# ON ENDOMORPHISM ALGEBRAS OF GELFAND–GRAEV REPRESENTATIONS II

Tzu-Jan Li and Jack Shotton

Abstract. Let $G$ be a connected reductive group defined over a finite field
$\mathbb{F}_{q}$ of characteristic $p$, with Deligne–Lusztig dual $G^{\ast}$.
We show that, over $\overline{\mathbb{Z}}[1/pM]$ where $M$ is the product of
all bad primes for $G$, the endomorphism ring of a Gelfand–Graev
representation of $G(\mathbb{F}_{q})$ is isomorphic to the Grothendieck ring
of the category of finite-dimensional
$\overline{\mathbb{F}}_{q}$-representations of $G^{\ast}(\mathbb{F}_{q})$.

######  0.

Introduction. — Let $G$ be a connected reductive group defined over a finite
field $\mathbb{F}_{q}$ of characteristic $p$, let $F$ be the associated
Frobenius endomorphism of $G$, and let $\Lambda$ be a subring of
$\overline{\mathbb{Q}}$ containing $\overline{\mathbb{Z}}[\frac{1}{p}]$. Let
$B_{0}$ be an $F$-stable Borel subgroup of $G$ with (necessarily $F$-stable)
unipotent radical $U_{0}$, and let
$\psi:U_{0}^{F}\longrightarrow\Lambda^{\times}$ be a regular (also called
nondegenerate) character. The Gelfand–Graev representation

$\Gamma_{G,\psi}:=\mathrm{Ind}_{U_{0}^{F}}^{G^{F}}\psi$

is an important representation of $G^{F}$ (already studied in [DeLu, Sec. 10]
and [DLM]). Its endomorphism ring

$\Lambda\mathsf{E}_{G}:=\mathrm{End}_{\Lambda G^{F}}(\Gamma_{G,\psi})$

is commutative, independent of the choice of $\psi$ up to isomorphism and,
over $\overline{\mathbb{Q}}$, may be identified with the ring of
$\overline{\mathbb{Q}}$-valued class functions on $G_{\mathrm{ss}}^{\ast
F^{\ast}}$, where $(G^{\ast},F^{\ast})$ is a chosen Deligne–Lusztig
${\mathbb{F}}_{q}$-dual of $(G,F)$ (see [Cu]). Such an identification only
depends on choices of group homomorphisms
$(\mathbb{Q}/\mathbb{Z})_{p^{\prime}}\simeq\overline{\mathbb{F}}_{q}^{\times}\hookrightarrow\overline{\mathbb{Q}}^{\times}$,
which we fix from now on.

There are then (at least) two natural $\Lambda$-lattices in
$\overline{\mathbb{Q}}\mathsf{E}_{G}$: $\Lambda\mathsf{E}_{G}$ and the lattice
$\Lambda\mathsf{K}_{G^{\ast}}$ spanned by Brauer characters of irreducible
representations of $G^{\ast F^{\ast}}$; here, $\mathsf{K}_{G^{\ast}}$ is the
Grothendieck group of the category of finite-dimensional
$\overline{\mathbb{F}}_{q}G^{\ast F^{\ast}}$-modules, and it is a ring with
tensor product as multiplication. Denoting by $G_{\mathrm{ss}}^{\ast
F^{\ast}}\\!/\\!\sim$ the set of semisimple conjugacy classes in $G^{\ast
F^{\ast}}$, we may then identify

$\overline{\mathbb{Q}}\mathsf{E}_{G}=\overline{\mathbb{Q}}^{G^{\ast
F^{\ast}}_{\mathrm{ss}}\\!/\sim}=\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}.$
(0.1)

The main result of this paper may now be stated as follows:

Main theorem. If all bad primes for $G$ are invertible in $\Lambda$, then the
two $\Lambda$-lattices $\Lambda\mathsf{E}_{G}$ and
$\Lambda\mathsf{K}_{G^{\ast}}$ of $\overline{\mathbb{Q}}\mathsf{E}_{G}$ are
equal.

Here, we use the notion of “bad primes for $G$” from [Sp]. Denoting by $R$ the
root system of $G$, a prime number $\ell$ is called bad for $G$ if one of the
following three conditions holds: (i) $\ell=2$, and $R$ has an irreducible
factor other than type $A$; (ii) $\ell=3$ or 5, and $R$ has an irreducible
factor of exceptional type ($G_{2}$, $F_{4}$, $E_{6}$, $E_{7}$, or $E_{8}$);
(iii) $\ell=7$, and $R$ has an irreducible factor of type $E_{8}$.

In this theorem, the assumption on the bad primes for $G$ is due to the usage
of almost characters in Lusztig’s work on unipotent characters, where bad
primes appear in the denominators of the “Fourier transform matrix.” We expect
that the theorem remains true without this assumption, though our present
method cannot prove it.

The above theorem improves the equality
$\overline{\mathbb{Z}}[\frac{1}{p|W|}]\mathsf{E}_{G}=\overline{\mathbb{Z}}[\frac{1}{p|W|}]\mathsf{K}_{G^{\ast}}$
(where $W$ is the Weyl group of $G$) in [Li, Thm. 2.9] whenever the adjoint
group of $G$ is simple of type other than $F_{4}$ or $G_{2}$ (in these two
excluded types, the bad primes and the primes dividing the order of the Weyl
group coincide). Moreover, via the $\mathbb{Z}$-model $\mathsf{E}_{G}$ of
$\Lambda\mathsf{E}_{G}$ from [Li, Sec. 1.7], if we denote by $M$ is the
product of all bad primes for $G$, then the above theorem implies that
$\mathbb{Z}[\frac{1}{pM}]\mathsf{E}_{G}=\mathbb{Z}[\frac{1}{pM}]\mathsf{K}_{G^{\ast}}$.

Relation with invariant theory. Let $\mathsf{B}_{G^{\vee}}$ be the ring of
functions of the $\mathbb{Z}$-scheme $(T^{\vee}\sslash W)^{F^{\vee}}$, where
$(G^{\vee},T^{\vee})$ is the split $\mathbb{Z}$-dual of $(G,T)$ with $T$ an
$F$-stable maximal torus of $G$, $W=N_{G^{\vee}}(T^{\vee})/T^{\vee}$ is the
Weyl group of $(G^{\vee},T^{\vee})$, and $F^{\vee}:T^{\vee}\longrightarrow
T^{\vee}$ is induced by the action of $F$ on $Y(T^{\vee})=X(T)$. If $G^{\ast}$
has simply-connected derived subgroup, then $\Lambda\mathsf{B}_{G^{\vee}}$ is
also a $\Lambda$-lattice of $\overline{\mathbb{Q}}\mathsf{E}_{G}$ and appears
to be significant for the local Langlands correspondence in families. Indeed,
for $\mathrm{GL}_{n}$, in the course of constructing this correspondence in
joint work with Moss [HeMo], Helm proved in [Hel, Thm. 10.1] the equality
$\Lambda\mathsf{E}_{\mathrm{GL}_{n}}=\Lambda\mathsf{B}_{\mathrm{GL}_{n}^{\vee}}$
for $\Lambda$ being the ring of Witt vectors of $\overline{\mathbb{F}}_{\ell}$
with $\ell\neq p$. In our current context ($G$ a connected reductive group
over $\mathbb{F}_{q}$), when $G^{\ast}$ has simply-connected derived subgroup,
it is known that $\mathsf{B}_{G^{\vee}}=\mathsf{K}_{G^{\ast}}$ (see [Li, Thm.
3.22]), so that our main theorem yields the equalities

$\Lambda\mathsf{E}_{G}=\Lambda\mathsf{K}_{G^{\ast}}=\Lambda\mathsf{B}_{G^{\vee}}$

for $\Lambda=\overline{\mathbb{Z}}[\frac{1}{pM}]$. In particular, for
$\mathrm{GL}_{n}$, $M=1$ and so we provide an alternative proof of Helm–Moss’s
equality.

On the proof of the main theorem. Identify
$\Lambda\mathsf{E}_{G}=e_{\psi}\Lambda G^{F}e_{\psi}\subset\Lambda G^{F}$
where $e_{\psi}:=\frac{1}{|U_{0}^{F}|}\sum_{u\in U_{0}^{F}}\psi(u^{-1})u$ is
the primitive central idempotent of $\Lambda U_{0}^{F}$ associated with
$\psi$. We may then consider the symmetrizing form

$\tau=\tau_{G}:=|U_{0}^{F}|\mathrm{ev}_{1_{G^{F}}}:\Lambda\mathsf{E}_{G}\longrightarrow\Lambda$

($\mathrm{ev}_{1_{G^{F}}}$ denotes the evaluation map at $1_{G^{F}}$; recall
that a symmetrizing form on a finite projective $\Lambda$-algebra $A$ is a map
$\tau:A\to\Lambda$ such that the map $(a,b)\mapsto\tau(ab)$ is a perfect
symmetric bilinear form), and denote its $\overline{\mathbb{Q}}$-linear
extension again by $\tau$. It has been shown in [Li, Sec. 2.8] that
$\tau(\mathsf{K}_{G^{\ast}})\subset\mathbb{Z}$ and that
$\tau|_{\Lambda\mathsf{K}_{G^{\ast}}}:\Lambda\mathsf{K}_{G^{\ast}}\longrightarrow\Lambda$
is a symmetrizing form. Therefore, the equality
$\Lambda\mathsf{E}_{G}=\Lambda\mathsf{K}_{G^{\ast}}$ will hold if

$\tau(h\pi)\in\Lambda\mbox{\;\;for all\;\;}h\in\Lambda\mathsf{E}_{G}\mbox{ and
}\pi\in\Lambda\mathsf{K}_{G^{\ast}}.$ (0.2)

Indeed, (0.2) shows that each of $\Lambda\mathsf{E}_{G}$ and
$\Lambda\mathsf{K}_{G^{\ast}}$ is contained in the dual of the other; as each
is self-dual, they are equal.

To study (0.2), we will need a family of special elements
$\\{\pi_{\lambda}\\}$ of $\mathsf{K}_{G^{\ast}}$ constructed as follows. Let
$\mathsf{K}(G^{\ast}\mbox{-mod})$ be the Grothendieck group of the category of
finite-$\overline{\mathbb{F}}_{q}$-dimensional algebraic $G^{\ast}$-modules;
it is a ring with multiplication given by tensor product. As shown in [Ja, Ch.
II.2], for every maximal torus $T^{\ast}$ of $G^{\ast}$, the associated formal
character map gives a ring isomorphism

$\mathrm{ch}:\mathsf{K}(G^{\ast}\mbox{-mod})\xrightarrow{\;\;\sim\;\;}\mathbb{Z}[X(T^{\ast})]^{W}$

where $X(T^{\ast})=\mathrm{Hom}_{\mathrm{alg}}(T^{\ast},\mathbb{G}_{m})$ is
the character group of $T^{\ast}$ and $W$ is the Weyl group of
$(G^{\ast},T^{\ast})$. For $\lambda\in X(T^{\ast})$, set

$r_{\lambda}=r_{G,\lambda}:=\sum\limits_{\mu\in
W\lambda}\mu\in\mathbb{Z}[X(T^{\ast})]^{W}\mbox{\quad
and\quad}\pi_{\lambda}=\pi_{G,\lambda}:=\mathrm{ch}^{-1}(r_{\lambda})|_{G^{\ast
F^{\ast}}}\in\mathsf{K}_{G^{\ast}},$

where for $\lambda\in X(T^{\ast})$, $W\lambda$ denotes the $W$-orbit of
$\lambda$. Note that the $\mathbb{Z}$-module $\mathbb{Z}[X(T^{\ast})]^{W}$ is
generated by $\\{r_{\lambda}:\lambda\in X(T^{\ast})\\}$.

After preparations on Deligne-Lusztig characters and Curtis homomorphisms
(Section 1), we will reduce (0.2) to the study of the condition
“$\tau(h\pi_{\lambda})\in\Lambda$” by fitting $G^{\ast}$ into a central
extension (Section 2) and studying related compatibility questions (Sections 3
and 4). To study the condition “$\tau(h\pi_{\lambda})\in\Lambda$,” we will
extend the definition of $\tau(h\pi_{\lambda})$ to $h\in G^{F}$ (Section 5),
reduce the discussion to the case where the semisimple part $s$ of $h$ is
central in $G$ (Section 6), and finally deal with the case of central $s$
(Section 7).

Acknowledgements. The first author thanks Professor Jean-François Dat, his PhD
thesis advisor, for his constant support and enlightening opinions on this
work. The second author thanks Robert Kurinczuk for bringing the work [Li] of
the first author to his attention. We thank Jay Taylor for providing a helpful
reference.
This project has received funding from the European Union’s Horizon 2020
research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
754362.

######  1.

Preliminaries. — In this section, we recall some properties of Deligne–Lusztig
characters and Curtis homomorphisms that we will need later on.

Deligne–Lusztig characters. Let $S$ be an $F$-stable maximal torus of $G$, let
$P$ be a Borel subgroup containing $S$, and let $V$ be the unipotent radical
of $P$. Then we have the Deligne–Lusztig variety (see [DiMi2, Def. 9.1.1])

$DL_{S\subset P}^{G}=\\{gV\in G/V:g^{-1}F(g)\in V\cdot F(V)\\},$

which admits a (left) $G^{F}\times(S^{F})^{\mathrm{op}}$-action. When there is
no need to specify the chosen Borel subgroup $P$, we will write $DL_{S\subset
P}^{G}$ simply as $DL_{S}^{G}$.

We consider the virtual $\ell$-adic cohomology $H^{*}_{c}(\cdot)=\sum_{j\geq
0}(-1)^{j}H^{j}_{c}(\cdot)$, for $\ell$ a prime distinct from $p$. For every
character $\chi:S^{F}\longrightarrow\overline{\mathbb{Q}}^{\times}$, upon
choosing a field embedding
$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{\ell}$, we have
the corresponding Deligne–Lusztig character

$R_{S}^{G}(\chi)(-):=\mathrm{Tr}(-|H_{c}^{\ast}(DL_{S\subset
P}^{G})\otimes_{\overline{\mathbb{Q}}_{\ell}S^{F}}\chi)=\frac{1}{|S^{F}|}\sum_{s\in
S^{F}}\mathrm{Tr}((-,s)|H_{c}^{\ast}(DL_{S\subset P}^{G}))\chi(s^{-1}),$

which is independent of the choice of $P$ and which takes values in
$\overline{\mathbb{Q}}_{\ell}$ à priori; but by [DeLu, Prop. 3.3], for any
$(g,s)\in G^{F}\times S^{F}$, the trace
$\mathrm{Tr}((g,s)|H^{*}_{c}(DL_{S\subset P}^{G}))$ is an integer independent
of $\ell$, so in fact $R_{S}^{G}(\chi)$ takes values in
$\overline{\mathbb{Q}}$, and it can be verified that ${R}_{S}^{G}(\chi)$ is
independent of the choices of $\ell$ and of the embedding
$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{\ell}$.

Curtis homomorphisms. For an $F$-stable maximal torus $S$ of $G$, we consider
the Curtis homomorphism

$\mathrm{Cur}^{G}_{S}:\overline{\mathbb{Q}}\mathsf{E}_{G}\longrightarrow\overline{\mathbb{Q}}S^{F}$

defined as in [Li, Sec. 1.11] (see also [Cu, Thm. 4.2]). In terms of the
Deligne-Lusztig dual, the map $\mathrm{Cur}^{G}_{S}$ is simply a “restriction
map to a dual torus”: indeed, upon fixing an $F^{\ast}$-stable maximal torus
$S^{\ast}$ of $G^{\ast}$ dual to $S$ (whence a duality
$\mathrm{Irr}_{\overline{\mathbb{Q}}}(S^{F})\simeq S^{\ast F^{\ast}}$ and thus
a ring isomorphism
$\overline{\mathbb{Q}}S^{F}\simeq\overline{\mathbb{Q}}^{S^{\ast F^{\ast}}}$),
the map $\mathrm{Cur}^{G}_{S}$ is the unique ring homomorphism making the
following diagram commutative (see [Li, Lem. 1.13]):

${\overline{\mathbb{Q}}\mathsf{E}_{G}}$${\overline{\mathbb{Q}}^{G^{\ast
F^{\ast}}_{\mathrm{ss}}\\!/\sim}}$${\overline{\mathbb{Q}}S^{F}}$${\overline{\mathbb{Q}}^{S^{\ast
F^{\ast}}}}$${(\ref{EK-
identify})}$${\sim}$${\mathrm{Cur}^{G}_{S}}$${\mathrm{Res}}$${\sim}$ (1.1)

We will later need the following formula of Bonnafé–Kessar ([BoKe, Prop. 2.5],
with the missing sign factor corrected):

$\mathrm{Cur}_{S}^{G}(h)=\frac{\epsilon_{G}\epsilon_{S}}{|S^{F}|}\sum_{s\in
S^{F}}\mathrm{Tr}((h,s)|H_{c}^{\ast}(DL_{S\subset
P}^{G}))s^{-1}\in\overline{\mathbb{Q}}S^{F}\quad(h\in\overline{\mathbb{Q}}\mathsf{E}_{G}\subset\overline{\mathbb{Q}}G^{F}).$
(1.2)

Here, as usual, $\epsilon_{G}=(-1)^{\mathrm{rk}_{\mathbb{F}_{q}}(G)}$ for $G$
any reductive group over $\mathbb{F}_{q}$. Observe that (1.2) shows that
$\mathrm{Cur}^{G}_{S}$ is independent of the choice of $S^{\ast}$.

######  2.

On central extensions. — For our group $G$, we can fit its Deligne–Lusztig
dual $G^{\ast}$ into an $F^{\ast}$-equivariant exact sequence of reductive
groups

$1\longrightarrow Z^{\ast}\longrightarrow H^{\ast}\longrightarrow
G^{\ast}\longrightarrow 1$ (2.1)

where the derived subgroup of $H^{\ast}$ is simply-connected and $Z^{\ast}$ is
a torus central in $H^{\ast}$.

We fix a choice of $F$-equivariant exact sequence of reductive groups

$1\longrightarrow G\longrightarrow
H\xrightarrow{\;\;\kappa\;\;}Z\longrightarrow 1$ (2.2)

which is dual to (2.1). Let $T_{H}$ be an $F$-stable maximal torus of $H$, let
$B_{H}$ be a Borel subgroup of $H$ containing $T_{H}$, and let $V$ be the
unipotent radical of $B_{H}$. Then

$DL_{T_{H}\subset B_{H}}^{H}=\bigsqcup_{z\in Z^{F}}(DL_{T_{H}\subset
B_{H}}^{H})(z)$

where for each $z\in Z^{F}$ we have set

$(DL_{T_{H}\subset B_{H}}^{H})(z):=\\{hV\in DL_{T_{H}\subset
B_{H}}^{H}:\kappa(h)=z\\}.$

Let $T_{G}=\mathrm{ker}(\kappa|_{T_{H}}:T_{H}\twoheadrightarrow Z)$ (resp.
$B_{G}=\mathrm{ker}(\kappa|_{B_{H}}:B_{H}\twoheadrightarrow Z)$), which is an
$F$-stable maximal torus of $G$ (resp. a Borel subgroup of $G$). Then
$T_{G}\subset B_{G}$, and the unipotent radical of $B_{G}$ is also $V$. As
$T_{G}$ is connected, we have $\kappa(T_{H}^{F})=Z^{F}$, so for each $z\in
Z^{F}$ we may choose a $\dot{z}\in T_{H}^{F}$ such that $\kappa(\dot{z})=z$.
Under the inclusion $G\subset H$, for each $z\in Z^{F}$ we have

$DL_{T_{H}\subset B_{H}}^{H}(z)=DL_{T_{G}\subset B_{G}}^{G}\cdot\dot{z}\subset
H/V,$

so that

$DL_{T_{H}\subset B_{H}}^{H}(z)\simeq DL_{T_{G}\subset B_{G}}^{G}\mbox{ as
}(G^{F}\times(T_{G}^{F})^{\mathrm{op}})\mbox{-varieties}.$

In terms of virtual $\ell$-adic cohomology we therefore have

$H_{c}^{\ast}(DL_{T_{H}\subset B_{H}}^{H})=\sum_{z\in
Z^{F}}H_{c}^{\ast}(DL_{T_{H}\subset B_{H}}^{H}(z)),$

and the $H^{F}\times(T_{H}^{F})^{\mathrm{op}}$-action on
$H_{c}^{\ast}(DL_{T_{H}\subset B_{H}}^{H})$ satisfies:

$\left\\{\begin{aligned} &\mbox{ for every }(h,t)\in
H^{F}\times(T_{H}^{F})^{\mathrm{op}},(h,t)\cdot H_{c}^{\ast}(DL_{T_{H}\subset
B_{H}}^{H}(z))\subset H_{c}^{\ast}(DL_{T_{H}\subset
B_{H}}^{H}(\kappa(ht)z));\\\ &\mbox{ for every }z\in
Z^{F},\;H_{c}^{\ast}(DL_{T_{H}\subset B_{H}}^{H}(z))\simeq
H_{c}^{\ast}(DL_{T_{G}\subset B_{G}}^{G})\mbox{ as
}G^{F}\times(T_{G}^{F})^{\mathrm{op}}\mbox{-modules}.\end{aligned}\right.$

In particular, we obtain the following trace formulae: for $(h,t)\in
H^{F}\times(T_{H}^{F})^{\mathrm{op}}$,

$\left\\{\begin{aligned} \kappa(ht)\neq
1&\Longrightarrow\mathrm{Tr}((h,t)|H_{c}^{\ast}(DL_{T_{H}\subset
B_{H}}^{H}))=0;\\\ (h,t)\in
G^{F}\times(T_{G}^{F})^{\mathrm{op}}&\Longrightarrow\mathrm{Tr}((h,t)|H_{c}^{\ast}(DL_{T_{H}\subset
B_{H}}^{H}))=|Z^{F}|\cdot\mathrm{Tr}((h,t)|H_{c}^{\ast}(DL_{T_{G}\subset
B_{G}}^{G})).\end{aligned}\right.$ (2.3)

We will later need the compatibility (for
$\chi:T_{H}^{F}\longrightarrow\overline{\mathbb{Q}}^{\times}$):

$R_{T_{H}}^{H}(\chi)|_{G^{F}}=R_{T_{G}}^{G}(\chi|_{T_{G}^{F}}).$ (2.4)

This follows immediately from the defining formula of $R_{T_{H}}^{H}(\chi)$
and (2.3). (See also [DiMi2, Prop. 11.3.10]).

######  3.

A compatibility lemma. — Notation as in Section 2. We extend the $F$-stable
Borel subgroup $B_{0}$ of $G$ in Section  ‣  0. (used to determine the
Gelfand–Graev module $\Gamma_{G,\psi}$) to the $F$-stable Borel subgroup
$B_{0}^{\prime}$ of $H$, so that $B_{0}^{\prime}/B_{0}=Z$ under (2.2); the
unipotent radical of $B_{0}^{\prime}$ is then equal to $U_{0}$ (the unipotent
radical of $B_{0}$), and the inclusion $G^{F}\subset H^{F}$ induced by (2.2)
gives rise to a $\Lambda$-algebra inclusion

$\Lambda\mathsf{E}_{G}=e_{\psi}\Lambda G^{F}e_{\psi}\hookrightarrow
e_{\psi}\Lambda H^{F}e_{\psi}=\Lambda\mathsf{E}_{H}.$ (3.1)

On the other hand, (2.1) yields the identification

$(G^{\ast F^{\ast}}_{\mathrm{ss}}\\!/\\!\sim)=(H^{\ast
F^{\ast}}_{\mathrm{ss}}\\!/\\!\sim)/Z^{\ast F^{\ast}},$ (3.2)

which enables us to regard functions on $G^{\ast
F^{\ast}}_{\mathrm{ss}}/\\!\sim$ as functions on $H^{\ast
F^{\ast}}_{\mathrm{ss}}/\\!\sim$ which are constant on each $Z^{\ast
F^{\ast}}$-orbit.

Let us prove the following “compatibility lemma”:

Lemma. The following diagram of rings is commutative:

${\overline{\mathbb{Q}}\mathsf{E}_{G}}$${\overline{\mathbb{Q}}^{G^{\ast
F^{\ast}}_{\mathrm{ss}}\\!/\sim}}$${\overline{\mathbb{Q}}\mathsf{E}_{H}}$${\overline{\mathbb{Q}}^{H^{\ast
F^{\ast}}_{\mathrm{ss}}\\!/\sim}}$${(\ref{EK-
identify})}$${\sim}$${(\ref{i-inclusion})}$${(\ref{ss-ZHG})}$${(\ref{EK-
identify})}$${\sim}$ (3.3)

Proof. Let $T_{G}$ and $T_{H}$ be as in Section 2, and choose a
$F^{\ast}$-stable maximal torus $T_{G}^{\ast}$ of $G^{\ast}$ dual to $T_{G}$
(resp. $T_{H}^{\ast}$ of $H$ dual to $T_{H}$) such that
${T_{H}^{\ast}/Z^{\ast}=T_{G}^{\ast}}$. Then the Weyl groups of $(G,T_{G})$,
$(G^{\ast},T_{G}^{\ast})$, $(H,T_{H})$ and $(H^{\ast},T_{H}^{\ast})$ are all
the same, and we denote this common Weyl group by $W$. For each $w\in W$,
choose an $F$-stable maximal torus $T_{G,w}$ of $G$ whose $G^{F}$-conjugacy
class corresponds to the $F$-conjugacy class of $w$ in $W$ (with respect to
$T_{G}$, so that we may choose $T_{G,1}=T_{G}$); choose $T_{G,w}^{\ast}\subset
G^{\ast}$, $T_{H,w}\subset H$ and $T_{H,w}^{\ast}\subset H^{\ast}$ in a
similar way.

In the toric case where $(G,H)=(T_{G},T_{H})$, the commutativity of (3.3)
follows from toric dualities.

For the general case of $(G,H)$, we use the Curtis embeddings
$\mathrm{Cur}^{G}=(\mathrm{Cur}_{T_{G,w}}^{G})_{w\in W}$ and
$\mathrm{Cur}^{H}=(\mathrm{Cur}_{T_{H,w}}^{H})_{w\in W}$ (see Section 1) to
embed (3.3) into the following cubic diagram of rings:

${\prod\limits_{w\in W}\overline{\mathbb{Q}}T_{G,w}^{F}}$${\prod\limits_{w\in
W}\overline{\mathbb{Q}}^{T_{G,w}^{\ast
F^{\ast}}}}$${\overline{\mathbb{Q}}\mathsf{E}_{G}}$${\overline{\mathbb{Q}}^{G_{\mathrm{ss}}^{\ast
F^{\ast}}\\!/\sim}}$${\prod\limits_{w\in
W}\overline{\mathbb{Q}}T_{H,w}^{F}}$${\prod\limits_{w\in
W}\overline{\mathbb{Q}}^{T_{H,w}^{\ast
F^{\ast}}}}$${\overline{\mathbb{Q}}\mathsf{E}_{H}}$${\overline{\mathbb{Q}}^{H_{\mathrm{ss}}^{\ast
F^{\ast}}\\!/\sim}}$${\sim}$${\sim}$${\mathrm{Cur}^{G}}$${\mathrm{Res}}$${\sim}$${\sim}$${\mathrm{Cur}^{H}}$${\mathrm{Res}}$
(3.4)

In (3.4), the right face is clearly commutative; the top and the bottom faces
are commutative by (1.1); the back face is the toric case of (3.3) and is
hence commutative. So to prove the commutativity of (3.3), it remains to show
that the left face in (3.4) is commutative.

Using (1.2) and the relation
$\epsilon_{H}\epsilon_{T_{H,w}}=\epsilon_{G}\epsilon_{T_{G,w}}$, the
commutativity of the left face in (3.4) is equivalent to the property that,
for all
$h\in\overline{\mathbb{Q}}\mathsf{E}_{G}\subset\overline{\mathbb{Q}}G^{F}$ and
all $w\in W$, we have

$\frac{1}{|T_{H,w}^{F}|}\sum_{t\in
T_{H,w}^{F}}\mathrm{Tr}((h,t)|H_{c}^{\ast}(DL_{T_{H,w}}^{H}))t^{-1}=\frac{1}{|T_{G,w}^{F}|}\sum_{t\in
T_{G,w}^{F}}\mathrm{Tr}((h,t)|H_{c}^{\ast}(DL_{T_{G,w}}^{G}))t^{-1}.$ (3.5)

By (2.3) and the fact that $T_{H,w}^{F}/T_{G,w}^{F}=Z^{F}$, we see that (3.5)
is true for all $h\in G^{F}$, so the left face in (3.4) commutes. This
completes the proof of the lemma.∎

######  4.

Reduction to the study of $\tau(h\pi_{\lambda})$. — We keep notations in
Section 2. As $Z^{\ast F^{\ast}}$ is central in $H^{\ast F^{\ast}}$, the
association of each irreducible $\overline{\mathbb{F}}_{q}G^{\ast
F^{\ast}}$-module to the restriction to $Z^{\ast F^{\ast}}$ of its central
character induces a $\widehat{Z^{\ast F^{\ast}}}$-graded decomposition

$\mathsf{K}_{H^{\ast}}=\bigoplus\limits_{\lambda\in\widehat{Z^{\ast
F^{\ast}}}}(\mathsf{K}_{H^{\ast}})_{\lambda}\mbox{\;\;with\;\;}\mathsf{K}_{G^{\ast}}=(\mathsf{K}_{H^{\ast}})_{0}.$
(4.1)

In particular, we have a ring inclusion
$\mathsf{K}_{G^{\ast}}\subset\mathsf{K}_{H^{\ast}}$, and it is evident that
the following diagram of rings is commutative (where $\mathrm{br}$ denotes the
Brauer character map):

${\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}}$${\overline{\mathbb{Q}}^{G^{\ast
F^{\ast}}_{\mathrm{ss}}\\!/\sim}}$${\overline{\mathbb{Q}}\mathsf{K}_{H^{\ast}}}$${\overline{\mathbb{Q}}^{H^{\ast
F^{\ast}}_{\mathrm{ss}}\\!/\sim}}$${(\ref{K-grading})}$${\mathrm{br}}$${\sim}$${(\ref{ss-
ZHG})}$${\mathrm{br}}$${\sim}$ (4.2)

Recall that we have identified
$\overline{\mathbb{Q}}\mathsf{E}_{G}=\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}$
in (0.1). We now establish another “compatibility lemma”:

Lemma. For $h\in\Lambda\mathsf{E}_{G}\subset\Lambda\mathsf{E}_{H}$ and
$\pi\in\mathsf{K}_{G^{\ast}}\subset\mathsf{K}_{H^{\ast}}$, we have
$\tau_{G}(h\pi)=\tau_{H}(h\pi)$.

Proof. For $T_{G}$ and $T_{H}$ as in Section 2, $(G,T_{G})$ and $(H,T_{H})$
have the same Weyl group $W$. Let $T_{G,w}$ and $T_{H,w}$ be associated with
$w\in W$ as in the proof of (3.3). Let $h\in\Lambda\mathsf{E}_{G}$ and
$\pi\in\mathsf{K}_{G^{\ast}}$. By [BoKe, Eq. 3.5], we have:

$\displaystyle\tau_{G}(h\pi)$ $\displaystyle=\frac{1}{|W|}\sum_{w\in
W}\mathrm{ev}_{1_{T_{G,w}^{F}}}(\mathrm{Cur}_{T_{G,w}}^{G}(h\pi));$
$\displaystyle\tau_{H}(h\pi)$ $\displaystyle=\frac{1}{|W|}\sum_{w\in
W}\mathrm{ev}_{1_{T_{H,w}^{F}}}(\mathrm{Cur}_{T_{H,w}}^{H}(h\pi)).$

Via the commutative diagrams (3.3) and (4.2), we identify $h\pi$ consistently
as an element of $\overline{\mathbb{Q}}\mathsf{E}_{G}$,
$\overline{\mathbb{Q}}\mathsf{E}_{H}$,
$\overline{\mathbb{Q}}^{G_{\mathrm{ss}}^{\ast F^{\ast}}\\!/\sim}$,
$\overline{\mathbb{Q}}^{H_{\mathrm{ss}}^{\ast F^{\ast}}\\!/\sim}$,
$\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}$ or
$\overline{\mathbb{Q}}\mathsf{K}_{H^{\ast}}$; we then have
$\mathrm{Cur}_{T_{G,w}}^{G}(h\pi)=(h\pi)|_{T_{G,w}^{\ast F^{\ast}}}$ (see
(1.1)). In addition, for an $F$-stable maximal torus $S$ of $G$ with a chosen
dual torus $S^{\ast}$ (an $F^{\ast}$-stable maximal torus of $G^{\ast}$), upon
denoting by $\widehat{\cdot}:S^{\ast
F^{\ast}}\xrightarrow{\;\sim\;}\mathrm{Irr}_{\overline{\mathbb{Q}}}(S^{F})$
the associated duality and hence identifying $\overline{\mathbb{Q}}^{S^{\ast
F^{\ast}}}=\overline{\mathbb{Q}}S^{F}$:

$\mbox{for every }f\in\overline{\mathbb{Q}}^{S^{\ast F^{\ast}}}\mbox{ and
every }t\in S^{F},\;\mathrm{ev}_{t}(f)=\frac{1}{|S^{F}|}\sum_{s\in S^{\ast
F^{\ast}}}f(s)\widehat{s}(t^{-1}).$ (4.3)

Then the following calculation will complete the proof of the lemma:

$\displaystyle\mathrm{ev}_{1_{T_{G,w}^{F}}}(\mathrm{Cur}_{T_{G,w}}^{G}(h\pi))$
$\displaystyle=\frac{1}{|T_{G,w}^{F}|}\sum_{z\in T_{G,w}^{\ast
F^{\ast}}}(h\pi)(z)\quad(\mbox{by (\ref{S-lemma})})$
$\displaystyle=\frac{|Z^{\ast F^{\ast}}|}{|T_{H,w}^{F}|}\sum_{z\in
T_{G,w}^{\ast F^{\ast}}}(h\pi)(z)\quad(|T_{G,w}^{F}|=|T_{G,w}^{\ast
F^{\ast}}|=\frac{|T_{H,w}^{\ast F^{\ast}}|}{|Z^{\ast
F^{\ast}}|}=\frac{|T_{H,w}^{F}|}{|Z^{\ast F^{\ast}}|})$
$\displaystyle=\frac{1}{|T_{H,w}^{F}|}\sum_{z\in T_{H,w}^{\ast
F^{\ast}}}(h\pi)(z)\quad(h\pi\mbox{ is constant on every $Z^{\ast
F^{\ast}}$-orbit})$
$\displaystyle=\mathrm{ev}_{1_{T_{H,w}^{F}}}(\mathrm{Cur}_{T_{H,w}}^{H}(h\pi))\quad\;\;(\mbox{again
by (\ref{S-lemma})}).$ ∎

A reduction. As $H^{\ast}$ has simply-connected derived subgroup, the
restriction map

$\mathrm{Res}:\mathsf{K}(H^{\ast}\mbox{-mod})\longrightarrow\mathsf{K}_{H^{\ast}}$

is surjective ([St, Thm. 7.4] and [Her, Thm. 3.10]), so that the
$\mathbb{Z}$-module $\mathsf{K}_{H^{\ast}}$ is generated by
$\\{\pi_{H,\lambda}:\lambda\in X(T_{H}^{\ast})\\}$; each
$\pi\in\mathsf{K}_{G^{\ast}}\subset\mathsf{K}_{H^{\ast}}$ is then of the form
$\pi=\sum\limits_{j=1}^{m}n_{j}\pi_{H,\lambda_{j}}$ for some
$m\in\mathbb{N}^{\ast}$, $n_{j}\in\mathbb{Z}$ and $\lambda_{j}\in
X(T_{H}^{\ast})$, and then the previous lemma gives

$\tau_{G}(h\pi)=\tau_{H}(h\pi)=\sum_{j=1}^{m}n_{j}\tau_{H}(h\pi_{H,\lambda_{j}}).$
(4.4)

Formula (4.4) reduces the study of the condition (0.2) to the study of
$\tau_{H}(h\pi_{H,\lambda})$; that is, if we can prove that
$\tau_{H}(h\pi_{H,\lambda})\in\Lambda$ for every $h\in\Lambda\mathsf{E}_{H}$
and every $\lambda\in X(T_{H}^{\ast})$, then we have
$\tau_{G}(h\pi)\in\Lambda$ for every $h\in\Lambda\mathsf{E}_{G}$ and
$\pi\in\Lambda\mathsf{K}_{G^{\ast}}$.

Note however that in the study of the condition
“$\tau_{H}(h\pi_{H,\lambda})\in\Lambda$,” we won’t need the property that
$H^{\ast}$ has simply-connected derived subgroup; in the following, we shall
thus return to the group $G$ and study the condition
“$\tau_{G}(h\pi_{G,\lambda})\in\Lambda$.”

######  5.

An extension $\widetilde{\tau}$ for $\tau(h\pi)$. — We return to the group $G$
(the derived subgroup of $G^{\ast}$ may not be simply-connected) and write
$\tau_{G}=\tau$. Let $T$ be an $F$-stable maximal torus of $G$, let
$W=N_{G}(T)/T$ be the Weyl group of $(G,T)$, and let $T_{w}$ be an $F$-stable
maximal torus of $G$ associated with $w\in W$ (with respect to $T$) as in the
proof of (3.3). Recall the identification
$\overline{\mathbb{Q}}\mathsf{E}_{G}=\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}$
from (0.1). Then, for $h\in\overline{\mathbb{Q}}\mathsf{E}_{G}$ and
$\pi\in\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}$:

$\displaystyle\tau(h\pi)$ $\displaystyle=\frac{1}{|W|}\sum_{w\in
W}\mathrm{ev}_{1_{T_{w}^{F}}}(\mathrm{Cur}_{T_{w}}^{G}(h\pi))\quad(\mbox{by
\cite[cite]{[\@@bibref{}{Bonnafe-Kessar}{}{}, Eq.\;3.5]}})$
$\displaystyle=\frac{1}{|W|}\sum_{w\in
W}\mathrm{ev}_{1_{T_{w}^{F}}}(\mathrm{Cur}_{T_{w}}^{G}(h)\cdot\mathrm{Cur}_{T_{w}}^{G}(\pi))\quad\mbox{($\mathrm{Cur}_{T_{w}}^{G}$is
a ring homomorphism)}$ $\displaystyle=\frac{1}{|W|}\sum_{w\in W}\sum_{t\in
T_{w}^{F}}\mathrm{Cur}_{T_{w}}^{G}(h)(t^{-1})\cdot\mathrm{Cur}_{T_{w}}^{G}(\pi)(t)$
$\displaystyle=\frac{1}{|W|}\sum_{w\in W}\sum_{t\in
T_{w}^{F}}\frac{\epsilon_{G}\epsilon_{T_{w}}}{|T_{w}^{F}|}\cdot\mathrm{Tr}((h,t)|H_{c}^{\ast}(DL_{T_{w}}^{G}))\cdot\mathrm{Cur}_{T_{w}}^{G}(\pi)(t)\quad\mbox{(by
(\ref{BK-Curtis}))}$ $\displaystyle=\frac{1}{|W|}\sum_{w\in
W}\frac{\epsilon_{G}\epsilon_{T_{w}}}{|T_{w}^{F}|}\sum_{t\in
T_{w}^{F}}\sum_{\chi\in\mathrm{Irr}_{\overline{\mathbb{Q}}}(T_{w}^{F})}R_{T_{w}}^{G}(\chi)(h)\cdot\chi(t)\cdot\mathrm{Cur}_{T_{w}}^{G}(\pi)(t)\quad\mbox{(trace
formula)}$ $\displaystyle=\frac{1}{|W|}\sum_{w\in
W}\frac{\epsilon_{G}\epsilon_{T_{w}}}{|T_{w}^{F}|}\sum_{t\in
T_{w}^{F}}\sum_{\chi\in\mathrm{Irr}_{\overline{\mathbb{Q}}}(T_{w}^{F})}R_{T_{w}}^{G}(\chi)(h)\cdot\chi(\mathrm{Cur}_{T_{w}}^{G}(\pi)).$
(5.1)

Using the formula (5.1), we can extend the $\overline{\mathbb{Q}}$-bilinear
map

$\overline{\mathbb{Q}}\mathsf{E}_{G}\times\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}\longrightarrow\overline{\mathbb{Q}},\quad(h,\pi)\longmapsto\tau(h\pi),$

to a $\overline{\mathbb{Q}}$-bilinear map
$\widetilde{\tau}(\cdot,\cdot):\overline{\mathbb{Q}}G^{F}\times\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}\longrightarrow\overline{\mathbb{Q}}$
by setting

$\widetilde{\tau}(h,\pi):=\frac{1}{|W|}\sum_{w\in
W}\frac{\epsilon_{G}\epsilon_{T_{w}}}{|T_{w}^{F}|}\sum_{\chi\in\mathrm{Irr}_{\overline{\mathbb{Q}}}(T_{w}^{F})}R_{T_{w}}^{G}(\chi)(h)\cdot\chi(\mathrm{Cur}_{T_{w}}^{G}(\pi))\quad(h\in\overline{\mathbb{Q}}G^{F},\;\pi\in\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}).$
(5.2)

We then have

$\tau(h\pi)=\widetilde{\tau}(h,\pi)\;\mbox{for
all}\;h\in\overline{\mathbb{Q}}\mathsf{E}_{G}\;\mbox{and
all}\;\pi\in\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}.$ (5.3)

The formula (5.2) for $\widetilde{\tau}$ involves choices of $T$ and $T_{w}$;
we now derive an intrinsic formula for $\widetilde{\tau}$ as follows.

Let $\mathcal{T}_{G}$ be the set of $F$-stable maximal tori of $G$, and let
$\mathcal{T}_{G}/G^{F}$ be the set of $G^{F}$-conjugacy classes in
$\mathcal{T}_{G}$. For each $S\in\mathcal{T}_{G}$, let $W_{G}(S)=N_{G}(S)/S$.
Since the isomorphism class of $T_{w}$ depends only on the $F$-twisted
conjugacy class of $w\in W$, and the stabiliser of $w\in W$ under $F$-twisted
conjugacy may be identified with $W_{G}(T_{w})^{F}$, we have that there are
$\frac{|W|}{|W_{G}(S)^{F}|}$ elements $w\in W$ such that $T_{w}$ is
$G^{F}$-conjugate to $S$. By (5.2), for
$h\in\overline{\mathbb{Q}}\mathsf{E}_{G}$ and
$\pi\in\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}$, we have:

$\displaystyle\widetilde{\tau}(h,\pi)$
$\displaystyle=\sum_{S\in\mathcal{T}_{G}/G^{F}}\frac{\epsilon_{G}\epsilon_{S}}{|W_{G}(S)^{F}|}\frac{1}{|S^{F}|}\sum_{\chi\in\mathrm{Irr}_{\overline{\mathbb{Q}}}(S^{F})}R_{S}^{G}(\chi)(h)\cdot\chi(\mathrm{Cur}_{S}^{G}(\pi))$
$\displaystyle=\frac{1}{|G^{F}|}\sum_{S\in\mathcal{T}_{G}}\epsilon_{G}\epsilon_{S}\sum_{\chi\in\mathrm{Irr}_{\overline{\mathbb{Q}}}(S^{F})}R_{S}^{G}(\chi)(h)\cdot\chi(\mathrm{Cur}_{S}^{G}(\pi)).$
(5.4)

######  6.

Reduction to the case of central $s$. — From now on, let $h=su\in G^{F}$ with
$s\in G^{F}$ (resp. $u\in G^{F}$) the semisimple (resp. unipotent) part in the
Jordan decomposition of $h$. Recall Deligne–Lusztig’s character formula [DeLu,
Thm. 4.2] for each $F$-stable maximal torus $S$ of $G$: (notation:
$\mathrm{ad}(g)x={}^{g}x=gxg^{-1}$)

$R_{S}^{G}(\chi)(h)=\frac{1}{|C_{G}(s)^{\circ
F}|}\sum_{\begin{subarray}{c}g\in G^{F}\\\ g^{-1}sg\in
S^{F}\end{subarray}}Q_{\mathrm{ad}(g)S}^{C_{G}(s)^{\circ}}(u)\cdot({}^{g}\chi)({s})$
(6.1)

where
$Q_{S}^{G}=R_{S}^{G}(\mathrm{id})\cdot\mathbf{1}_{G_{\mathrm{unip}}^{F}}$
denotes the Green function and $C_{G}(s)^{\circ}$ is the identity component of
the centralizer of $s$ in $G$.

We shall write $\widetilde{\tau}=\widetilde{\tau}_{G}$ to specify the group
$G$. Substituting (6.1) into (5.4), we obtain: (below,
$\pi\in\mathsf{K}_{G^{\ast}}$)

$\displaystyle\widetilde{\tau}_{G}(h,\pi)$
$\displaystyle=\frac{1}{|G^{F}|}\sum_{S\in\mathcal{T}_{G}}\epsilon_{G}\epsilon_{S}\sum_{\chi\in\mathrm{Irr}_{\overline{\mathbb{Q}}}(S^{F})}\frac{1}{|C_{G}(s)^{\circ
F}|}\sum_{\begin{subarray}{c}g\in G^{F}\\\ g^{-1}sg\in
S^{F}\end{subarray}}Q_{\mathrm{ad}(g)S}^{C_{G}(s)^{\circ}}(u)\cdot\chi({}^{g^{-1}}s\cdot\mathrm{Cur}_{S}^{G}(\pi))$
$\displaystyle=\frac{1}{|G^{F}|}\frac{1}{|C_{G}(s)^{\circ
F}|}\sum_{S\in\mathcal{T}_{G}}\epsilon_{G}\epsilon_{S}|S^{F}|\sum_{\begin{subarray}{c}g\in
G^{F}\\\ g^{-1}sg\in
S^{F}\end{subarray}}Q_{\mathrm{ad}(g)S}^{C_{G}(s)^{\circ}}(u)\cdot\mathrm{Cur}_{S}^{G}(\pi)({}^{g^{-1}}(s^{-1}))$

(where we have applied the orthogonality of characters)

$\displaystyle=\frac{1}{|G^{F}|}\frac{1}{|C_{G}(s)^{\circ F}|}\sum_{g\in
G^{F}}\sum_{\begin{subarray}{c}S\in\mathcal{T}_{G}\\\ (\mathrm{ad}(g)S)^{F}\ni
s\end{subarray}}\epsilon_{G}\epsilon_{S}|S^{F}|\cdot
Q_{\mathrm{ad}(g)S}^{C_{G}(s)^{\circ}}(u)\cdot\mathrm{Cur}_{\mathrm{ad}(g)S}^{G}(\pi)(s^{-1})$

(where we have used
$\mathrm{Cur}_{\mathrm{ad}(g)S}^{G}(\pi)({}^{g}x)=\mathrm{Cur}_{S}^{G}(\pi)(x)$
for $g\in G^{F}$)

$\displaystyle=\frac{1}{|C_{G}(s)^{\circ
F}|}\sum_{\begin{subarray}{c}S\in\mathcal{T}_{G}\\\ S^{F}\ni
s\end{subarray}}\epsilon_{G}\epsilon_{S}|S^{F}|\cdot
Q_{S}^{C_{G}(s)^{\circ}}(u)\cdot\mathrm{Cur}_{S}^{G}(\pi)(s^{-1})\quad(S\longmapsto\mathrm{ad}(g^{-1})S)$
$\displaystyle=\frac{1}{|C_{G}(s)^{\circ
F}|}\sum_{\begin{subarray}{c}S\in\mathcal{T}_{C_{G}(s)^{\circ}}\\\ S^{F}\ni
s\end{subarray}}\epsilon_{G}\epsilon_{S}|S^{F}|\cdot
Q_{S}^{C_{G}(s)^{\circ}}(u)\cdot\mathrm{Cur}_{S}^{G}(\pi)(s^{-1}),$ (6.2)

where the last equality holds because for $S\in\mathcal{T}_{G}$, if $S^{F}$
contains $s$ then $S\subset C_{G}(s)^{\circ}$.

Consider

$\mathsf{K}_{G^{\ast}}^{\circ}:=\\{\widetilde{\pi}|_{G^{\ast
F^{\ast}}}:\widetilde{\pi}\in\mathsf{K}(G^{\ast}\mbox{-mod})\\}\subset\mathsf{K}_{G^{\ast}};$

observe that $\mathsf{K}_{G^{\ast}}^{\circ}$ is generated by
$\\{\pi_{G,\lambda}:\lambda\in X(T^{\ast})\\}$ as a $\mathbb{Z}$-module (see
Section  ‣  0.). Let $\Lambda_{0}$ be a subring of $\overline{\mathbb{Q}}$,
and consider the following statement:

$\widetilde{\tau}_{G}(h,\pi)\in\Lambda_{0}\mbox{ for all }h=su\in G^{F}\mbox{
and all }\pi\in\mathsf{K}_{G^{\ast}}^{\circ}.$ (6.3)

Now we establish the following “reduction lemma” which will reduce the study
of (6.3) for general $G$ to the case of central $s$ in $G$:

Lemma. Fix a choice of $G$. Suppose that (6.3) is true whenever $G$ therein is
replaced by $C_{G}(s)^{\circ}$ with $s$ a semisimple element in $G^{F}$. Then
(6.3) is true for $G$.

Proof. Fix any $h=su\in G^{F}$, and choose an $F$-stable maximal torus $T$ of
$G$ containing $s$, so that $T$ is also an $F$-stable maximal torus of
$C_{G}(s)^{\circ}$. As $u\in C_{G}(s)^{\circ}$ [DiMi, Prop. 1.27], we have
$h\in C_{G}(s)^{\circ F}$. To verify (6.3) for the chosen $h$, it suffices to
show that $\widetilde{\tau}_{G}(h,\pi_{G,\lambda})\in\Lambda_{0}$ for all
$\lambda\in X(T^{\ast})$, since $\\{\pi_{G,\lambda}:\lambda\in X(T^{\ast})\\}$
generates the $\mathbb{Z}$-module $\mathsf{K}_{G^{\ast}}^{\circ}$.

Let $\lambda\in X(T^{\ast})$, and let $S$ be an $F$-stable maximal torus of
$G$. Choose an $F^{\ast}$-stable maximal torus $S^{\ast}$ of $G^{\ast}$ dual
to $S$ and with a duality
$\widehat{\cdot}:S^{F}\xrightarrow{\;\sim\;}\mathrm{Irr}_{\overline{\mathbb{F}}_{q}}(S^{\ast
F^{\ast}})$, and fix a choice of $g\in G^{\ast}$ such that
$S^{\ast}={}^{g}T^{\ast}$. This duality and the fixed embedding
$\overline{\mathbb{F}}^{\times}_{q}\hookrightarrow\overline{\mathbb{Q}}^{\times}$
allow us to identify $\overline{\mathbb{Q}}S^{F}$ with
$\overline{\mathbb{Q}}^{S^{\ast F^{\ast}}}$. For each $\mu\in X(T^{\ast})$,
set $\mu_{S^{\ast}}={}^{g}\mu\in X(S^{\ast})$, and define $\phi_{S}(\mu)\in
S^{F}$ by the relation $\mu_{S^{\ast}}|_{S^{\ast
F^{\ast}}}=\widehat{\phi_{S}(\mu)}\in\mathrm{Irr}_{\overline{\mathbb{F}}_{q}}(S^{\ast
F^{\ast}})$. We then have a map $\phi_{S}:X(T^{\ast})\longrightarrow S^{F}$
which extends to a ring homomorphism

$\phi_{S}:\overline{\mathbb{Q}}[X(T^{\ast})]\longrightarrow\overline{\mathbb{Q}}S^{F}=\overline{\mathbb{Q}}^{S^{\ast
F^{\ast}}}.$

The following diagram then commutes (where
$W=W_{G^{\ast}}(T^{\ast})=N_{G^{\ast}}(T^{\ast})/T^{\ast}$):

${\overline{\mathbb{Q}}\mathsf{K}(G^{\ast}\mbox{-mod})}$${\overline{\mathbb{Q}}[X(T^{\ast})]^{W}}$${\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}}$${\overline{\mathbb{Q}}^{G^{\ast
F^{\ast}}}}$${\overline{\mathbb{Q}}^{S^{\ast
F^{\ast}}}}$${\mathrm{ch}}$${\sim}$${\mathrm{Res}_{G^{\ast
F^{\ast}}}^{G^{\ast}}}$${\phi_{S}}$${\mathrm{br}}$${\sim}$${\mathrm{Res}^{G^{\ast
F^{\ast}}}_{S^{\ast F^{\ast}}}}$

Combining this with (1.1) we see that the following diagram of rings also
commutes:

${\overline{\mathbb{Q}}\mathsf{K}(G^{\ast}\mbox{-mod})}$${\overline{\mathbb{Q}}[X(T^{\ast})]^{W}}$${\overline{\mathbb{Q}}\mathsf{K}_{G^{\ast}}}$${\overline{\mathbb{Q}}\mathsf{E}_{G}}$${\overline{\mathbb{Q}}S^{F}}$${\mathrm{Res}_{G^{\ast
F^{\ast}}}^{G^{\ast}}}$${\mathrm{ch}}$${\sim}$${\phi_{S}}$${(\ref{EK-
identify})}$${\mathrm{Cur}_{S}^{G}}$ (6.4)

The commutative diagram (6.4) gives the relation

$\mathrm{Cur}_{S}^{G}(\pi_{G,\lambda})=\phi_{S}(r_{G,\lambda}),$ (6.5)

where we recall that $r_{G,\lambda}=\sum\limits_{\mu\in
W_{G^{\ast}}(T^{\ast})\lambda}\mu\in\mathbb{Z}[X(T^{\ast})]^{W_{G^{\ast}}(T^{\ast})}.$
Via the identifications

$W_{C_{G}(s)^{\circ\ast}}(T^{\ast})=W_{C_{G}(s)^{\circ}}(T)\leq
W_{G}(T)=W_{G^{\ast}}(T^{\ast}),$

we may write
$W_{G^{\ast}}(T^{\ast})\lambda=\bigsqcup\limits_{\lambda^{\prime}\in\Omega}W_{C_{G}(s)^{\circ\ast}}(T^{\ast})\lambda^{\prime}$
for some finite subset $\Omega$ of $W_{G^{\ast}}(T^{\ast})\lambda$, so that
$r_{G,\lambda}=\sum\limits_{\lambda^{\prime}\in\Omega}r_{C_{G}(s)^{\circ},\lambda^{\prime}}$
and then
$\mathrm{Cur}_{S}^{G}(\pi_{G,\lambda})=\sum\limits_{\lambda^{\prime}\in\Omega}\mathrm{Cur}_{S}^{C_{G}(s)^{\circ}}(\pi_{C_{G}(s)^{\circ},\lambda^{\prime}})$
by (6.5). Applying (6.2) to $\pi=\pi_{G,\lambda}$, we thus deduce that

$\widetilde{\tau}_{G}(h,\pi_{G,\lambda})=\epsilon_{G}\epsilon_{C_{G}(s)^{\circ}}\sum_{\lambda^{\prime}\in\Omega}\widetilde{\tau}_{C_{G}(s)^{\circ}}(h,\pi_{C_{G}(s)^{\circ},\lambda^{\prime}}).$
(6.6)

Thanks to (6.6) and the assumption in the lemma, for our choice of $h=su\in
G^{F}$ and $T$ (with $s\in T$), we have
$\widetilde{\tau}_{G}(h,\pi_{\lambda})\in\Lambda_{0}$ for all $\lambda\in
X(T^{\ast})$, whence $\widetilde{\tau}_{G}(h,\pi)\in\Lambda_{0}$ for all
$\pi\in\mathsf{K}_{G^{\ast}}^{\circ}$. As $h\in G^{F}$ was arbitrary, (6.3) is
true for $G$. ∎

######  7.

The case of central $s$. — Keep the notation $T$, $W$ and $T_{w}$ as in
Section 5. Let $h=su\in G^{F}$ be as in Section 6, and suppose furthermore
that $s$ lies in the centre of $G$. Then $C_{G}(s)^{\circ}=G$, and (6.1)
becomes $R_{S}^{G}(\chi)(h)=Q_{S}^{G}(u)\chi(s)$, so that (5.2) yields

$\displaystyle\widetilde{\tau}_{G}(h,\pi_{\lambda})$
$\displaystyle=\frac{1}{|W|}\sum_{w\in
W}\frac{\epsilon_{G}\epsilon_{T_{w}}}{|T_{w}^{F}|}\sum_{\chi\in\mathrm{Irr}_{\overline{\mathbb{Q}}}(T_{w}^{F})}Q_{T_{w}}^{G}(u)\cdot\chi(s)\cdot\chi(\mathrm{Cur}_{T_{w}}^{G}(\pi_{\lambda}))$
$\displaystyle=\frac{1}{|W|}\sum_{w\in
W}\epsilon_{G}\epsilon_{T_{w}}Q_{T_{w}}^{G}(u)\langle\pi_{\lambda}|_{T_{w}^{\ast
F^{\ast}}},\widehat{s^{-1}}\rangle_{T_{w}^{\ast
F^{\ast}}}\quad(\mbox{orthogonality of characters})$
$\displaystyle=\langle\pi_{\lambda},\gamma\rangle_{G^{\ast F^{\ast}}}$ (7.1)

where (using [DeLu, Prop. 7.3])

$\gamma:=\frac{1}{|W|}\sum_{w\in
W}\epsilon_{G}\epsilon_{T_{w}}Q_{T_{w}}^{G}(u)\mathrm{Ind}_{T_{w}^{\ast
F^{\ast}}}^{G^{\ast F^{\ast}}}\widehat{s^{-1}}=\frac{1}{|W|}\sum_{w\in
W}Q_{T_{w}}^{G}(u)R_{T_{w}^{\ast}}^{G^{\ast}}\widehat{s^{-1}}\otimes\mathrm{St}_{G^{\ast}}.$

As $s$ lies in the centre of $G$, $\widehat{s^{-1}}$ is in fact a
multiplicative character of $G^{\ast F^{\ast}}$, so

$\displaystyle\gamma$
$\displaystyle=\gamma^{\prime}\otimes\widehat{s^{-1}}\otimes\mathrm{St}_{G^{\ast}}$
with $\displaystyle\gamma^{\prime}$ $\displaystyle:=\frac{1}{|W|}\sum_{w\in
W}Q_{T_{w}}^{G}(u)R_{T_{w}^{\ast}}^{G^{\ast}}(\mathrm{id}).$ (7.2)

We need some facts from the theory of almost characters, following [Lu, Ch.
3-4]; in the notation of that book, we are considering the case $n=1$ and $L$
trivial. See also [Ca, Sec. 7.3] for a concise exposition, but with some
extraneous hypotheses. Let $c$ be the order of the automorphism $F$ on $W$
(when $G$ is split, we have $c=1$); denote by $\widehat{W}_{\mathrm{ex}}$ the
set of all $\phi\in\mathrm{Irr}_{\mathbb{Q}}(W)$ which can be extended to a
$\mathbb{Q}$-valued irreducible character of $\widehat{W}:=W\rtimes\langle
F\rangle$ (by [Sp2, Cor. 1.15], every irreducible representation of $W$ over a
characteristic 0 field is defined over $\mathbb{Q}$); for each
$\phi\in\widehat{W}_{\mathrm{ex}}$, there exists such an extension (in fact,
exactly two) $\widetilde{\phi}\in\mathrm{Irr}_{\mathbb{Q}}(\widehat{W})$.
Fixing a choice of such $\tilde{\phi}$, we then call

$R_{\widetilde{\phi}}^{G^{\ast}}:=\frac{1}{|W|}\sum_{w\in
W}\widetilde{\phi}(wF)R_{T_{w}^{\ast}}^{G^{\ast}}(\mathrm{id})$

an almost character of $G^{\ast F^{\ast}}$.

Recall from Section  ‣  0. the definition of bad primes for $G$. Note that a
prime is bad for $G$ if and only if it is bad for $G^{*}$. Define

$\mbox{$M_{G}=$ product of all bad primes for $G$}.$ (7.3)

Using Lusztig’s work on unipotent characters,

each almost character $R_{\widetilde{\phi}}^{G^{\ast}}$ is a
$\overline{\mathbb{Z}}[\frac{1}{M_{G}}]$-linear combination (7.4)
$\displaystyle\mbox{of irreducible $\overline{\mathbb{Q}}$-valued unipotent
characters of $G^{\ast F^{\ast}}$}.$

Indeed, if $G^{\ast}$ has connected centre, then [Lu, Thm. 4.23] expresses
$R_{\widetilde{\phi}}$ as a linear combination of unipotent characters of
$G^{\ast F^{\ast}}$. By [Lu, (4.21.7)], the denominators divide the orders of
certain groups $\mathcal{G}_{\mathcal{F}}$ of the form
$\prod\mathcal{G}_{\mathcal{F}_{i}}$ where the product is over the irreducible
factors of the root system of $G^{\ast}$. Each $\mathcal{G}_{\mathcal{F}_{i}}$
is defined in a case-by-case fashion, in a way depending only on the
corresponding irreducible factor of the root system, in [Lu, 4.4–4.13], and
has order divisible only by bad primes for that factor. If $G^{\ast}$ does not
have connected centre then we choose a short exact sequence

$1\to G^{\ast}\to H^{\ast}\to Z^{\ast}\to 1$

as in (2.2) (with the roles of $G^{\ast}$ and $G$ reversed). Extending the
chosen maximal $F^{\ast}$-stable torus and Borel from $G^{\ast}$ to $H^{\ast}$
as in Section 2, we may identify the Weyl groups of $G^{\ast}$ and $H^{\ast}$.
Using (2.4) (with $\chi=\mathbf{1}$ therein), we then have

$R_{\widetilde{\phi}}^{H^{\ast}}|_{G^{\ast
F^{\ast}}}=R_{\widetilde{\phi}}^{G^{\ast}},$

whence $R_{\widetilde{\phi}}^{G^{\ast}}$ is
$\overline{\mathbb{Z}}[\frac{1}{M_{G}}]$-linear combination of restrictions to
$G^{\ast F^{\ast}}$ of unipotent characters of $H^{\ast F^{\ast}}$. However,
the restriction to $G^{\ast F^{\ast}}$ of a unipotent character of $H^{\ast
F^{\ast}}$ is a unipotent character by [DiMi2, Prop. 11.3.8], so (7.4)
follows.

Now we prove the following lemma:

Lemma. The character $\gamma^{\prime}$ in (7.2) is a finite
$\overline{\mathbb{Z}}[\frac{1}{M_{G}}]$-linear combination of almost
characters of $G^{\ast F^{\ast}}$.

Proof. We have

$\displaystyle\gamma^{\prime}$ $\displaystyle=\frac{1}{|W|}\sum_{w\in
W}R_{T_{w}}^{G}(\mathrm{id})(u)R_{T_{w}^{\ast}}^{G^{\ast}}(\mathrm{id})$
$\displaystyle=\frac{1}{|W|}\sum_{w\in
W}R_{T_{w}}^{G}(\mathrm{id})(u)\sum_{\phi\in\widehat{W}_{\mathrm{ex}}}\widetilde{\phi}(wF)R_{\widetilde{\phi}}^{G^{\ast}}\quad(\mbox{see
\cite[cite]{[\@@bibref{}{Carter}{}{}, p.\;76]}})$
$\displaystyle=\sum_{\phi\in\widehat{W}_{\mathrm{ex}}}R_{\widetilde{\phi}}^{G}(u)R_{\widetilde{\phi}}^{G^{\ast}};$

by (7.4) and the fact that character values of representations of finite
groups are algebraic integers, all $R_{\widetilde{\phi}}^{G}$ must take values
in $\overline{\mathbb{Z}}[\frac{1}{M_{G}}]$.∎

Remark. In the above lemma, the character $\gamma^{\prime}$ in (7.2) can
indeed be written as a finite $\overline{\mathbb{Z}}$-linear combination of
almost characters of $G^{\ast F^{\ast}}$; to see this, one will need to use a
theorem of Shoji ([Sh, Thm. 5.5]; see also [DiMi2, Thm. 13.2.3]) which says
that $Q_{T_{w}}^{G}(u)=\mathrm{Tr}(wF|H_{c}^{\ast}(\mathcal{B}_{u}))$ for all
$w\in W$, where $\mathcal{B}_{u}$ is the variety of Borel subgroups of $G$
containing $u$. We won’t need this stronger property of $\gamma^{\prime}$
later and so we omit its proof here.

Using the previous lemma, (7.1), (7.2) and (7.4), we get the following
proposition:

Proposition. We have
$\widetilde{\tau}_{G}(h,\pi_{\lambda})\in\overline{\mathbb{Z}}[\frac{1}{M_{G}}]$
for all $\lambda\in X(T^{\ast})$ and all $h\in G^{F}$ whose semisimple part
$s$ is central in $G$. ($M_{G}$ is as in (7.3).)

End of proof of the main theorem in Section  ‣  0.. From now on, we remove the
assumption that $s$ is central in $G$.

Observe that a prime number that is bad for $C_{G}(x)^{\circ}$ with $x$ a
semisimple element of $G^{F}$ is also bad for $G$; indeed, this follows from
the definition of bad primes in Section  ‣  0. and from the following two
facts: (i) if $G$ is simple of type $A$ (resp. of classical type), then the
centralizer of every semisimple element of $G$ has only factors of type $A$
(resp. of classical type); (ii) if $G$ is simple of type $G_{2}$, $F_{4}$,
$E_{6}$ or $E_{7}$, then the centralizer of every semisimple element of $G$
cannot have factors of type $E_{8}$ (for dimensional reasons).

Therefore, the previous proposition and the lemma in Section 6 together imply
that
$\widetilde{\tau}_{G}(h,\pi_{\lambda})\in\overline{\mathbb{Z}}[\frac{1}{M_{G}}]$
for all $h\in G^{F}$ and all $\lambda\in X(T^{\ast})$. We then deduce from
(5.3) that

$\tau_{G}(h\pi_{\lambda})=\widetilde{\tau}_{G}(h,\pi_{\lambda})\in\Lambda[\frac{1}{M_{G}}]$
for all $h\in\Lambda\mathsf{E}_{G}$ and all $\lambda\in X(T^{\ast})$. (7.5)

Now fit $G$ into the exact sequence (2.2); as $H$ therein has the same type of
root datum as $G$, we have $M_{H}=M_{G}$, so (7.5) applied to $H$ gives
$\tau_{H}(h\pi_{H,\lambda})\in\Lambda[\frac{1}{M_{G}}]$ for all
$h\in\Lambda\mathsf{E}_{H}$ and all $\lambda\in X(T_{H}^{\ast})$; for our $G$,
(4.4) then tells us that (0.2) is true when $\Lambda$ therein is replaced by
$\Lambda[\frac{1}{M_{G}}]$. Consequently, when all bad prime numbers for $G$
are invertible in $\Lambda$, we have $\Lambda[\frac{1}{M_{G}}]=\Lambda$ and
$\Lambda\mathsf{E}_{G}=\Lambda\mathsf{K}_{G^{\ast}}$. ∎

## References

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Tzu-Jan Li: Institut de Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG),
4 place Jussieu, 75252 Paris cedex 05, FRANCE
Email address<EMAIL_ADDRESS>
Jack Shotton: Department of Mathematical Sciences, Mathematical Sciences &
Computer Science Building, Durham University Upper Mountjoy Campus, Stockton
Road, Durham DH1 3LE UNITED KINGDOM
Email address<EMAIL_ADDRESS>


Titanium based oxides are abundant, chemically stable, non-toxic, and highly
versatile materials, with applications ranging from photovoltaics to
catalysis. For rutile and anatase phases of TiO2, the bandgap ranges from
3.0–3.2 eV, which are too large to absorb in the visible range (400 nm - 700
nm), resulting in poor photo-catalytic efficiency. Nitrogen doping into TiO2
has been able to narrow it’s bandgap, forming an absorption tail in the
visible-light region. However, TiO2 has limits to which it can be doped,
suggesting investigations of the oxygen-deficient Ti2O3. Using the state-of-
the-art Density Functional Theory (DFT) as implemented in the Quantum ESPRESSO
package, we report on the structural and electronic properties of corundum-
type Ti2N2O (an example TinN2O2n-3 compound with n=2). The most stable sample
of the oxynitride (Ti2N2O-P1), has a bandgap of 2.2 eV, which is clearly near
the middle of visible light, and has no in-gap states, suggesting that they
are more efficient visible-light-driven materials for photocatalytic
applications compared to TiO2, TiO2: N and Ti2O3.

Key words: DFT, Ti2O3,Ti2N2O, structural and electronic properties.

# The Impact of Nitrogen Doping on Structural and Electronic Properties of
Titanium Sesquioxide, Ti2O3 : An ab-initio Study

Lynet Allan<EMAIL_ADDRESS>Department of Physics, School of Physical
Sciences, University of Nairobi, P.O.Box 30197-00100 Nairobi Kenya.  George O
Amolo<EMAIL_ADDRESS>Materials Modeling Group, School of Physics
and Earth Sciences, The Technical University of Kenya, 52428-00200, Nairobi,
Kenya.  Julius Mwabora  Department of Physics, School of Physical Sciences,
University of Nairobi, P.O.Box 30197-00100 Nairobi Kenya.  Silas Mureramanzi
Department of Physics, School of Physical Sciences, University of Nairobi,
P.O.Box 30197-00100 Nairobi Kenya.

##  I Introduction

Due to increasing global energy demands and the harmful environmental
pollution caused by the combustion of fossil fuels, the quest for sustainable
alternative energy resources has become a major area of researchShaffer
(2019). Ti-based oxides have been intensively investigated as photo-catalysts
due to their wide applications in environmental remediation and solar energy
conversion Daghrir  _et al._ (2013); Agustina  _et al._ (2005). Despite having
wide bandgaps of 3.0 eV and 3.2 eV, respectively, rutile and anatase phases of
titanium dioxide (TiO2) still attracts considerable attention Aoki  _et al._
(2019); Agustina  _et al._ (2005); Atambo  _et al._ (2019); Carlo ; Fujishima
_et al._ (2000); Li (2016); Wu  _et al._ (2013); Shirai  _et al._ (2018). To
improve the solar application efficiency of Ti-based oxides, there has been an
attempt to narrow their bandgap to absorb a major part of the solar spectrum
Khan  _et al._ (2012). Doping is considered to be one of the promising methods
of narrowing the bandgap of the Ti-based oxides Guo  _et al._ (2011).
Previous ab-initio works have been reported on doped TiO2 compounds. V, Cr,
Mn, Fe, and Ni have been doped into TiO2 at the Ti sitesInturi  _et al._
(2014); Umebayashi  _et al._ (2002), such cationic doping leads to the
localized d-states deep in the bandgap of TiO2, which would act as the
recombination centers for photoexcited electrons and holes, lowering
photocatalytic activity. An anion doping in TiO2 with N, C, S, and B has been
equally explored Morikawa  _et al._ (2001); Fujishima  _et al._ (2008);
Sakthivel and Kisch (2003); Schneider  _et al._ (2014); Morikawa  _et al._
(2001), such resulted in the p-states near the valence band being similar to
other deep donor levels in the semiconductorSchneider  _et al._ (2014). Owing
to the finite number of oxygen vacancies, TiO2 has limits to which it can be
dopedShevlin and Woodley (2010). In this work, we increase nitrogen
concentrations in titanium oxide systems by considering compounds having the
composition Tin O2n-1, rather than TiO2 with a finite number of oxygen
vacancies. These systems can be viewed as ordered oxygen-deficient TiO2
variants with a lot of oxygen vacancies which allows nitrogen substitutions,
hence, should be able to accommodate far greater nitrogen concentrations than
TiO2. In search of novel photocatalytic materials that are stable, efficient,
and capable of being visible light driven, Wu and his co-workersWu  _et al._
(2013), using high throughput screening method within density functional
theory (DFT)Kohn and Sham (1965), discovered TinN2O2n-3 compound for the n=3
case (Ti3N2O3), which they predicted to be synthesizable, suitable for visible
light (or UV light) absorption, and energetically favorable to catalyze the
water-splitting reaction.
Corundum Ti2O3, a Ti based oxide with nominal Tin O2n-1, n=2 configuration
together with its oxynitrides TinN2O2n-3, n=2, (Ti2N2O), has been much less
investigated. Here, we explore the structural and electronic properties of
Corundum Ti2O3 using Density Functional Theory (DFT) and Density Functional
Theory with the inclusion of Hubbard U correction (DFT$\boldsymbol{+}$U)
Himmetoglu  _et al._ (2014) studies. In particular, we employ nitrogen
substitutional doping at oxygen sites. The structural and electronic
properties of the pristine and doped oxides, with and without the inclusion of
Hubbard U, are simulated and analyzed. This study provides complementary
theoretical data on structural and electronic properties of anionic doping of
Ti2O3 with nitrogen.

##  II Computational details

We have carried out simulations on trigonal Ti2O3 with a= b=c,
$\alpha=\beta=\gamma\neq\ $900 within density functional theory (DFT)Kohn and
Sham (1965) using the generalized gradient approximation (GGA) as implemented
in the Quantum ESPRESSO(QE) codeGiannozzi  _et al._ (2009). The electron-ion
interactions are described by ultra-soft pseudo-potentialsKresse and Joubert
(1999) drawn from the 2.0.1 version of the library of Dal Corso et
al,(2001)Carlo . For the exchange-correlation interaction, the enhanced
Perdew-Burke- Ernzerhof functional of the generalized gradient approximation,
for solids (GGA-PBEsol) Perdew  _et al._ (1996) are employed. This exchange-
correlation is selected because it is computationally efficient and no
adjustable parameter is required. An extra Hubbard U term is included to
account for the strong electron correlations, with U values tested in the
range of 1 eV - 7 eV, applied only to the Ti-3d orbitals in Ti2O3. The wave
functions for valence electrons are expanded through a plane-wave basis set
within the energy cut-off of 30 Ry. The atomic coordinates are relaxed until
the forces are less than 0.01 eV/Å. The crystal structures are then viewed
using the X-window Crystalline and Molecular Structure Visualization
(XCrysDen)Kokalj and Causà (2001) program where the lattice parameters a and c
for the pristine and doped structures with and without the inclusion of the
hubbard U, are determined. A Monkhorst-Pack mesh sampling Monkhorst and Pack
(1976) equivalent to $5\times 5\times 5$ are employed.For band structure
calculations, explicit positions along the high symmetry axes are described on
the path
${\Gamma}$$\rightarrow{Z}$$\rightarrow{F}$$\rightarrow$${\Gamma}$$\rightarrow{L}$Setyawan
and Curtarolo (2010) shown in Fig 1.

Figure 1: Brillouin zone integration for Corundum Ti2O3 showing the origin at
$\Gamma$

##  III Results and discussions

###  III.1 Structural Properties: Lattice Parameters

By fitting the energy volume relationship to the Murnaghan equation of
stateMurnaghan (1944), the volume that yields the minimum energy value are
obtained. From table 1 the calculated lattice parameters a and c (at U=0 eV)
lie slightly above the experimental dataLi  _et al._ (2018). It is well known
that GGA tends to overestimate the lattice parametersNiu  _et al._ (2011). The
slight deviations are attributed to the fact that simulations have not taken
into account the existence of intrinsic and other defects occurring in actual
situations. To check on the feasibility of the doped systems, the structural
properties of pure and doped Ti2O3 are compared. After the doping process, the
lattice constants of doped systems are found to be slightly deformed, showing
lattice strain due to the difference in radius between the dopant N atom(s)
and the substituted O atom(s). The optimized lattice parameters for undoped
Ti2O3 are a=b=c 5.274 Å, while for doped systems are a=b=c 5.271 Å. This
result is consistent with the experimental resultsLi  _et al._ (2018), as well
as other theoretical studiesAoki  _et al._ (2019), demonstrating the accuracy
and dependability of the structural optimization parameters and method used.
To account for the strong electron correlations, an extra Hubbard-U term is
added, with a Ueff of 1 eV - 7 eV applied only to the Ti-3d orbitals. As seen
in Table 1, there are some slight changes in the lattice parameter. To be more
exact, the c/a ratio is seen to be decreasing with increasing values of U at
some point. The inclusion of the U values in this computation is likely to
have resulted in changes in the lattice parameters due to the correction of
hybridization between O-2p and Ti-3d orbitalsPersson and Mirbt (2006). As a
result, the ideal value of U is determined by seeking a fair agreement with
experimental data for both structural and electronic properties.

U Values |  |  |
---|---|---|---
(eV) | Lattice Parameters | Band gaps (eV)
| a(Å) | c/a |
0 | 5.27 | 1.03 | No gap
1 | 5.22 | 1.03 | 0.11
2 | 5.24 | 1.02 | 0.49
3 | 5.25 | 1.02 | 0.87
4 | 5.19 | 1.01 | 1.36
5 | 5.16 | 1.00 | 1.63
6 | 5.38 | 1.00 | 1.76
7 | 5.35 | 1.03 | 1.86
ExptLi  _et al._ (2018) | 5.15 | 1.01 | 1.65
Table 1: Variation of lattice parameters and calculated band gaps with
increasing Hubbard U Values for Corundum Ti2O3

###  III.2 Choice of Hubbard U for Ti2O3

Corundum Ti2O3, appeared metallic, as shown in Fig 7(a), this is consistent
with other DFT findings Aoki  _et al._ (2019). Experimental studies by Li et
al (2018)Li  _et al._ (2018), however, show that Corundum Ti2O3 has a bandgap
of 1.65 eV. As expected, Kohn-Sham DFT calculations could not predict this
small gap since it is known to underestimate the bandgaps of systems with d
and f orbitals Hu and Metiu (2011). A moderated Hubbard potential,U, applied
to the calculations of electronic structures leads to more precise electronic
band gaps. The U values are optimized empirically by tuning the values of U
when seeking an agreement with the experimental results of the properties of
the Ti2O3Hu and Metiu (2011), as well as checking on the impact on lattice
parameters as indicated in Table 1. One thing that came out clear was that the
value of the energy gap kept increasing in an almost linearly manner with the
U values as shown in Fig 2. Corundum Ti2O3 opened a 1.638 eV band gap by DFT+U
(U=5 eV) studies, supporting the experimental data that this structure has
semiconducting properties. The lattice parameter at U=5 eV presented in Table
1 is also in good agreement with the experimental lattice parameterLi  _et
al._ (2018) of 5.15 eV. The value, U = 5 eV, is therefore considered as the
optimal value and inferred for improved prediction of the band gap of the
corundum based Ti2N2O structures.

Figure 2: Variations of band gaps with increasing values of U

###  III.3 Nitrogen substituted Ti2O3 (Ti2N2O)

By substituting two of the ${2n-1}$ oxygen atoms of TinO2n-1 compounds with
nitrogen, a series of Ti2N2O oxynitrides are modelled. Four out of six oxygen
atoms are substituted with nitrogen to obtain Ti2N2O compounds. Due to the
symmetry of the parent material, only three types (P1, P2, and P3) shown in
Fig 3 are considered.

Figure 3: (Color online) (P0) Pure Corundum Ti2O3 [R3c (No. 167),
trigonal],(P1)-(P3) Different orientations of Nitrogen substitutions in Ti2O3
structure. P2 has an inversion symmetry while P1 and P3 do not have.

The other alternative is to model the oxynitrides in the hexagonal
convectional cells of Ti2O3 containing 30 atoms (twelve titanium and eight
oxygen atoms)Aoki  _et al._ (2019). This is as well considered and viewed at
three different angles of projection in Fig 4. Twelve out of eighteen oxygen
atoms are replaced by nitrogen in seven different ways to form seven samples
of Ti2N2O oxynitrides, projected in the a+b direction (for a clearer view),
and arranged according to their stability as S1-S7, (S1 being the most stable
sample) as shown in Fig 5.

Figure 4: (Color online) Three different angles of projection of Ti2O3
hexagonal convectional cell (a) Bird’s eye view (b) Top view, and (c)
Projected in the direction of a+b. Figure 5: (Color online) Seven samples of
Ti2N2O (S-structures)

The stability of Ti2N2O (P and S) corundum based structures, are then compared
as shown in Fig 6.

Figure 6: Plots for energy/cell (eV) against lattice parameter for both S- and
P-structures.

P-structures have lower formation energies than S-structures from Fig 6, hence
more stable, with P1 being the most stable sample of all. This is well in
agreement with the theory of Aoki et al.(2019)Aoki  _et al._ (2019), where GW
studies were employed to study the structures. Therefore, the P structures are
considered for further analysis of the band structures and projected density
of state (PDOS) shown in Fig 7.

###  III.4 Band structures and Projected Density of States

To explore the effect of doping on the electronic properties of Ti2O3 and
clarify the origin of improved visible light photo activity, the band
structures and projected density of states (PDOS) are calculated as given in
Fig 7. For ease of comparison, the Fermi level (Ef) is set at zero. With
Ti-3d/ O-2p orbitals dominant in the entire band structure, Corundum Ti2O3
appeared metallic, as shown in Fig 7(a).

Fig 7(b)-(d) shows that nitrogen substitution into corundum Ti2O3 opens up a
gap of 2.2 eV by DFT+U studies with U=5 eV in the P1 sample (most stable)
while the P2 and P3 samples have band gaps of 1.98 eV and 1.92 eV,
respectively. The electronic structure of Ti2O3 changes as the lattice
parameters and deformation around the dopants changes, modifying their
photocatalytic activity. In all three (P1, P2 and P3) samples, indirect
bandgaps with bandgap character of Ti-3d/N-2p orbitals are observed. Some
hybridization between N-2p and Ti-3d states is also observed in the valence
band, although there are no in-gap states within the intrinsic gap, as there
are in TiO2 doped nitrogen Persson and Mirbt (2006), thus the absorption
threshold is unlikely to be affected. We have obtained a Ti-based oxide with a
reduced band gap compared to the highly investigated TiO2, without necessarily
creating the in-gap states which are known to serve as recombination centers
for the photo-excited electrons and holes. This study suggests that, relative
to pristine TiO2, Ti2O3 and TiO2: N compounds, Ti2N2O structures are more
promising photo-catalytic/photo-voltaic materials. Table 2 offers a
description of a comparison of the calculated and experimental bandgap for the
oxides.

Figure 7: Band structures and Pdos for Corundum-based Ti2N2O_P structures using DFT+U (U=5 eV) method. Table 2: Summary of the comparison of the calculated and experimental bandgap for the structures MATERIAL | ENERGY BAND GAP (eV)
---|---
|  | THIS WORK
---
(GGA)
| THEORETICAL
---
(GGA)
| THIS WORK
---
(DFT+U)
EXPERIMENTAL
Corundum Ti2O3 | No gap | No gap Aoki  _et al._ (2019) | 1.64(U=5 eV) | 1.65Li  _et al._ (2018)
| Oxynitrides
---
Ti2N2O (P1)
Ti2N2O (P2)
Ti2N2O (P3)
|  |  | 2.20 (U=5 eV)
---
1.98 (U=5 eV)
1.92 (U=5 eV)

##  IV Conclusion

Using the DFT and DFT+U approach, we have investigated the effect of nitrogen
substitution on the geometry and electronic structure of corundum Ti2O3 in
order to enhance its photo-catalytic performance. Significant structure
information which may be useful in guiding experimental work as well as
identifying appropriate applications for the material has been obtained.
Structural and electronic properties of corundum Ti2O3 and the oxynitrides
Ti2N2O have been analysed in this work. With regard to the impact of the
Hubbard U parameter on both structural and electronic properties, the best
value of U is found to be 5 eV. Even though these calculations assume defect-
free structures, the lattice parameters and band gap obtained for this value
of U are in good agreement with other DFT results and experimental data. The
oxygen deficient corundum Ti2O3, which showed metal-like properties opens up a
bandgap of 1.638 eV by DFT+U (U= 5 eV) studies, agreeing well with reported
experimental data. To enhance photo-activity of Ti-based oxides, without
creating mid-gap states, we aimed at reducing the band gaps of the oxides.
This was achieved in Ti2N2O structures, which had a bandgap of 2.2 eV (563.56
nm), in the most stable sample_P1, and had no in-gap states. In principle,
Ti2N2O materials are found to be better for photo-catalytic/ photo-voltaic
applications, compared to TiO2, TiO2:N and Ti2O3 materials.

##  V Acknowledgments

This research was supported by the International Science Program (UON
500-661-127, PI- Prof. Aduda Bernard O.), through condensed matter thematic
group of the Department of Physics, University of Nairobi. The Centre of High-
Performance Computing (CHPC) through the project MATS862, Rosebank Cape Town
Republic of South Africa is appreciated for providing access to the HPC
cluster facility used in this research. Much appreciation to the Materials
Modeling Group at the Technical University of Kenya for support.

##  VI conflicts of interest

There is no conflict of interest.

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