\mathbf{M}\cdot\mathbf{M}^{\operatorname{tr}}=\mathbf{M}^{\operatorname{tr}}\cdot\mathbf{M}=1$.

  4. (F4)

Let $\lambda_{0}$ be the row index of the almost character corresponding to
the special character in $\mathscr{F}$. Then all entries in this row are non-
zero.

  5. (F5)

The structure constants

$a_{\lambda\mu}^{\nu}:=\sum_{\kappa\in\operatorname{Uch}(\mathscr{F})}\frac{a_{\lambda\kappa}a_{\mu\kappa}a_{\nu\kappa}}{a_{\lambda_{0}\kappa}}$

are rational integers for all
$\lambda,\mu,\nu\in\operatorname{Alm}(\mathscr{F})$.

  6. (F6)

If $\sigma=1$ then $\mathbf{M}=\mathbf{M}^{\operatorname{tr}}$,
$\Delta\cdot\mathbf{M}=\mathbf{M}\cdot\Delta$, and
$(\mathbf{F}\cdot\Delta\cdot\mathbf{M})^{3}=1$.

  7. (F6’)

If $\sigma\neq 1$ and $\sigma^{2}=1$, then
$(\mathbf{F}_{2}\cdot\mathbf{M}^{\operatorname{tr}}\cdot\mathbf{F}_{1}^{-1}\cdot\mathbf{M})^{2}=1$.

  8. (F6”)

If $\sigma\neq 1$ and $\sigma^{3}=1$, then
$(\mathbf{F}_{2}\cdot\mathbf{M}^{\operatorname{tr}}\cdot\mathbf{F}_{1}^{-1}\cdot\mathbf{M})^{3}=1$.

We can now define the pairing $\\{-,-\\}_{W}$ for finite Coxeter groups by

$\\{\chi,\phi\\}_{W}=\begin{cases}\mathbf{M}_{\chi,\phi}&\text{if $\chi$ and
$\phi$ are in the same family}\\\ 0&\text{otherwise}\end{cases}$

where the Fourier matrices are taken for $\sigma=1$. That is, the pairing
corresponds to the block diagonal matrix consisting of Fourier matrices on
families restricted to the principal series. Property (T1) follows from (F1),
(F3), and (F6). Property (T2) follows from (F6), and property (T3) follows
from Theorem 5.17.

###  6.4. The Fourier transform for $G(m,1,n)$

This construction follows [Mal95, §4A]. Let $\mathscr{F}$ be a family of
unipotent characters of $G(m,1,n)$, and fix some $S\in\mathscr{F}$. Let $Y$ be
a totally ordered set with $m\ell+1$ elements, $m>0$, and let $\Psi=\Psi(Y)$
be the set of functions

$\psi:Y\to\\{0,1,\dots,m-1\\}\quad\text{such that}\quad\sum_{y\in
Y}\psi(y)\equiv\ell\binom{m}{2}\operatorname{mod}m.$

Let $\pi:Y\to\mathbb{Z}_{\geq 0}$ be a function such that
$|\pi^{-1}(k)|=\\#\\{i:k\in S_{i}\\}$. This function gives rise to an
equivalence relation on $\Psi$ with $\phi\sim_{\pi}\psi$ if
$\pi\circ\phi^{-1}(i)=\pi\circ\psi^{-1}(i)$ for $0\leq i\leq m-1$. We’ll say
that $\psi\in\Psi$ is $\pi$-admissible if $\pi(y)=\pi(y^{\prime})$ and
$\psi(y)=\psi(y^{\prime})$ implies $y=y^{\prime}$. Denote by $[\psi]_{\pi}$
the equivalence class under this relation. The map from equivalence classes of
$\pi$-admissible elements of $\Psi$ to $\mathscr{F}$ given by

$\kappa:[\psi]_{\pi}\mapsto S^{[\psi]_{\pi}},\quad\text{where}\quad
S_{i}^{[\psi]_{\pi}}:=\pi(\psi^{-1}),$

is a well-defined bijection independent of the choice of $S\in\mathscr{F}$.

We then define a pairing
$\\{-,-\\}:\operatorname{Uch}\times\operatorname{Uch}\to\mathbb{C}$ by

$\\{S,S^{\prime}\\}=\frac{(-1)^{\ell(m-1)}}{\tau(m)^{\ell}}\sum_{\nu\in[\kappa^{-1}(S)]_{\pi}}\epsilon(\nu)\epsilon(\kappa^{-1}(S^{\prime}))\prod_{y\in
Y}\zeta_{m}^{-\nu(y)\kappa^{-1}(S^{\prime})(y)}$

if $S,S^{\prime}$ belong to the same family, where
$\epsilon(\psi)=(-1)^{\mathfrak{c}(\psi)}$,

$\mathfrak{c}(\psi)=\\#\\{(y,y^{\prime})\in Y\times Y:y<y^{\prime}\text{ and
}\psi(y)<\psi(y^{\prime})\\},$

and $\\{S,S^{\prime}\\}=0$ otherwise.

The restriction $\\{-,-\\}_{G(m,1,n)}$ of this pairing to the irreducible
characters of $G(m,1,n)$ then satisfies (T1), (T2), and (T3) by [Las12, Remark
5.3.26 and Corollary 5.3.33].

###  6.5. The Fourier transform for $G(m,m,n)$ and the exceptional groups

In [Las12, Conjecture 5.4.34], Lasy conjectures the existence of such a
Fourier transform for $G(m,m,n)$ and describes its relation to a “pre-Fourier”
transform which is a slight modification of the construction in [Mal95, §6C].
This conjectured transform will satisfy (T1), (T2), and (T3) [Las12, Corollary
5.4.38].

The Fourier transform for the exceptional groups (and for the families
$G(m,1,n)$ and $G(m,m,n)$) are contained in the data for GAP3, but the
properties of these matrices have not yet appeared in publication. See [Mal98]
and [BMM14].

##  7\. Rational Catalan numbers

For an irreducible complex reflection group $W$ with Coxeter number $h$,
define rational $W$-Catalan numbers [GG12]

$\operatorname{Cat}_{p}(W)=\prod_{i=1}^{n}\frac{p+(pe_{i}\operatorname{mod}h)}{d_{i}},$

and their $q$-deformations

$\operatorname{Cat}_{p}(W;q)=\prod_{i=1}^{n}\frac{[p+(pe_{i}\operatorname{mod}h)]_{q}}{[d_{i}]_{q}}.$

######  Theorem 7.1.

Let $W$ be an irreducible spetsial complex reflection group with Coxeter
number $h$, and let $c$ be a $\zeta_{h}$-regular element of $W$. Let
$\mathbf{c}\in B(W)$ be a lift of $c$ such that $\mathbf{c}^{h}=\bm{\pi}$.
Then

$\tau_{q}(T_{\mathbf{c}}^{-p})=q^{-np}(1-q)^{n}\operatorname{Cat}_{p}(W;q).$

###### Proof.

Starting from Proposition 3.16, then using Lemma 6.1, Theorem 3.17, Example
2.9, and Example 2.15:

$\displaystyle\tau_{q}(T_{\mathbf{c}}^{-p})$
$\displaystyle=\frac{1}{P_{W}}\sum_{\chi\in\operatorname{Irr}(W)}q^{(h_{\chi}-nh)p/h}\operatorname{Feg}_{\chi}(e^{2\pi
ip/h})\operatorname{Deg}_{\chi}(q)$
$\displaystyle=\frac{1}{P_{W}}\sum_{\chi\in\operatorname{Irr}(W)}q^{(h_{\chi}-nh)p/h}\operatorname{Feg}_{\chi}(q)\operatorname{Deg}_{\chi}(e^{2\pi
ip/h})$
$\displaystyle=\frac{1}{P_{W}}\sum_{k=0}^{n}(-1)^{k}q^{(k-n)p}\sum_{i_{1}<\cdots<i_{k}}q^{e_{i_{1}}(V^{\sigma_{p}})+\cdots+e_{i_{k}}(V^{\sigma_{p}})}$
$\displaystyle=\frac{1}{P_{W}}q^{-np}\prod_{i=1}^{n}\left(1-q^{p+e_{i}(V^{\sigma_{p}})}\right)$
$\displaystyle=q^{-np}(1-q)^{n}\prod_{i=1}^{n}\frac{[p+e_{i}(V^{\sigma_{p}})]_{q}}{[d_{i}]_{q}}$
$\displaystyle=q^{-np}(1-q)^{n}\operatorname{Cat}_{p}(W;q).\qed$

######  Remark 7.2.

This proof for $G(m,m,n)$ relies on Conjecture 6.2. For the exceptional
groups, the result can be established by computer [MW23] without assuming the
conjecture. ∎

######  Remark 7.3.

This result is not true for the groups that are well-generated but not
spetsial, and one might reasonably ask where this proof fails. For these
groups, it is still true that

$\tau_{q}(T_{\mathbf{c}}^{-p})=\frac{1}{P_{W}}\sum_{\chi\in\operatorname{Irr}(W)}q^{(h_{\chi}-nh)p/h}\operatorname{Feg}_{\chi}(e^{2\pi
ip/h})\operatorname{Deg}_{\chi}(q)$

and

$q^{-np}(1-q)^{n}\operatorname{Cat}_{p}(W;q)=\frac{1}{P_{W}}\sum_{\chi\in\operatorname{Irr}(W)}q^{(h_{\chi}-nh)p/h}\operatorname{Feg}_{\chi}(q)\operatorname{Deg}_{\chi}(e^{2\pi
ip/h}),$

but the right-hand-sides of these equations are not equal to each other. So
there is no Fourier transform for these groups. ∎

These “trace techniques” also allow us to extend the algebraic part of
[GLTW22, Corollary 6.15] to spetsial groups.

######  Corollary 7.4.

For $W$ an irreducible spetsial complex reflection group, let $\mathcal{B}$ be
a basis of the spetsial Hecke algebra $\mathcal{H}_{q}(W)$ (adapted to the
Wedderburn decomposition), and let $\mathbf{c}$ be a lift of a
$\zeta_{h}$-regular element such that $\mathbf{c}^{h}=\mathbf{\pi}$. Then

$\sum_{b\in\mathcal{B}}\tau_{q}(b^{\vee}T_{\mathbf{c}^{p}}b)=(q-1)^{n}[p]_{q}^{n}.$

###### Proof.

Using [GP00, pg. 226],

$\displaystyle\sum_{b\in\mathcal{B}}\tau_{q}(b^{\vee}T_{\mathbf{c}^{p}}b)$
$\displaystyle=\sum_{b\in\mathcal{B}}\sum_{\chi\in\operatorname{Irr}(W)}\frac{1}{S_{\chi}(q)}\chi_{q}(b^{\vee}T_{\mathbf{c}^{p}}b)=\sum_{\chi\in\operatorname{Irr}(W)}\frac{1}{S_{\chi}(q)}\sum_{b\in\mathcal{B}}\chi_{q}(b^{\vee}T_{\mathbf{c}^{p}}b)$
$\displaystyle=\sum_{\chi\in\operatorname{Irr}(W)}(\operatorname{dim}\chi)\cdot\chi_{q}(T_{\mathbf{c}^{p}})=\sum_{\chi\in\operatorname{Irr}(W)}q^{(nh-
h_{\chi})p/h}\operatorname{Deg}_{\chi}(1)\operatorname{Feg}_{\chi}(e^{-2\pi
ip/h})$
$\displaystyle=\sum_{k=0}^{n}q^{p(n-k)}(-1)^{k}\binom{n}{k}=(q^{p}-1)^{n}=(q-1)^{n}[p]_{q}^{n}.$

∎

This corollary motivates future work:

######  Question 7.5.

Are there noncrossing parking objects for spetsial complex reflection groups
analogous to those in [GLTW22] whose enumeration is related to the sum
$\sum_{b\in\mathcal{B}}\tau_{q}(b^{\vee}T_{\mathbf{c}^{p}}b)$?

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