# Non-Abelian nonsymmorphic chiral symmetries

Yi Yang<EMAIL_ADDRESS>Department of Physics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA Department of Physics,
University of Hong Kong, Pokfulam, Hong Kong, China Hoi Chun Po  Department of
Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA Department of Physics, Hong Kong University of Science and
Technology, Clear Water Bay, Hong Kong, China Vincent Liu  Department of
Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA Department of Physics, University of California, Berkeley,
California 94720, USA John D. Joannopoulos  Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Liang Fu  Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA Marin Soljačić  Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(27th August 2024)

###### Abstract

The Hofstadter model exemplifies a large class of physical systems
characterized by particles hopping on a lattice immersed in a gauge field.
Recent advancements on various synthetic platforms have enabled highly-
controllable simulations of such systems with tailored gauge fields featuring
complex spatial textures. These synthetic gauge fields could introduce
synthetic symmetries that do not appear in electronic materials. Here, in an
${\rm SU}(2)$ non-Abelian Hofstadter model, we theoretically show the
emergence of multiple nonsymmorphic chiral symmetries, which combine an
internal unitary anti-symmetry with fractional spatial translation. Depending
on the values of the gauge fields, the nonsymmorphic chiral symmetries can
exhibit non-Abelian algebra and protect Kramer quartet states in the bulk band
structure, creating general four-fold degeneracy at all momenta. These
nonsymmorphic chiral symmetries protect double Dirac semimetals at zero
energy, which become gapped into quantum confined insulating phases upon
introducing a boundary. Moreover, the parity of the system size can determine
whether the resulting insulating phase is trivial or topological. Our work
indicates a pathway for creating topology via synthetic symmetries emergent
from synthetic gauge fields.

The quantum Hall Cage  _et al._ (2012) and quantum anomalous Hall Haldane
(1988); Chang  _et al._ (2013) effects represent the earliest examples of
topological phases of matter. However, such phases with robust chiral edge
modes are only realizable under stringent conditions, like a strong breaking
of time-reversal symmetry, either though external magnetic fields or suitable
intrinsic magnetic order. The topological landscape changed completely with
the advent of topological insulators Hasan and Kane (2010); Qi and Zhang
(2011). A key insight from the early studies was how time-reversal symmetry
could protect new forms of nontrivial topology and this greatly enlarges the
physical setups in which topological phases could emerge. Along with the
particle-hole and chiral symmetries, the time-reversal symmetry represents one
of the three internal symmetries relevant for the classification of
topological phases, and general classification results were soon obtained
under the ten-fold way Kitaev (2009); Chiu  _et al._ (2016). The
classification was then further refined in the presence of symmorphic Fu
(2011); Hsieh  _et al._ (2012); Tanaka  _et al._ (2012); Dziawa  _et al._
(2012); Ando and Fu (2015) and nonsymmorphic Fang and Fu (2015); Liu  _et al._
(2014); Shiozaki  _et al._ (2015); Parameswaran  _et al._ (2013); Varjas  _et
al._ (2015); Shiozaki  _et al._ (2016); Liang  _et al._ (2017); Chang  _et
al._ (2017); Ma  _et al._ (2017); Schoop  _et al._ (2016, 2018); Wieder  _et
al._ (2016, 2018); Wang  _et al._ (2016); Armitage  _et al._ (2018) spatial
symmetries. Such successive extension of the symmetry setting has led to a
comprehensive understanding of the diverse set of phases protected by the 230
spatial symmetry groups Kruthoff  _et al._ (2017); Po  _et al._ (2017);
Bradlyn  _et al._ (2017); Zhang  _et al._ (2019); Vergniory  _et al._ (2019);
Tang  _et al._ (2019), and the results were further extended to magnetic
materials Watanabe  _et al._ (2018); Elcoro  _et al._ (2021); Xu  _et al._
(2020) in which time-reversal can also combine nontrivially with partial
translation into a symmetry of the magnetic order.

Exhaustive as it may seem, the systematic treatment of (magnetic) spatial
symmetries has thus far focused on symmetries that are relevant to electronic
materials. Engineered physical platforms Aidelsburger  _et al._ (2018), like
cold-atomic, photonic, and acoustic systems, could inherently feature
synthetic symmetries that would have been unnatural or fine-tuned for
electronic problems El-Ganainy  _et al._ (2018); Peng  _et al._ (2016); Miri
_et al._ (2013); Zhao  _et al._ (2020); Li  _et al._ (2014); Xue  _et al._
(2022); Arkinstall  _et al._ (2017); Kremer  _et al._ (2020). Here we show
that a non-Abelian Hofstadter model with ${\rm SU}(2)$ gauge fields,
potentially realizable in engineered systems, calls for a further extension of
symmetry analysis. A key new ingredient is the coexistence of multiple
nonsymmorphic chiral symmetries, which combine site-dependent, local phase
factors with fractional translation. As the nonsymmorphic chiral symmetry is
not generally respected in electronic materials, it is not included as part of
the comprehensive (magnetic) space-group symmetry analysis on electronic
topological band theory. Yet its presence has been recognized in the study of
specific models, including a minimal two-band model for nonsymmorphic
topological crystalline insulators Shiozaki  _et al._ (2015), certain
antiferromagnetic semimetals Brzezicki and Cuoco (2017), the low-energy states
in the SnTe material class Brzezicki  _et al._ (2019), and, as a theoretical
construction via the square-root operation from parent Hamiltonians Arkinstall
_et al._ (2017). So far, efforts have been mostly dedicated to systems that
obey a single nonsymmorphic chiral symmetry.

In this work, we show multiple coexisting nonsymmorphic chiral symmetries
could be non-Abelian, lead to intriguing symmetry algebras, and, consequently,
protect unusual band degeneracy and topology, as illustrated in the non-
Abelian Hofstadter model. More concretely, we analyze the associated algebras
of the multiple nonsymmorphic chiral symmetries and reveal their dependence on
the parity of the two non-Abelian gauge fields that are assumed rational. In
particular, when both of the rational gauge fields have even denominators, and
one and only one of the denominators is an integer multiple of four, their
algebra becomes non-Abelian and gives rise to generic Kramer quartets, i.e.
four-fold degeneracy at all momenta, which are jointly protected by inversion
and time-reversal. The nonsymmorphic chiral symmetries turn the magnetic
Brillouin torus into a real projective plane, and therein protect double Dirac
semimetals at half filling. For relatively small systems relevant to
engineered physical platforms, we further show that the semimetals get gapped
and become an insulator upon the introduction of a boundary, which necessarily
breaks some of the nonsymmorphic chiral symmetries. The resulting insulator
can be tuned to be either trivial or topological, depending on the parity of
the system size.

Figure 1:  Non-Abelian Hofstadter model (a) and its inhomogeneous real-space
magnetic fields (b-d). Zero, one, and two nonsymmorphic chiral symmetries
involving a half-lattice translation appear in b, c and d, respectively, as
evident from the patterns of the magnetic fields. Four magnetic unit cells
(divided by bold black lines) are shown. The real space magnetic fields are
characterized by the gauge-invariant, real-space Wilson loop
$W=\mathrm{Tr}\exp\left(\mathrm{i}\oint\boldsymbol{A}\mathop{}\mathrm{d}{l}\right)$
for each unit plaquette.

$\mathbf{U}\subset\mathbf{G}$ | $q_{x}\in 2\mathbb{Z}+1$ | $q_{x}\in 4\mathbb{Z}+2$ | $q_{x}\in 4\mathbb{Z}$
---|---|---|---
$q_{y}\in 2\mathbb{Z}+1$ |  | $\mathbb{Z}_{1}\subset\mathbb{Z}_{2}$ | $\mathbb{Z}_{1}\subset\mathbb{Z}_{2}$
$q_{y}\in 4\mathbb{Z}+2$ | $\mathbb{Z}_{1}\subset\mathbb{Z}_{2}$ | $\pagecolor{gray!30}\mathbb{Z}_{4}\times\mathbb{Z}_{2}\subset\leavevmode\color[rgb]{0,0,1}\mathbb{D}_{8}\times\mathbb{Z}_{2}$ | $\pagecolor{gray!30}\leavevmode\color[rgb]{0,0,1}\mathbb{D}_{8}\subset\leavevmode\color[rgb]{0,0,1}\mathbb{D}_{8}\times\mathbb{Z}_{2}$
$q_{y}\in 4\mathbb{Z}$ | $\mathbb{Z}_{1}\subset\mathbb{Z}_{2}$ | $\pagecolor{gray!30}\leavevmode\color[rgb]{0,0,1}\mathbb{D}_{8}\subset\leavevmode\color[rgb]{0,0,1}\mathbb{D}_{8}\times\mathbb{Z}_{2}$ | $\pagecolor{gray!30}\mathbb{K}_{4}\subset\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$

$H$ | $q_{x}\in 2\mathbb{Z}+1$ | $q_{x}\in 4\mathbb{Z}+2$ | $q_{x}\in 4\mathbb{Z}$
---|---|---|---
$q_{y}\in 2\mathbb{Z}+1$ | AII | DIII | CII
$q_{y}\in 4\mathbb{Z}+2$ | DIII | $\pagecolor{gray!30}\mathrm{AII}\oplus\mathrm{AII}\oplus\mathrm{AII}\oplus\mathrm{AII}$ | $\pagecolor{gray!30}\mathrm{CII}\oplus\mathrm{CII}$
$q_{y}\in 4\mathbb{Z}$ | CII | $\mathrm{CII}\oplus\mathrm{CII}$ | $\pagecolor{gray!30}\mathrm{CII}\oplus\mathrm{CII}\oplus\mathrm{CII}\oplus\mathrm{CII}$

Table 1:  Algebra of chiral symmetries and the resulting classification of the
non-Abelian Hofstadter problem in the symmetric gauge. $\mathbf{U}$ (left), a
subgroup of $\mathbf{G}$, contains only unitary symmetries generated by the
chiral symmetries $\mathbf{S}$ and results in different classifications of the
systems (right). Shaded entries correspond to cases where multiple
nonsymmorphic chiral symmetries appear. Blue color indicates non-Abelian
groups.

_Non-Abelian Hofstadter problem._ A less-known fact about the well-known
Abelian Hofstadter model, featuring U(1) gauge fields, is that it obeys chiral
symmetries that are nonsymmorphic (see Sec. LABEL:sec:abelian). Nevertheless,
its nonsymmorphic chiral operators can always be transformed into a local
basis and their algebra is always trivial (see Sec. LABEL:sec:abelian), which
relates to the equivalence between its formulations in the Landau and the
symmetric gauges. However, these conditions can get modified in non-Abelian
Hofstadter models Yang  _et al._ (2020). Let us consider a Hofstader–Harper-
like, SU(2) gauge fields

$\displaystyle\boldsymbol{A}=(-y\phi_{x}\sigma_{x},x\phi_{y}\sigma_{y},0).$
(1)

The associated Hamiltonian $H(\phi_{x}/2\pi,\phi_{y}/2\pi)$ on a square
lattice is given by

$\displaystyle
H=-\sum_{x,y}t_{x}c^{\dagger}_{x+1,y}\mathrm{e}^{-\mathrm{i}y\phi_{x}\sigma_{x}}c_{x,y}+t_{y}c^{\dagger}_{x,y+1}\mathrm{e}^{\mathrm{i}x\phi_{y}\sigma_{y}}c_{x,y}+\text{H.c.}$
(2)

Here $t_{x}$ and $t_{y}$ are the real hopping terms in the $x$ and $y$
directions and we restrict ourselves to $t_{x}=t_{y}=t$. $c_{x,y}$ and
$c^{\dagger}_{x,y}$ are the annihilation and creation operators at site
$(x,y)$. $\phi_{x}$ and $\phi_{y}$ are the SU(2) Peierls phases. The
Hamiltonian has a spin-rotation symmetry and stays invariant for arbitrary
choices of distinct Pauli matrices in Eq. (1). If $\phi_{x}$ and $\phi_{y}$
are both rational, i.e. they can be written as $\phi_{x}=2\pi p_{x}/q_{x}$ and
$\phi_{y}=2\pi p_{y}/q_{y}$ for integers $p_{x}$, $q_{x}$, $p_{y}$, and
$q_{y}$. The Hamiltonian can be solved in the $q_{y}\times q_{x}$ super-cell
with the magnetic Brillouin zone (MBZ) defined as $k_{x}\in[0,2\pi/q_{y})$ and
$k_{y}\in[0,2\pi/q_{x})$.

This model is reminiscent of but distinct from the symmetric-gauge, Abelian
Hofstadter problem. Evidently, $H$ reduces to two decoupled Abelian
counterparts with opposite homogeneous magnetic fluxes when either $\phi_{x}$
or $\phi_{y}$ vanishes. In contrast, under gauge fields that meet the genuine
non-Abelian condition Yang  _et al._ (2020), the associated magnetic fields
become spatially inhomogeneous, as characterized by the real-space Wilson loop
$W=\mathrm{Tr}\exp\left(\mathrm{i}\oint\boldsymbol{A}\mathop{}\mathrm{d}{l}\right)$
(see Fig. 1).

_Symmetry algebra._ For $\left\\{\mu,\nu\right\\}=\left\\{x,y\right\\}$, a
chiral symmetry $S_{\mu}$ appears when $q_{\nu}\in 2\mathbb{Z}$. Its explicit
form is given by

$\displaystyle\bra{\mathbf{r}}S_{\mu}(\mathbf{k})\ket{\mathbf{r^{\prime}}}$
$\displaystyle=(-1)^{\mu}(\mathrm{i})^{q_{\nu}/2}\exp(\mathrm{i}k_{\mu}q_{\nu}/2)\sigma_{0}\delta_{\mu+q_{\nu}/2,\mu^{\prime}}\delta_{\nu,\nu^{\prime}},$
(3)

where $\mathbf{r}=(x,y)$ and $\sigma_{0}$ operates on the spin degree of
freedom. This chiral operator contains site-dependent phase factors and a half
translation along a single dimension in the magnetic unit cell (see Fig. 2a
and b), satisfies $S_{\mu}^{2}(\mathbf{k})=\exp(\mathrm{i}k_{\mu}q_{\nu})$
that restores a full translation, and thus is an order-two nonsymmorphic
symmetry with $d_{\parallel}=0$ Shiozaki  _et al._ (2016) for the associated
Bloch Hamiltonian.

If $\left(q_{x}\in 2\mathbb{Z},q_{y}\in 2\mathbb{Z}\right)$, multiple chiral
symmetries appear. First, two nonsymmorphic chiral operators $S_{x}$ and
$S_{y}$ emerge because we can apply Eq. (3) to both $x$ and $y$ directions
(see Fig. 2a and b). It is noted that they have no counterparts in the Landau-
gauge non-Abelian Hofstadter model Yang  _et al._ (2020), where gauge fields
are arranged along a single spatial dimension. Second, because the magnetic
unit cell becomes bipartite under the same condition, the conventional local
chiral symmetry $S_{0}$ exists:

$\displaystyle\bra{\mathbf{r}}S_{0}\ket{\mathbf{r^{\prime}}}=(-1)^{x+y}\delta_{x,x^{\prime}}\delta_{y,y^{\prime}}\sigma_{0}.$
(4)

After we quotient away the translational part in $S_{x}$ and $S_{y}$,
$\mathbf{S}\equiv\left\\{S_{0},S_{x},S_{y}\right\\}$ becomes the generator of
a finite unitary group $\mathbf{G}$. There always exists a proper index-2
subgroup $\mathbf{U}\subset\mathbf{G}$ such that $\mathbf{U}$ only contains
unitary symmetries of the hamiltonian, i.e. $\mathbf{U}\equiv S_{\mu}S_{\nu}$.
$\mathbf{U}$ contains symmetries that are reminiscent of the projective
translational symmetry Zhao  _et al._ (2020), which are shown to protect
doubly-degenerate bands and Dirac points at single momenta Zhao  _et al._
(2020), as recently demonstrated in acoustic lattices Li  _et al._ (2022); Xue
_et al._ (2022).

The parity of the gauge field, in particular the denominators, strongly
affects the symmetry algebra and thereby the classification of this non-
Abelian model (see Table 1), even in the continuum limit with weak fields
(i.e. $\phi_{x},\phi_{y}\to 0$). If $\left(q_{x}\in 2\mathbb{Z}+1,q_{y}\in
2\mathbb{Z}+1\right)$, no other internal symmetry beside time reversal
$T_{0}=\mathrm{i}\sigma_{y}K$ exists, $\mathbf{U}$ is empty, and $H$ belongs
to class AII in Table 1. If
$\left(q_{\mu}=2\mathbb{Z}+1,q_{\nu}=2\mathbb{Z}\right)$, a nonsymmorphic
chiral symmetry $S_{\mu}$ exists. Although $T_{0}^{2}=-1$ and $S_{\mu}^{2}=1$
are ensured, the square of their associated nonsymmorphic particle-hole
symmetry $C_{\mu}=T_{0}^{-1}S_{\mu}$ is uncertain—$C_{\mu}^{2}=1$ if
$q_{\nu}=4\mathbb{Z}+2$ and $C_{\mu}^{2}=-1$ if $q_{\nu}=4\mathbb{Z}$, which
corresponds to class DIII and CII in Table 1, respectively. For both of the
DIII and CII classes, $\mathbf{G}=\\{S_{\mu},1\\}$ is $\mathbb{Z}_{2}$ and
$\mathbf{U}=\\{1\\}$ is the trivial group $\mathbb{Z}_{1}$.

Richer symmetry algebra appears for $\left(q_{x}\in 2\mathbb{Z},q_{y}\in
2\mathbb{Z}\right)$. In this case, $T_{0}$ and $S_{0}$ enable a local
particle-hole symmetry $C_{0}=T_{0}^{-1}S_{0}$ that satisfies $C_{0}^{2}=-1$.
Although $\commutator{S_{x}}{S_{y}}=0$ is ensured, $S_{0}$ and $S_{\mu}$ may
commute or anticommute. Specifically, $\commutator{S_{0}}{S_{\mu}}=0$ when
$q_{\nu}=4\mathbb{Z}$ and $\anticommutator{S_{0}}{S_{\mu}}=0$ when
$q_{\nu}=4\mathbb{Z}+2$. There are three resulting scenarios. First, when
$\left(q_{x}\in 4\mathbb{Z},q_{y}\in 4\mathbb{Z}\right)$, $\mathbf{S}$ is an
Abelian generator. $\mathbf{G}$ is an elementary Abelian group
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ and
$\mathbf{U}=\left\\{1,S_{0}S_{x},S_{0}S_{y},S_{x}S_{y}\right\\}$ is the Klein
four-group $\mathbb{K}_{4}$. $H$ is consequently diagosed as
$\mathrm{CII}\oplus\mathrm{CII}\oplus\mathrm{CII}\oplus\mathrm{CII}$, with
each subspace chiral symmetric. Second, when $\left(q_{x}\in
4\mathbb{Z}+2,q_{y}\in 4\mathbb{Z}+2\right)$,
$\mathbf{G}=\mathbb{D}_{8}\times\mathbb{Z}_{2}$ becomes non-Abelian. However,
its subgroup $\mathbf{U}=\mathbb{Z}_{4}\times\mathbb{Z}_{2}$ is still Abelian,
which diagnoses $H$ as
$\mathrm{AII}\oplus\mathrm{AII}\oplus\mathrm{AII}\oplus\mathrm{AII}$, forming
two pairs of chiral partners. Third, when $\left(q_{\mu}\in
4\mathbb{Z}+2,q_{\nu}\in 4\mathbb{Z}\right)$,
$\mathbf{G}=\mathbb{D}_{8}\times\mathbb{Z}_{2}$ is the same as that in the
previous scenario. However, they differ in their detailed symmetry algebra
(Sec. LABEL:sec:sym_diff); accordingly, the only non-Abelian subgroup of
$\mathbf{U}=\mathbb{D}_{8}$ appears in Table 1, which leads to a
classification of $\mathrm{CII}\oplus\mathrm{CII}$ and the appearance of the
Kramer quartet states, as we describe below.

Figure 2:  Nonsymmorphic chiral symmetries and wavefunctions of Kramer
quartets. Nonsymmorphic chiral symmetries [a and b; see Eq. (3)] appear as the
product between half translations and site-dependent phase factors
(represented by light and dark shadings for $\pm\mathrm{i}$ and $\pm 1$ in a
and b, respectively). When $q_{x}=4\mathbb{Z}$ and $q_{y}=4\mathbb{Z}+2$, two
Kramer doublets (c and d) can be constructed by occupying each bipartite
sublattices respectively with $\alpha^{\pm}$ [ red and blue dots label the
$\pm$ superscript; see Eq. (9)] or vice versa. The non-Abelian group
$\mathbb{D}_{8}$ protects the degeneracy between the two Kramer doublets and
thus enables a Kramer quartet, i.e. a four-fold degeneracy at all momenta and
all energies. Here, $H\left(1/4,1/6\right)$ is illustrated as an example.
Figure 3:  Double Dirac points on a real projective plane protected by chiral
symmetries. a. Bulk bands and the four subspaces (b-e) of $H(1/4,1/4)$ in the
MBZ. All subspaces are aperiodic in the original MBZ, which becomes a real
projective plane due to the nonsymmorphic symmetries along both $k_{x}$ and
$k_{y}$ directions. The double Dirac point at $\Gamma$ of $H$ is ‘shared’ by
its four subspaces at $\Gamma$, $2X$, $2Y$, and $2M$.

_Kramer quartets._ A consequence of the non-Abelian symmetry algebra is the
existence of generic Kramer quartet states, i.e. a global four-fold degeneracy
in the entire MBZ for the case of $\left(q_{\mu}\in 4\mathbb{Z}+2,q_{\nu}\in
4\mathbb{Z}\right)$. Here, time reversal (can be nonsymmorphic, as constructed
from the nonsymmorphic chiral symmetries) and inversion protect the
conventional Kramer doublets, while the non-Abelian chiral symmetry group
ensures that they are degenerate Mobius partners. Without loss of generality,
we can decompose $H$ into two CII subspaces, $H_{\mathrm{a}}$ and
$H_{\mathrm{b}}$, by diagonalizing the unitary symmetry $U_{xy}=S_{x}S_{y}$
such that both $H_{\mathrm{a}}$ and $H_{\mathrm{b}}$ also obey inversion and a
nonsymmorphic time-reversal symmetry $T_{\mu}\equiv S_{\mu}C_{0}^{-1}$ (see
Sec. LABEL:sec:kramer), meaning that spins at one site need to perform
fractional translation to find their time-reversal partners. Therefore, both
$H_{\mathrm{a}}$ and $H_{\mathrm{b}}$ are Kramer doublets.

Mathematically, the anticommutation relation in the non-Abelian group
$\mathbf{U}$ enforces the degeneracy between the two CII subsystems
$H_{\mathrm{a}}$ and $H_{\mathrm{b}}$ Zhao  _et al._ (2020). Physically, we
explicitly construct their wavefunctions by considering the transformation:

$\displaystyle\left(\begin{array}[]{c}\alpha^{+}_{mn}\\\
\alpha^{-}_{mn}\end{array}\right)=\exp(\mathrm{i}\tau_{x}\pi/4)\left(\begin{array}[]{c}u_{mn}\\\
u_{m+q_{y}/2,n+q_{x}/2}\end{array}\right),$ (9)

where $u$ is the wavefunction in the original basis, $\tau_{x}$ acts on the
lattice sites, and we drop the spin index $\sigma$ since the transformation
does not act on the spin. $\alpha^{\pm}_{mn}$ are eigenstates of $U_{xy}$ with
eigenvalues $\pm 1$ because
$U_{xy}\alpha^{\pm}_{mn}=(-1)^{m+n}\mathrm{i}^{(q_{x}+q_{y})/2\mp
1}\alpha^{\pm}_{mn}.$ Since $(q_{x}+q_{y})/2=2\mathbb{Z}+1$, the eigenstate of
$H_{\mathrm{a}}$ ($H_{\mathrm{b}}$) occupies each bipartite sublattice with
$\alpha^{\pm}$ ($\alpha^{\mp}$), respectively (see Fig. 2c and d). The two
subspaces must be similar because $U_{xy}$ consists of two half translation
operations, each along $x$ and $y$ directions, which map between the two
sublattices (see Fig. 2a and b). Taken together, the non-Abelian subgroup
$\mathbf{U}$, inversion, and nonsymmorphic time-reversal symmetry jointly
protect the generic Kramer quartet states at arbitrary momenta.

Figure 4:  Trivial and topological quantum-confined phases dictated by the
system parity. a. Projected bulk bands of $H(1/4,1/6)$ near half-filling. b-g.
quantum-confined bandgaps (shaded grey) and edge states (red and blue, for
chiral and anti-chiral subspaces, respectively) appear upon introducing a
boundary that breaks a nonsymmorphic chiral symmetry $S_{y}$, which protects
the bulk double Dirac points at $\Gamma$ and $X$. The quantum-confined
insulating phase can be trivial ($\mathbb{Z}_{2}$-even; b-d) and topological
($\mathbb{Z}_{2}$-odd; e-g) in both the full manifold and its chiral and anti-
chiral subspaces when the open $y$ direction of the cylinder geometry contains
an even or odd number of unit cells. Shaded yellow and green indicates half
MBZ for the full manifold and the two subspaces.

_Quantum confinement effects on the real projective plane._ When
$\left(q_{x}\in 2\mathbb{Z},q_{y}\in 2\mathbb{Z}\right)$ (shaded entries in
Table 1), the multiple NCS symmetries $\mathbf{S}$ render the MBZ of the full
system a real projective plane, i.e. a 2D generalization of the non-orientable
Mobius strip. For the subspaces of the system, there are two sets of momentum
labels
$\left[\pm\exp\left(\mathrm{i}k_{x}q_{y}/2\right),\pm\exp\left(\mathrm{i}k_{y}q_{x}/2\right)\right]$,
independent of each other (except for the the Kramer-quartet case, as proved
earlier). As a result, the subspaces are defined on an enlarged MBZ (doubled
size in both dimensions) and they exchange these momentum labels at the
original MBZ boundary of the full system.

The chiral symmetries $\mathbf{S}$ are also essential for stabilizing the
degeneracies at half filling. It is established that the existence of a chiral
and a spatial symmetry leads to a lower bound $N_{S}(\mathbf{k})$ on the
number of chiral zero modes at momentum $\mathbf{k}$ Koshino  _et al._ (2014)
(also see Sec. LABEL:sec:index). In our case, the spatial symmetry is
inversion, which always commutes with all chiral symmetries (see Sec.
LABEL:sec:kramer). In the presence of multiple chiral symmetries, the lower
bound can be calculated for each of them, i.e.
$\boldsymbol{N_{S}}(\mathbf{k})\equiv\left(N_{S_{0}}(\mathbf{k}),N_{S_{x}}(\mathbf{k}),N_{S_{y}}(\mathbf{k})\right)$.
As we will show below, $\boldsymbol{N_{S}}(\mathbf{k})$ depends on the choice
of gauge fields.

In the case of $H(1/4,1/4)$ (Fig. 3), the full system can be decomposed into
four chiral-symmetric subspaces that display pairwise Mobius-type band
connections along both the $x$ and $y$ directions at the zone boundary.
Notably, its double Dirac point (eight-fold degenerate; Fig. 3a) at $\Gamma$
is, in fact, shared by its four subspaces at the four quadrants. Each of them
hosts a double Dirac point at $\Gamma$, $2X$, $2Y$, and $2M$ (Fig. 3b-e and
also see Fig. LABEL:fig:1414), respectively. The chiral zero mode indices at
the $\Gamma$ point are $\boldsymbol{N_{S}}(\Gamma)=\left(8,8,8\right)$, which
indicates that inversion, and both of the local and the nonsymmorphic chiral
symmetries jointly protect the double Dirac points at $\Gamma$. This
protection is confirmed by an $x$-periodic inversion-symmetric cylinder
calculation in Fig. LABEL:smfig:edge1414, which breaks $S_{y}$ but preserves
$S_{0}$ and $S_{x}$. Therein, the double Dirac point in the bulk bands reduces
to a single Dirac point in the edge spectra at the cost of the reduced
symmetries.

There are situations when components of $\boldsymbol{N_{S}}$ are not
identical, which cause consequences of quantum confinement. We again use
$H(1/4,1/6)$ as an example. Its chiral zero mode indices are
$\boldsymbol{N_{S}}(\mathbf{k})=(0,0,8)$, which indicates $S_{y}$ being
crucial in stabilizing the double Dirac points in the bulk bands (Fig. 4a).
Thus, an inversion-symmetric cylinder geometry, which is open in $x$ but
periodic in $y$, still preserves $S_{y}$ and remains gapless (an example shown
in Fig. LABEL:smfig:edge1416). In contrast, a cylinder geometry open in $y$,
even if inversion-symmetric, can be gapped because the nonsymmorphic chiral
symmetry $S_{y}$ is unavoidably broken. This is confirmed in Fig. 4 with the
appearance of a band gap (shaded gray) and edge states (red and blue).
Introducing the boundary also lifts the degeneracy of the Kramer quartets
because the violation of $S_{y}$ alters the symmetry algebra. Specifically,
$\mathbf{U}\subset\mathbf{G}$ is modified as
$\mathbb{Z}_{4}\subset\mathbb{D}_{8}$ for this edge Hamiltonian (cf.
$\mathbb{D}_{8}\subset\mathbb{D}_{8}\times\mathbb{Z}_{2}$ for the bulk in
Table 1). Evidently, bulk boundary correspondence is violated, a typical
feature of topological crystalline phases but in this case originating from
chiral symmetries: the bulk phase in Fig. 4a is a double Dirac semimetal
protected by inversion and nonsymmorphic chiral symmetries, while an edge that
violates $S_{y}$ renders it a quantum-confined insulator with a vanishingly
small band gap in the thermodynamic limit.

Moreover, this quantum-confined, insulating phase can be either trivial or
topological, depending on the parity of the number of unit cells in the
cylinder geometry, as illustrated in Fig. 4b-g. Because the cylinder still
respects $S_{0}$ and $S_{x}$, we can still decompose the full manifold into
two $4X$-periodic, chiral and anti-chiral subspaces that are Mobius partners
to each other. Although chiral symmetry does not hold for each subspace, there
remains a sublattice symmetry that maps $E(k_{x})\to-E(k_{x}+2X)$, typical in
Hofstadter problems. For an even (odd) number of unit cells in the open
direction, each subspace contains an even (odd) number of Kramer partners.
Taken together, there must be an even (odd) number of crossings at zero energy
for each subspace. Therefore, the entire manifold and its two subspaces are
simultaneously $\mathbb{Z}_{2}$-even or $\mathbb{Z}_{2}$-odd depending on the
parity of the unit cells along the open boundary direction. For other gapped
phases at non-half fillings, there is no such dependence. Instead,
$\mathbb{Z}_{2}$ time-reversal-invariant insulating phases with a Mobius
structure can appear for the edge spectra (Fig. LABEL:fig:gapped1416), where
chiral-partnered edge states exchange their nonsymmorphic symmetry labels at
the original MBZ boundary.

_Conclusion._ We have shown that multiple nonsymmorphic chiral symmetries
naturally emerge in a non-Abelian generalization of the Hofstadter model with
${\rm SU}(2)$ gauge fields. The nontrivial commutation relations between these
symmetries lead to several topological consequences: the Kramer quartet
states, semimetals at half filling, and quantum-confined insulators whose
topology depends on the system parity. Richer non-Abelian algebras and
topology are expected in higher dimensions, e.g. the three-dimensional non-
Abelian Hofstadter model where each dimension hosts a unique nonsymmorphic
chiral symmetry Liu  _et al._ (2021) (see Sec. LABEL:sec:3d). In particular,
our results showcase the diverse possibilities in which internal symmetries,
classified according to the ten-fold way, could be combined with spatial
symmetries in physical systems beyond electronic band theory. A particularly
interesting question concerns whether similar effective symmetries could be
relevant to the parton description of spin liquid candidates with an emergent
${\rm SU}(2)$ gauge field.

The proposed non-Abelian Hofstadter system could be simulated with photons and
cold atoms. In optics, relevant candidate platforms are anisotropic or
bianisotropic materials with electromagnetic duality Liu and Li (2015); He
_et al._ (2016); Cheng  _et al._ (2016); Chen  _et al._ (2019), and
$\mathcal{PTD}$-symmetric systems Silveirinha (2017). In cold atoms, the two
spatially-dependent gauge potentials could be realized by existing
methods—such as laser-assisted tunneling Aidelsburger  _et al._ (2013); Miyake
_et al._ (2013), lattice shading Hauke  _et al._ (2012), or magnetic wires
with spatially-modulated currents Goldman  _et al._ (2010); Anderson  _et al._
(2013)—along the two spatial dimensions.

## Acknowledgments

This material is supported in part by the Air Force Office of Scientific
Research under the awards number FA9550-20- 1-0115, and FA9550-21-1-0299, as
well as in part by the US Office of Naval Research (ONR) Multidisciplinary
University Research Initiative (MURI) grant N00014-20-1-2325 on Robust
Photonic Materials with High- Order Topological Protection. This material is
also based upon work supported in part by the U. S. Army Research Office
through the Institute for Soldier Nanotechnologies at MIT, under Collaborative
Agreement Number W911NF-18-2-0048. Y. Y. thanks the support from the start-up
fund of the University of Hong Kong and the National Natural Science
Foundation of China Excellent Young Scientists Fund (HKU 12222417).

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# A short ODE proof of the Fundamental Theorem of Algebra

Ramona Anton1, Nicolae Mihalache2 and François Vigneron3

###### Abstract

We propose a short proof of the Fundamental Theorem of Algebra based on the
ODE that describes the Newton flow and the fact that the value $|P(z)|$ is a
Lyapunov function. It clarifies an idea that goes back to Cauchy.

The Fundamental Theorem of Algebra is one of the major results of mathematics;
historical reviews can be found e.g. in [7], [2, Chap. II] or [6]. In
practice, the roots of polynomials are computed with a variety of different
algorithms (see e.g. [5] for an exhaustive review) and, notably, with Newton-
Raphson’s method [1]. It is well known that the different basins of attraction
of the iterative Newton method are deeply intricate. The existence of
universal sets of starting points such that the closure of all Newton’s
iterates will always contain all roots is remarkable though highly non-trivial
(see Hubbard-Schleicher-Sutherland [4]). On the other hand, the flow of
Newton’s ODE presented below in (1) is topologically much simpler and suitable
for a short constructive proof of the fundamental Theorem of Algebra that we
present in this note.

This idea can be found, at a higher level of generality, in Hirsch-Smale [3].
However, the proof below is of great practical importance because, in addition
to the general considerations of [3], we are able to control very explicitly
the speed of convergence of each Newton trajectory. For example, (2) implies
that, away from roots and critical points, each step of Newton’s method
approximately divides the value of the polynomial by $e$ (see Figure 1), which
is a property that, to the best of our knowledge, had not been noticed before.
In [5] we use this idea to split a non-trivial polynomial of record degree
$10^{12}$. We believe it is important to highlight it.

Given a polynomial $P\in\mathbb{C}[z]$ of degree $d\geq 1$, we denote by
$\mathcal{Z}(P)=P^{-1}(0)$ the set of its zeros and by
$\operatorname{Crit}(P)=\mathcal{Z}(P^{\prime})$ the set of its critical
points, a finite set (by Euclidean division). Let also
$\mathcal{V}(P)=P(\operatorname{Crit}(P))$ the set of critical values of $P$.

The Newton flow of $P$ is defined by the ODE

$\varphi_{z}(0)=z\quad\text{and}\quad\varphi_{z}^{\prime}(t)=-\frac{P(\varphi_{z}(t))}{P^{\prime}(\varphi_{z}(t))}\cdotp$
(1)

The set $\mathcal{Z}(P)$ is composed of stationary solutions of (1). Thanks to
the Cauchy-Lipschitz theorem, the Newton flow is locally well defined starting
from any point $z\notin\operatorname{Crit}(P)$ and can be extended locally as
long as $\varphi_{z}(t)\notin\operatorname{Crit}(P)$. For
$z\notin\operatorname{Crit}(P)$, let us denote by $T(z)\in(0,+\infty]$ the
maximal forward time of existence of $\varphi_{z}$. One has

$\forall z\notin\operatorname{Crit}(P),\quad\forall t\in[0,T(z))\qquad
P(\varphi_{z}(t))=e^{-t}P(z)$ (2)

because Leibniz’s rule and (1) imply

$\frac{d}{dt}\left[e^{t}P(\varphi_{z}(t))\right]=e^{t}P(\varphi_{z}(t))+e^{t}P^{\prime}(\varphi_{z}(t))\varphi_{z}^{\prime}(t)=0.$

In particular, from (2) we see that Newton’s flow is iso-angle i.e. $\arg
P(\varphi_{z}(t))=\arg P(z)$ if $P(z)\neq 0$. Moreover, for $t\in(0,T(z))$,
$\varphi_{z}(t)\in\mathcal{B}(z)=\\{y\in\mathbb{C}\,;\,|P(y)|\leq|P(z)|\\}$,
which is a compact set because $\lim\limits_{|z|\to\infty}|P(z)|=+\infty$.

######  Theorem 1.

Every non-constant polynomial $P\in\mathbb{C}[z]$ has at least one root in
$\mathbb{C}$.

###### Proof.

Our strategy to find a root of $P$ is to choose a starting point $z_{0}$ with
$T(z_{0})=+\infty$ and then pass to the limit in (2). We will see that it is
enough that $\arg(P(z_{0}))$ avoids the finite set $\arg(\mathcal{V}(P))$.
Observe that $0\notin\mathcal{V}(P)$, otherwise $P$ has a root in
$\operatorname{Crit}(P)$ and we stop here.

If $T(z)<\infty$ for some $z\in\mathbb{C}\setminus\operatorname{Crit}(P)$,
then $P(z)\neq 0$. As $t\to T(z)$, the orbit $\varphi_{z}(t)$ leaves any
compact subset of $\mathbb{C}\backslash\operatorname{Crit}(P)$. As
$\varphi_{z}(t)\in\mathcal{B}(z)$, a compact set, there is a sequence
$t_{n}\to T(z)$ such that $\varphi_{z}(t_{n})$ converges to some point
$c\in\operatorname{Crit}(P)$. The continuity of $P$ at $c$ implies that $\arg
P(z)=\arg P(\varphi_{z}(t_{n}))=\arg(P(c))\in\arg(\mathcal{V}(P))$.

Let us find $z_{0}$ with $\arg P(z_{0})\notin\arg(\mathcal{V}(P))$. As
$\lim\limits_{|z|\rightarrow\infty}z^{-d}P(z)=\alpha\in\mathbb{C}^{*}$, taking
$z_{0}=re^{i\vartheta}$ with
$d\vartheta+\arg\alpha\notin\arg(\mathcal{V}(P))\mod 2\pi$ and $r>0$ large
enough ensures that $\arg(P(z_{0}))\notin\arg(\mathcal{V}(P))$. Therefore
$T(z_{0})=+\infty$ and the bounded sequence
$\left(\varphi_{z_{0}}(n)\right)_{n\in\mathbb{N}}\subset\mathcal{B}(z_{0})$
admits an accumulation point $z_{\ast}$. The continuity of $P$ at $z_{\ast}$
and (2) imply that $P(z_{\ast})=\lim\limits_{n\to\infty}e^{-n}P(z_{0})=0$. ∎

## Acknowledgement.

The second author would like to thank Dierk Schleicher for several insightful
discussions about the Newton method and the Newton flow.

## References

  * [1] A. Galantai, The theory of Newton’s method. J. of Comp. App. Math. 124 (2000), 25-44.
  * [2] J. Guérindon and J. Dieudonné, L’Algèbre et la géométrie jusqu’en 1840 in Abrégé d’histoire des Mathématiques. Hermann, 1986.
  * [3] M.-W. Hirsch, S. Smale, On algorithms for solving $f(x)=0$. Communications in Pure and Applied Mathematics 32, 3 (1979), 281-312.
  * [4] J.H. Hubbard, D. Schleicher and S. Sutherland, How to find all roots of complex polynomials by Newton’s method. Invent. math., 146 (2001), 1-33.
  * [5] N. Mihalache, F. Vigneron. How to split a tera-polynomial. In preparation.
  * [6] R. Remmert, The Fundamental Theorem of Algebra in Graduate Texts in Mathematics 123, Numbers. Springer, 1991.
  * [7] S. Smale, The fundamental Theorem of Algebra and complexity theory. Bulletin of the American Mathematical Society 4-1 (1981), 1-36.

Figure 1:  Polynomial of degree 5, with roots (red) and critical points
(black). The level lines $p(z)=2^{n}$ with $n\in\mathbb{Z}$ and iso-angles
(solution of Newton’s flow $z^{\prime}=-p(z)/p^{\prime}(z)$ starting from the
points $z_{0}$ such that $p(z_{0})\in 2^{n_{0}}\\{\pm 1,\pm i\\}$ for some
$n_{0}\in\mathbb{Z}$) are indicated (gray). The Newton’s steps (white) follow
approximately Newton’s flow, away from the roots and the critical points.

1 Ramona Anton.
Sorbonne Université, IMJ-PRG, CNRS UMR 7586, F-75252 Paris. x
<EMAIL_ADDRESS>
2 Nicolae Mihalache.
Université Paris-Est Creteil, CNRS UMR 8050, LAMA, F-94010 Creteil, France and
Université Gustave Eiffel, LAMA, F-77447 Marne-la-Vallée. x
<EMAIL_ADDRESS>
3 François Vigneron.
Université de Reims Champagne-Ardenne, Laboratoire de Mathématiques de Reims,
UMR 9008 CNRS, Moulin de la Housse, BP 1039, F-51687 Reims. x
<EMAIL_ADDRESS>


# Two families of Dirac-like operators for Drinfeld’s Hecke algebra

Kieran Calvert

(July 2022)

###### Abstract

In this paper we define two generalisations of Dirac operators for Drinfeld’s
Hecke algebra. One generalisation, Parthasarathy operators inherit the notion
of the Dirac inequality. The second generalisation, Vogan operators, inherit
Dirac cohomology; if an operator has non-zero cohomology then it relates the
infinitesimal character with a character of the group $\widetilde{G}$. We
prove properties about these operators and give a family of operators in each
class.

##  1 Introduction

Dirac operators for real Lie groups, [1, 18, 20] have been significantly
utilised in the study of $(\mathfrak{g},K)$ representations. These ideas have
been generalised to graded affine Hecke algebra [3] and Drinfeld’s Hecke
algebras [11, 7]. There are many other algebras which have has associated
Dirac operators defined [10, 8, 15].

The main results for Dirac operators are the Dirac inequality and Dirac
cohomology. The Dirac inequality [20, 12] notes that the square of the Dirac
operator on a unitary module is non-negative. The square is equal to a sum of
two elements, both central in different algebras. In the case of Lie algebras,
these algebras are $\mathfrak{g}$ and the diagonal embedding of
$\mathfrak{k}$. In the case of graded affine Hecke algebras $\mathbb{H}$, the
first element is central in $\mathbb{H}$ and the second central in the
diagonal embedding of $\tilde{G}$. We define a generalisation of Dirac
operators for Drinfeld’s Hecke algebras, Parthasarathy operators (Definition
3.1). These operators are such that each Parthasarathy operator leads to an
associated inequality (Corollary 3.11). For graded affine Hecke algebras
associated to the symmetric group $\mathbb{H}(S_{n})$ we define a family of
Parthasarathy operators (Corollary 3.10) . Unfortunately, every inequality
given by this family is strictly weaker than the original Dirac inequality [2,
12] (Remark 3.12).

The second major result for Dirac operators is ‘Vogan’s conjecture’ and Dirac
cohomology. This states that if there exists non-zero Dirac cohomology [24]
for an irreducible representation $(X,\pi_{X})$ then the Dirac operator
relates the infinitesimal character of $X$ with a character of a diagonal
algebra. In the Lie algebra setting this diagonal algebra is $U(\mathfrak{k})$
and for graded affine Hecke algebras the diagonal algebra is
$\mathbb{C}\tilde{G}$. Inspired by an infinite family of Dirac operators for
the Dunkl angular momentum algebra [8] and the total Dunkl angular momentum
algebra [9], we define another generalisation of the Dirac operator, Vogan
operators (Definition 4.1). We give an infinite family of Vogan operators for
Drinfeld’s Hecke algebras (Definition 4.4 and Theorem 4.15). We then prove an
equivalent Vogan conjecture for the cohomology of these operators (Theorem
5.14). If there is non-zero cohomology for a Vogan operator
$\mathcal{D}_{\omega}$ (Definition 4.4) then there is a relation between the
infinitesimal character and a character for $\widetilde{G}$ (Theorem 5.17).
Furthermore, we prove a simple condition for an irreducible representation
$(X,\pi_{X})$ to have non-zero cohomology for at least one Vogan operator
$\mathcal{D}_{\omega}$ (Proposition 5.19). We show that this condition is
satisfied for every irreducible representation of $\mathbb{H}(S_{n})$.

It would be interesting to apply these ideas to Lie algebras [16] and
operators as defined by Flake [15], Chan [10] and the local/global theory of
for Cherednik algebras introduced by Ciubotaru and De Martino [13].
Application of these ideas to Lie algebras could lead to a generalisation of
Dirac cohomology [16] and an enlargement of the Dirac series for real Lie
groups [4].

##  2 Preliminaries

###  2.1 Drinfeld’s Hecke algebra

In this section we define Drinfeld algebras as introduced by Drinfeld. Given a
finite group $G$, anti-symmetric bilinear forms $b_{g}$ for $g\in G$ and a
representation $(V,\pi_{V})$ of G, then we construct an algebra

$\mathbb{H}=\mathbb{C}G\rtimes T(V)/R.$

Here $R$ is the two sided ideal of $\mathbb{C}G\rtimes T(V)$ generated by the
relations,

$g^{-1}vg=\pi_{V}(g)(v)\text{ for all }g\in G\text{ and }v\in V,$

and

$[u,v]=\sum_{g\in G}b_{g}(u,v)g\text{ for all }v,u\in V.$

We define a filtration on the algebra $\mathbb{C}G\rtimes T(V)/R$, a vector
$v$ has degree $1$ and a group element $g\in G$ has degree $0$.

######  Definition 2.1.

[14] An algebra of the form $\mathbb{H}=\mathbb{C}G\rtimes T(V)/R$ is a
Drinfeld algebra if it satisfies a PBW criterion. That is the associated
graded algebra is naturally isomorphic to

$\mathbb{C}G\rtimes S(V).$

We state the conditions on the bilinear forms $b_{g}$ such that $\mathbb{H}$
is a Drinfeld algebra. Define $G(b)=\\{g\in G:b_{g}\neq 0\\}$.

######  Theorem 2.2.

[14][21, Theorem 1.9] The algebra $\mathbb{H}$ is a Drinfeld algebra if and
only if for every $g,h\in G$ and $u,v\in V$, $h^{\prime}\in Z_{G}(g)$,
$g^{\prime}\in G(b)\setminus\ker\pi_{V}$,

  * •

$b_{g^{-1}hg}(u,v)=b_{h}(\pi_{V}(g)(u),\pi_{V}(g)(v))$

  * •

$\operatorname{Ker}b_{g^{\prime}}=V^{\pi_{V}(g^{\prime})}$ and
$\operatorname{dim}(V^{\pi_{V}(g^{\prime})})=\operatorname{dim}V-2,$

  * •

$\operatorname{det}(h^{\prime}|_{V^{\pi_{V}(g^{\prime})^{\perp}}})=1$,

where $V^{\pi_{V}(g)^{\perp}}=\\{v-\pi_{V}(g)(v):v\in V\\}$.

######  Example 2.3.

Let $G$ be the symmetric group $S_{n}$, acting by on the $n-1$ dimensional
root space $V$. The root space $V$ has a $S_{n}$ invariant form
$\langle\,,\,\rangle$. Consider roots $\Phi\subset V$ for $S_{n}$ and let c be
a parameter in $\mathbb{C}$. Define $b_{g}$ for $g\in S_{n}$

$b_{g}(u,v)=\begin{cases}0&\text{ if }g\text{ is not a three cycle},\\\
\textbf{c}\left(\langle\alpha,u\rangle\langle\beta,v\rangle-\langle\alpha,v\rangle\langle\beta,u\rangle\right)&\text{
for }g=s_{\alpha}s_{\beta}\text{ and }\alpha\beta\in\Phi^{+}\text{ such that
}\langle\alpha,\beta\rangle_{V^{*}}\neq 0.\end{cases}$

This defines the graded affine Hecke algebra associated to $S_{n}$ with
representation $\mathbb{C}^{n-1}$.

###  2.2 Clifford Algebra

We define some basics on Clifford algebras, for more details see [19]. We
assume that $V$ has a $G$-invariant non-degenerate bilinear form
$\langle\,,\,\rangle_{V}:V\times V\to\mathbb{C}.$ We consider the Clifford
algebra $\mathcal{C}\colonequals\mathcal{C}(V,\langle\,,\,\rangle_{V})$ with
canonical map $\gamma\colon V\to\mathcal{C}$. Let $\\{u_{i}\\}$ be a
$\langle\,,\,\rangle_{V}$-orthonormal basis of $V$, then $\mathcal{C}$ is
generated by $\\{e_{i}=\gamma(u_{i})\\}$ satisfying

$\\{e_{j},e_{k}\\}=e_{j}e_{k}+e_{k}e_{j}=2\langle
e_{j},e_{k}\rangle_{V}=2\delta_{jk}\,.$

The Clifford algebra is naturally $\mathbb{Z}_{2}$-graded with $\gamma(V)$
having degree $\bar{1}$. We extend this $\mathbb{Z}_{2}$ grading to
$\mathbb{H}\otimes\mathcal{C}$ by giving every element in $\mathbb{H}$ degree
$\bar{0}$.

######  Definition 2.4.

For a homogeneous element $a$ in $\mathbb{H}\otimes\mathcal{C}$ with
$\mathbb{Z}_{2}$-degree $|a|\in\mathbb{Z}_{2}$, we define
$\epsilon(a)=(-1)^{|a|}$.

In the Clifford algebra, there is a realisation of the group
$\mathsf{Pin}\colonequals\mathsf{Pin}(V,B)$, which is a double covering of the
orthogonal group $p\colon\mathrm{Pin}\to\mathsf{O}$. We define a double cover
$\widetilde{\pi_{V}G}=p^{-1}(\pi_{V}G)$. Note that $\widetilde{\pi_{V}(G)}$ is
not a double cover of $G$ but it is a double cover of $\pi_{V}(G)$. We
construct a cover of $G$. We define $\widetilde{G}$ to be the semi direct
product $\operatorname{Ker}\pi_{V}\rtimes\widetilde{\pi_{V}(G)}$. Given
$\widetilde{G}$ we can embed the group in $\mathbb{H}\otimes\mathcal{C}$ via

$\rho:\widetilde{G}\to\mathbb{H}\otimes\mathcal{C},$
$\rho(\tilde{g},h)=hp(\tilde{g})\otimes\tilde{g},\hskip
28.45274pt\tilde{g}\in\widetilde{\pi_{V}(G)},h\in\operatorname{Ker}\pi_{V}.$

For a reflection $s\in G$ let $\tilde{s}$ denote a preimage in $\tilde{G}$, so
$p(\tilde{s})=\pi_{V}s$. Let $\theta$ be the nontrivial preimage of $1$ in
$\widetilde{G}$. The element $\theta$ is central in $\widetilde{G}$ and has
order two: $\theta^{2}=1$.

######  Definition 2.5.

We define a character
$\operatorname{sgn}:\mathbb{C}\widetilde{G}\to\mathbb{C}$, such that
$\operatorname{sgn}(\tilde{g})=\det_{\pi_{V}}(p(\tilde{g}))$.

######  Example 2.6.

A Weyl group $W$ (with simple roots $\Delta$ and positive roots $\Phi^{+}$)
has presentations

$W=\langle s_{\alpha},\alpha\in\Phi^{+}\mid
s_{\alpha}^{2}=1,s_{\alpha}s_{\beta}s_{\alpha}=s_{\gamma},\gamma=s_{\alpha}(\beta)\rangle,$
$W=\langle
s_{\alpha},\alpha\in\Delta\mid(s_{\alpha}s_{\beta})^{m_{\alpha,\beta}}=1\rangle$

while the double-cover has presentations

$\widetilde{W}=\langle\theta,\tilde{s}_{\alpha},\alpha\in\Phi^{+}\mid\tilde{s}_{\alpha}^{2}=1=\theta^{2},\tilde{s}_{\alpha}\tilde{s}_{\beta}\tilde{s}_{\alpha}=\theta\tilde{s}_{\gamma},\gamma=s_{\alpha}(\beta),\theta\text{
central}\rangle,$
$\widetilde{W}=\langle\theta,\tilde{s}_{\alpha},\alpha\in\Delta\mid(\tilde{s}_{\alpha}\tilde{s}_{\beta})^{m_{\alpha,\beta}}=(\theta)^{m_{\alpha,\beta}-1},\theta\text{
central}\rangle.$

The group algebra $\mathbb{C}\widetilde{G}$ splits into two subalgebras

$\mathbb{C}\widetilde{G}=\frac{1}{2}(1+\theta)\mathbb{C}\widetilde{G}\oplus\frac{1}{2}(1-\theta)\mathbb{C}\widetilde{G}.$

The algebra $\frac{1}{2}(1+\theta)\mathbb{C}\widetilde{G}$ is isomorphic to
$\mathbb{C}G$. We shall denote the algebra $\frac{1}{2}(1\pm
z)\mathbb{C}\widetilde{G}$ by, respectively $\mathbb{C}\widetilde{G}_{\pm}$.
The algebra $\mathbb{C}G_{+}$ is isomorphic to $\mathbb{C}G$. Furthermore,
$\rho$ is an embedding of $\mathbb{C}\widetilde{G}_{-}$ into
$H\otimes\mathcal{C}$.

###  2.3 Hermitian forms

Let $\ast$ denote the anti-automorphism $\eta^{*}=\varepsilon(\eta^{t})$, for
all $\eta\in\mathcal{C}_{\mathbb{R}}$. Let also $\bullet$ be anti-linear form
on $\mathbb{H}$ by $v^{\bullet}=-v$ for $v\in V$, and $w^{\bullet}=w^{-1}$,
for all $w\in G$. We then define an anti-linear anti-involution $\star$ on
$\mathbb{H}\otimes\mathcal{C}$ by taking the tensor product of these two anti-
involutions.

######  Definition 2.7.

A Hermitian form $\langle\,,\,\rangle_{X}:X\times X\to\mathbb{C}$ is
$\mathbb{H}$ invariant if

$\langle hx_{1},x_{2}\rangle_{X}=\langle
x_{1},h^{\bullet}x_{2}\rangle_{X}\text{ for all }x_{1},x_{2}\in X\text{ and
}h\in\mathbb{H}.$

######  Definition 2.8.

A $\mathbb{H}$-module $X$ is unitary if there exists a positive definition
$\mathbb{H}$-invariant Hermitian form on $X$.

Now fix, once and for all, $(\sigma,S)$ an irreducible module (spinor module)
for $\mathcal{C}$.

######  Definition 2.9.

There exists a positive definite form $\langle\,,\,\rangle_{S}$ on $S$. such
that $\langle\gamma(v)s_{1},s_{2}\rangle=\langle s_{1},v^{*}s_{2}\rangle$ for
all $v\in V$ and $s_{1},s_{2}\in S$. This endows $S$ with a $\ast$-unitary
$\mathcal{C}$-structure.

For any $\bullet$-Hermitian module $(\pi,X)$ of $\mathbb{H}$ we endow
$X\otimes S$ with a $\star$-Hermitian structure $\langle x\otimes
s,x^{\prime}\otimes s^{\prime}\rangle_{X\otimes S}=\langle
x,x^{\prime}\rangle_{X}\langle s,s^{\prime}\rangle_{S}$ for all
$x,x^{\prime}\in X$ and $s,s^{\prime}\in S$. If $X$ is $\bullet$-unitary then
the $\star$-Hermitian form on $X\otimes S$ is also positive definite and hence
$\star$-unitary.

###  2.4 The original Dirac element

If $V$ has a $G$-invariant symmetric bilinear form then one can define a Dirac
operator $\mathcal{D}$. In [11] (resp. [7]) Dirac cohomology is defined for
faithful (resp. non-faithful Drinfeld algebra). In this section we recall
definitions and a formula for $\mathcal{D}^{2}$ from [7]. Given any basis
$\\{v_{i}\\}$ of $V$ and dual basis $\\{v^{i}\\}$ with respect to
$\langle\,,\,\rangle_{V}$ we define the Dirac element

$\mathcal{D}=\sum_{i}v_{i}\otimes v^{i}\in\mathbb{H}\otimes\mathcal{C}.$

For every $g\in G(b)$ set,

$\textbf{k}_{g}=\sum_{i,j}b_{g}(v_{i},v^{j})v^{i}v_{j}\in\mathcal{C},$

and

$\textbf{h}=\sum_{i}v_{i}v^{i}\in\mathbb{H}.$

Define the set $G(b)=\\{g\in G:b_{g}\neq 0\\}$, write $\widetilde{G}(b)$ for
the cover of this subset. For every coset representative $g\in
G(b)/\operatorname{Ker}(\pi_{V})$ define

$\tilde{g}=\alpha\beta\in\mathcal{C},\hskip
28.45274ptc_{\tilde{g}}=\frac{b_{g}(\alpha,\beta)}{1-\langle\alpha,\beta\rangle^{2}}\in\mathbb{C},\hskip
28.45274pte_{g}=\frac{b_{g}(\alpha,\beta)\langle\alpha\beta\rangle}{1-\langle\alpha,\beta\rangle^{2}}\in\mathbb{C}.$

Every $w\in\widetilde{g}(b)$ can be written as $h^{-1}gh$ where $g$ is a coset
representative of $\widetilde{G}(b)/\operatorname{Ker}\pi_{V}$ and
$h\in\operatorname{Ker}\pi_{V}$. Lemma [7, Lemma prod of alpha] gives
$g=s_{\alpha}s_{\beta}$ and $\tilde{g}=\alpha\beta\in\mathcal{C}$. We define,
for $w=h^{-1}gh\in\widetilde{G}$, define

$\tilde{g}=\alpha\beta\in\mathcal{C},\hskip
28.45274ptc_{\tilde{g}}=c_{\tilde{g}},\hskip 28.45274pte_{w}=he_{g}h^{-1}.$

Let us define the Casimir elements, $\Omega_{\mathbb{H}}$ in $\mathbb{H}$ and
$\Omega_{\widetilde{G}}$ in $\widetilde{G}$.

$\Omega_{\mathbb{H}}=\textbf{h}-\sum_{g\in
G(b)/\operatorname{Ker}\pi_{V}}e_{g}g\in\mathbb{H}^{G},$
$\Omega_{\widetilde{G}}=\sum_{\begin{subarray}{c}h\in\operatorname{Ker}\pi_{V}\\\
g\in
G(b)/\operatorname{Ker}\pi_{V}\end{subarray}}h^{-1}\tilde{g}hc_{\tilde{g}}\in\mathbb{C}[\widetilde{G}]^{\widetilde{G}}.$

We give a formula for $\mathcal{D}^{2}$. This is equivalent to [11, Theorem
2.7],. The only variation being that $\ker\pi_{V}$ replaces $1$.

######  Theorem 2.10.

[7, Theorem 2.4][11, c.f. Theorem 2.7] The square of the Dirac element can be
expressed as a sum of the two Casimir elements plus even terms from the
Clifford algebra:

$\mathcal{D}^{2}=-\Omega_{\mathbb{H}}\otimes
1+\pi_{V}(\Omega_{\widetilde{G}})+\frac{1}{2}\otimes\sum_{w\in\ker\pi_{V}}\textbf{k}_{w}.$

######  Lemma 2.11.

[11, Lemma 2.4]The operator $\mathcal{D}$ $\operatorname{sgn}$-commutes with
$\widetilde{G}$,

$\rho(\tilde{g})\mathcal{D}\rho(\tilde{g})^{-1}=\operatorname{sgn}(\tilde{g})\mathcal{D}$

for every $\tilde{g}\in\widetilde{G}$.

##  3 Parthasarathy Operators

######  Definition 3.1.

Let $\mathbb{H}$ be a Drinfeld algebra with group algebra $\mathbb{C}G$. We
say an operator $\mathcal{P}\in\mathbb{H}\otimes\mathcal{C}$ is a
Parthasarathy operator if the following holds:

  1. 1.

$\mathcal{P}^{*}=\mathcal{P}$

  2. 2.

$\mathcal{P}^{2}=z_{1}+z_{2}$, where $z_{1}\in Z(\mathbb{H})\otimes 1$ and
$z_{2}\in(Z(\rho(\mathbb{C}\widetilde{G}))$.

###  3.1 A family of Parthasarathy operators for $\mathbb{H}$

Let $\mathbb{H}$ be a Drinfeld algebra, with a Dirac operator
$\mathcal{D}\in\mathbb{H}\otimes\mathcal{C}$.

######  Definition 3.2.

For $\Xi\in\mathbb{C}\widetilde{G}_{-}$, we say that
$\Xi=\sum\lambda_{\tilde{g}}\tilde{g}$ is $P$-admissible if

  1. 1.

$\Xi^{\bullet}=\Xi$,

  2. 2.

$\operatorname{sgn}(\tilde{g})=-1$ for all $\lambda_{\tilde{g}}\neq 0$,

  3. 3.

$\Xi^{2}\in Z(\mathbb{C}\widetilde{G}_{-})$.

######  Remark 3.3.

The original Dirac operator (Section 2.4) is a Parthasarathy Dirac operator if
and only if $\kappa_{g}=0$ for every $g\in\operatorname{Ker}\pi_{V}$.

For the remainder of this section let us assume that $\kappa_{g}=0$ for all
$g\in\ker\pi_{V}$.

######  Theorem 3.4.

For all $P$-admissible elements $\Xi\in\mathbb{C}\widetilde{G}_{-}$, the
operator

$\mathcal{D}_{\Xi}=\mathcal{D}+\rho\Xi,$

is a Parthasarathy operator.

###### Proof.

Clearly, $\mathcal{D}_{\Xi}^{*}=\mathcal{D}_{\Xi}$. Furthermore,
$\mathcal{D}_{\Xi}^{2}=(\mathcal{D}+\rho\Xi)^{2}=\mathcal{D}^{2}+\rho\Xi^{2}+\mathcal{D}\rho\Xi+\rho\Xi\mathcal{D}$.
Note that $\rho\Xi\in\rho\mathbb{C}\widetilde{G}$, hence
$\rho\Xi\mathcal{D}=\sum\lambda_{\tilde{g}}\tilde{g}\mathcal{D}=\operatorname{sgn}(\tilde{g})\mathcal{D}\sum\lambda_{\tilde{g}}\tilde{g}=-\mathcal{D}\rho\Xi$.
We conclude that,

$\mathcal{D}_{\Xi}^{2}=\mathcal{D}^{2}+\rho\Xi^{2}=z_{1}+z_{2}$

where $z_{1}\in Z(\mathbb{H})$ and $z_{2}=\Omega_{\widetilde{G}}+\Xi^{2}\in
Z\rho(\widetilde{G})$.

∎

###  3.2 $P$-admissible elements

Throughout this section we only consider the symmetric group $S_{n}$. Recall
$S_{n}$ and $\tilde{S_{n}}$ have presentations:

$S_{n}=\langle s_{ij},{1\leq i<j\leq n}\mid(s_{ij}s_{kl})^{m_{ijkl}}=1\rangle$

where

$m_{ijkl}=\begin{cases}2&|\\{i,j,k,l\\}|=4,\\\ 3&|\\{i,j,k,l\\}|=3,\\\
1&|\\{i,j,k,l\\}|=2.\\\ \end{cases}$

Similary for $\widetilde{S}_{n}$

$\tilde{S_{n}}=\langle\theta,\tilde{s}_{ij},{1\leq i<j\leq
n}\mid(\tilde{s}_{ij}\tilde{s}_{kl})^{m_{ijkl}}=(\theta)^{m_{ijkl}-1},\theta\text{
central}\rangle.$

######  Definition 3.5.

[5, 3.1] The Jucys-Murphy elements in $\mathbb{C}S_{n}^{-}$ for $i=1,\ldots,n$
are,

$M_{j}=\sum_{i=1}^{j-1}\tilde{s}_{ij}.$

######  Remark 3.6.

[5, 3.1] The Jucys-Murphy elements anti-commute , that is

$M_{i}M_{j}=-M_{j}M_{i}\text{ if }i\neq j.$

######  Lemma 3.7.

[5, 3.2] The even centre $Z(\tilde{S_{n}}_{-})_{0}$ is spanned by the set of
symmetric polynomials of the Jucys-Murphy elements.

######  Proposition 3.8.

Every odd symmetric power-polynomial in the squares of the Jucys-Murphy
elements is a square of an odd symmetric polynomial in the Jucys-Murphy
elements.

###### Proof.

Suppose that $k$ is odd. Let $P_{k}=\sum_{i=1}^{n}(M_{i}^{2})^{k}$ and let
$Q_{k}=\sum_{i=1}^{n}(M_{i})^{k}$. We claim that $P_{k}=Q_{k}^{2}$.

$Q_{k}^{2}=(\sum_{i=1}^{n}(M_{i})^{k})^{2}=\sum_{i=1}^{n}(M_{i})^{k}\sum_{j=1}^{n}(M_{j})^{k}=\sum_{i=1}^{n}(M_{i})^{2k}+\sum_{i\neq
j}(M_{i})^{k}(M_{j})^{k}+(M_{j})^{k}(M_{i})^{k}$

Now since $k$ is odd and if $i\neq j$ then
$(M_{i})^{k}(M_{j})^{k}=-(M_{j})^{k}(M_{i})^{k}$. Hence we have shown that
$Q_{k}^{2}=P_{k}$.

∎

######  Theorem 3.9.

Let $G=S_{n}$, let $Q_{j}$ be an odd power polynomial in the Jucys-Murphy
elements $Q_{j}$. Then $\sqrt{-1}Q_{j}$ is $P$-admissible.

###### Proof.

The element, $Q_{j}$, consists of sums of odd polynomials in $\tilde{s}_{kl}$,
all of which have odd $sgn$ group elements. Each pseudo reflection
$\tilde{s}_{kl}$ is such that
$\rho\tilde{s}_{kl}^{\bullet}=-\rho\tilde{s}_{kl}$, hence $Q_{j}$ is skew-
adjoint and $\sqrt{-1}Q_{j}$ is self-adjoint. To complete the proof we note
that $Q_{j}^{2}$ is central in $\mathbb{C}\widetilde{G}_{-}$. ∎

######  Corollary 3.10.

Let $\mathbb{H}(S_{n})$ be a graded affine Hecke algebra with the symmetric
group. For every odd $j$, then the operator

$\mathcal{D}+\sqrt{-1}Q_{j}$

is a Parthasarathy operator.

###  3.3 A family of Dirac inequalities

Suppose that $X$ is unitary (Definition 2.7), then any Parthasarathy operator
$\mathcal{P}$ is self-adjoint. Furthermore, $X\otimes S$ has a
$\mathbb{H}\otimes\mathcal{C}$-invariant positive definite form. Hence,
$\mathcal{P}^{2}$ is a positive operator.

######  Corollary 3.11 (Generalised Parthasarathy inequality).

For every Parthasarathy operator $\mathcal{P}$ and unitary module
$(X,\pi_{X})$, then $\mathcal{P}^{2}$ is positive operator on $X\otimes S$ and

$\pi_{X}(z_{1})+\pi_{X}(z_{2})\geq 0.$

Here, using Definition 3.1, $\mathcal{P}^{2}=z_{1}+z_{2}$, with $z_{1}\in
Z(\mathbb{H})$, $z_{2}\in Z(\rho(\mathbb{C}\widetilde{G}))$.

######  Remark 3.12.

Unfortunately, every Parthasarathy operator defined in Definition 3.2 gives an
inequality weaker than the original Dirac inequality. Let $\mathcal{D}_{\Xi}$
be defined as in Definition 3.2, then $\Xi$ is self-adjoint. Therefore,
$\pi_{X}(\Xi)^{2}$ is a positive operator. The inequality associated to
$\mathcal{D}_{\Xi}$ is

$-\pi_{X}(\Omega_{\mathbb{H}})+\pi_{X}(\Omega_{\widetilde{G}})+\pi_{X}(\Xi)^{2}\geq
0.$

Since $\pi_{X}(\Xi)^{2}$ is positive this is less restrictive than the
original Dirac inequality,

$-\pi_{X}(\Omega_{\mathbb{H}})+\pi_{X}(\Omega_{\widetilde{G}})\geq 0.$

An interesting question, which the author intends to study, is whether there
are any Parthasarathy operators in $\mathbb{H}\otimes\mathcal{C}$ which lead
to new relations between the centre of $\mathbb{H}$ and the centre of
$Z(\widetilde{G})$.

##  4 Vogan operators

######  Definition 4.1.

Let $\mathbb{H}$ be a Drinfeld algebra, we say an operator
$\mathcal{V}\in\mathbb{H}\otimes\mathcal{C}$ is a Vogan operator if the
following holds:

  1. 1.

$\mathcal{V}^{*}=\mathcal{V}$

  2. 2.

The operator $\mathcal{V}$ is $\operatorname{sgn}$-invariant under the action
of $\rho\widetilde{G}$.

######  Definition 4.2.

Let $X$ be a $\mathbb{H}$-module and $S$ a spinor for $\mathcal{C}$, the
operator $\mathcal{V}$ acts on $X\otimes S$ as $\mathcal{V}_{X}$. Define
$H(X,\mathcal{V})$ as

$\operatorname{Ker}\mathcal{V}_{X\otimes
S}/\operatorname{Ker}\mathcal{V}_{X\otimes
S}\cap\operatorname{im}\mathcal{V}_{X\otimes S}.$

######  Proposition 4.3.

The cohomology of $\mathcal{V}$ is a $\widetilde{G}$ module.

###### Proof.

This follows from the fact that $\mathcal{V}$ is
$\operatorname{sgn}-\widetilde{G}$ invariant. ∎

###  4.1 A family of Vogan operators for $\mathbb{H}$

######  Definition 4.4.

We define the $\operatorname{sgn}$-centre of $\mathbb{C}\widetilde{G}_{-}$ to
be:

$Z_{\operatorname{sgn}}(\widetilde{G}_{-})=\\{g\in\mathbb{C}\widetilde{G}_{-}:gh=\operatorname{sgn}(h)hg\quad\text{
for all }h\in\mathbb{C}\widetilde{G}\\}.$

Furthermore, we say an element is $\operatorname{sgn}$-central if it is
contained in the $\operatorname{sgn}$-centre.

The ungraded centre of $Z^{ug}\mathbb{C}\widetilde{G}_{-}$ is equal to
$\operatorname{Hom}_{\widetilde{G}}(\operatorname{triv},\mathbb{C}\widetilde{G}_{-})$
and the $\operatorname{sgn}$-centre of $\mathbb{C}\widetilde{G}_{-}$ is equal
to
$\operatorname{Hom}_{\widetilde{G}}(\operatorname{sgn},\mathbb{C}\widetilde{G}_{-})$.

######  Definition 4.5.

A homogeneous element $\omega\in\mathbb{C}\widetilde{G}_{-}$ is called
$V$-admissible if $\omega$ is $\operatorname{sgn}$-central and
$\omega^{\bullet}=\omega$. For any $V$-admissible $\omega\in
Z_{\operatorname{sgn}}\widetilde{G}_{-}$, define

$\mathcal{D}_{\omega}:=\mathcal{D}+\rho\omega\in\mathbb{H}\otimes\mathcal{C}.$

######  Theorem 4.6.

The elements $\mathcal{D}_{\omega}$ are all Vogan operators. Furthermore, they
are precisely the modification of $\mathcal{D}$ by elements in
$\rho\mathbb{C}\widetilde{G}$ which are Vogan operators.

###  4.2 A formula for $\mathcal{D}_{\omega}^{2}$

######  Lemma 4.7.

The square of $\mathcal{D}_{\omega}$ is equal to a central element in
$\mathbb{H}$ plus a central element in $\widetilde{G}$, a linear term in
$\mathcal{D}_{\omega}$ and a correction quadratic term in $\mathcal{C}$;

$(\mathcal{D}_{\omega})^{2}=-\Omega_{\mathbb{H}}\otimes
1+\rho(\Omega_{\widetilde{G}}+\omega^{2})++(1+\operatorname{sgn}(\rho(\omega)))\rho(\omega)\mathcal{D}+\frac{1}{2}\otimes\sum_{w\in\ker\pi_{V}}\textbf{k}_{w}.$

###### Proof.

The following calculation is simple algebra,

$\displaystyle(\mathcal{D}_{\omega})^{2}$
$\displaystyle=(\mathcal{D}+\rho(\omega))^{2}$
$\displaystyle=(\mathcal{D})^{2}+\rho(\omega)^{2}+\mathcal{D}\rho(\omega)+\rho(\omega)\mathcal{D}$
$\displaystyle=(\mathcal{D})^{2}+\rho(\omega)^{2}+\operatorname{sgn}(\rho(\omega))\rho(\omega)\mathcal{D}+\rho(\omega)\mathcal{D}.$

Finishing the proof with application of Theorem 2.10. ∎

######  Lemma 4.8 ($\operatorname{sgn}$ invariance of $D_{\omega}$).

For every $\tilde{g}\in\widetilde{G}$, we have the invariance property:

$\rho(\tilde{g})\mathcal{D}_{\omega}\rho(\tilde{g})^{-1}=\operatorname{sgn}(p(\tilde{g}))\mathcal{D}_{\omega}.$

###### Proof.

Both $\mathcal{D}$ and $\omega$ are $\operatorname{sgn}$ invariant, hence so
it their sum $\mathcal{D}_{\omega}=\mathcal{D}+\rho(\omega)$ ∎

###  4.3 $V$-Admissible elements

######  Definition 4.9.

[9, Definition 6.3] Let us the define the $\theta$-centre of
$\mathbb{C}\widetilde{G}$,

$Z^{\theta}(\mathbb{C}\widetilde{G})=\\{a\in\mathbb{C}\widetilde{G}|a\tilde{g}=\theta^{\operatorname{sgn}(p(\tilde{g}))}\tilde{g}a\text{
for all }\tilde{g}\in\widetilde{G}\\}.$

######  Proposition 4.10.

[9, Proposition 6.4] The $\theta$-centre of $\mathbb{C}\widetilde{G}$ is
spanned by elements of the form

$C^{\theta}_{\tilde{g}}=\sum_{\tilde{h}\in\widetilde{G}}\theta^{|l(\tilde{h})|}\tilde{h}^{-1}\tilde{g}\tilde{h}=\sum_{\tilde{h}\in\widetilde{G}_{\overline{0}}}\tilde{h}^{-1}\tilde{g}\tilde{h}+\theta\sum_{\tilde{h}\in\widetilde{G}_{\overline{1}}}\tilde{h}^{-1}\tilde{g}\tilde{h}$

for any choice of $\tilde{g}\in\widetilde{G}$.

###### Proof.

Given any $\tilde{g}$, then $C^{\theta}_{\tilde{g}}$ is in
$Z^{\theta}\mathbb{C}\widetilde{G}$. Furthermore, any $a\in
Z^{\theta}\mathbb{C}\widetilde{G}$ that has a non-zero coefficient of
$\tilde{g}$, then there exists a non-zero scalar $t$ such that
$a-tC^{\theta}_{\tilde{g}}$ is $\theta$-central with no coefficient of
$\tilde{g}$. Continuing the process shows that $a$ is in the space spanned by
$C^{\theta}_{\tilde{g}}$. ∎

######  Theorem 4.11.

[9, Theorem 6.5]

The $\operatorname{sgn}$-centre of $\mathbb{C}\widetilde{G}_{-}$ is the
projection of the $\theta$-centre of $\mathbb{C}\widetilde{G}$

$Z^{\operatorname{sgn}}(\mathbb{C}\widetilde{G}_{-})=\frac{1-\theta}{2}Z^{\theta}\mathbb{C}\widetilde{G}.$

In particular, the $\operatorname{sgn}$-centre of
$\mathbb{C}\widetilde{G}_{-}$ is spanned by elements of the form

$C^{\operatorname{sgn}}_{\tilde{g}}=\sum_{\tilde{h}\in\widetilde{G}}(-1)^{|l(\tilde{h})|}\tilde{h}^{-1}\tilde{g}\tilde{h}\in\mathbb{C}\widetilde{G}_{-}.$

Denote by $\widetilde{G}_{\bar{0}}$ the even subgroup of $\widetilde{G}$.

######  Lemma 4.12.

[9, Lemma 6.7] Suppose that $\tilde{g}\in\widetilde{G}$ is even, define the
$\tilde{g}$ conjugacy class,
$C(\tilde{g})=\\{\tilde{w}\in\widetilde{G}:\tilde{g}=\tilde{h}^{-1}\tilde{g}\tilde{h},\tilde{h}\in\widetilde{G}\\}$,
then the element $Z^{\epsilon}_{\tilde{g}}$ is non zero if and only if the
conjugacy class $C(\tilde{g})$ splits into two conjugacy classes in
$\widetilde{G}_{\overline{0}}$.

######  Theorem 4.13.

[22] [23, Theorem 2.7] Let $\lambda$ be an even partition of $n$. The
$\tilde{S}_{n}$ conjugacy classes $C_{\lambda}$ (or $C_{\lambda}^{\pm})$ if
already split) split into two $\tilde{A}_{n}$ conjugacy classes if and only if
$\lambda\in DP_{n}^{+}$. Here $DP_{n}^{+}$ is the set of distinct partitions
of $n$ which are even.

######  Corollary 4.14.

The $sgn$-centre of $\mathbb{C}\widetilde{G}_{-}$ has basis

$\\{C^{\operatorname{sgn}}_{\tilde{g}}:C(\tilde{g})\text{ splits into two
conjugacy classes in }\widetilde{G}_{\overline{0}}\\}.$

Due to Corollary 4.14 we may assume that any $V$-admissible element in
$\mathbb{C}\widetilde{G}_{-}$ is even.

######  Theorem 4.15.

The $V$-admissible elements in $\mathbb{C}\widetilde{G}_{-}$ are equal to the
real-span of the set

$\\{C^{\operatorname{sgn}}_{\tilde{g}}+C^{\operatorname{sgn}}_{\tilde{g}^{-1}},\sqrt{-1}(C^{\operatorname{sgn}}_{\tilde{g}}-C^{\operatorname{sgn}}_{\tilde{g}^{-1}}):C(\tilde{g})\text{
splits into two conjugacy classes in }\widetilde{G}_{\overline{0}}\\}.$

###### Proof.

The $V$-admissible elements are elements in the $\operatorname{sgn}$-centre
which are self adjoint. Since $\rho\tilde{g}^{\bullet}$ =
$\rho\tilde{g}^{-1}$, then taking a basis for the $\operatorname{sgn}$-centre
from Corollary 4.14 and adding or subtracting
$C^{\operatorname{sgn}}_{\tilde{g}^{-1}})$ enforces this set to be self
adjoint (or skew adjoint respectively). Multiplying by $\sqrt{(}-1)$ forces
the skew adjoint operators to be self adjoint. This then spans all self-
adjoint operators in the $\operatorname{sgn}$-centre. ∎

######  Corollary 4.16.

Let $G=S_{n}$ and let $\\{g\\}$ be the set of elements in $S_{n}$ associated
to an even partition $\lambda$ which has distinct cycles. Then $C(\tilde{g})$
splits in $\tilde{A}_{n}=(\tilde{S}_{n})_{\overline{0}}$. Then from Lemma
4.12, $C^{\operatorname{sgn}}_{\tilde{g}}\neq 0$. The group
$\widetilde{S}_{n}$ is ambivalent, that is, $\tilde{g}^{-1}$ is always
conjugate to $\tilde{g}$. Therefore every nonzero element
$C^{\operatorname{sgn}}_{\tilde{g}}=\frac{1}{2}(C^{\operatorname{sgn}}_{\tilde{g}}+C^{\operatorname{sgn}}_{\tilde{g}^{-1}})$
is an admissible element and
$C^{\operatorname{sgn}}_{\tilde{g}}-C^{\operatorname{sgn}}_{\tilde{g}^{-1}}=0$
for every $C^{\operatorname{sgn}}_{\tilde{g}}$.

##  5 Vogan’s Dirac morphism

In this section we prove that there exits a Vogan morphism for each Vogan
operator we defined in Definition 4.4 and that Vogan’s conjecture holds for
each of these operators. Furthermore, we show that each operator
$\mathcal{D}_{\omega}$ defines a map between $\operatorname{Irr}\widetilde{G}$
and $\operatorname{Spec}\mathcal{B}$ for a suitably chosen abelian algebra
$\mathcal{B}$. We finish by showing that the defined family of Vogan operators
gives rise to non-zero cohomology of $(X,\pi_{X})$ whenever there is an
admissible element that is not in the kernel of $\pi_{X}$.

###  5.1 The linear map $d_{\omega}$

######  Definition 5.1.

Let $\mathcal{D}_{\omega}$ be a Vogan operator as defined in Definition 4.4.
We define a map from $\mathbb{H}\otimes\mathcal{C}$ to
$\mathbb{H}\otimes\mathcal{C}$.

$d_{\omega}:\mathbb{H}\otimes\mathcal{C}\to\mathbb{H}\otimes\mathcal{C},$

where $d_{\omega}(a)=\mathcal{D}_{\omega}a-\epsilon(a)\mathcal{D}_{\omega}$,
for $a\in\mathbb{H}\otimes\mathcal{C}$.

######  Remark 5.2.

The map $d_{\omega}$ is an odd derivation, i.e.,

$d_{\omega}(ab)=d_{\omega}(a)b+\epsilon(a)d_{\omega}(b)$

for all $a,b\in\mathbb{H}\otimes\mathcal{C}$.

######  Lemma 5.3.

The image under $\rho$ of $\mathbb{C}\widetilde{G}$ is in the kernel of
$d_{\omega}$

$\rho(\mathbb{C}\widetilde{G})\subset\ker d_{\omega}.$

###### Proof.

This follows from the fact that $\mathcal{D}_{\omega}$ is
$\operatorname{sgn}-\widetilde{G}$ invariant. ∎

The operator $\mathcal{D}_{\omega}$ intertwines the $\operatorname{sgn}$ and
$\operatorname{triv}$ $\widetilde{G}$ isotypic components
$\mathbb{H}\otimes\mathcal{C}^{\operatorname{triv}}$ and
$\mathbb{H}\otimes\mathcal{C}^{\operatorname{sgn}}$. We define
$d_{\omega}^{\operatorname{triv}}$ and $d_{\omega}^{\operatorname{sgn}}$ to be
$d_{\omega}$ restricted to the $\operatorname{triv}$ and $\operatorname{sgn}$
isotypic components respectively. Since the kernel of $d_{\omega}$, contains
$\rho(\mathbb{C}\widetilde{G})$, then the kernel of
$d_{\omega}^{\operatorname{triv}}$, contains the
$\operatorname{triv}$-isotypic component
$\rho(\mathbb{C}\widetilde{G}^{\widetilde{G}})$.

######  Theorem 5.4.

The kernel of $d_{\omega}^{\operatorname{triv}}$ equals:

$\ker
d_{\omega}^{\operatorname{triv}}=\operatorname{im}d_{\omega}^{\operatorname{sgn}}\oplus\rho(\mathbb{C}\widetilde{G}^{\widetilde{G}}).$

We prove this theorem in the following section.

###  5.2 The map $\bar{d_{\omega}}$

In this section we proof Theorem 5.4 by reducing to the associated graded
algebra. This method is identical method to [3, 11]. The algebra $\mathbb{H}$
is filtered with $g\in\mathbb{C}G$ degree zero and $v\in V$ degree one.
Similarly $\mathcal{C}$ is filtered with $\gamma(V)\in\mathcal{C}$ of degree
one. Hence $\mathbb{H}\otimes\mathcal{C}$ is a filtered algebra and has an
associated graded algebra
$\mathrm{gr}\mathbb{H}\otimes\mathrm{gr}\mathcal{C}$.

######  Definition 5.5.

The map $d_{\omega}$ defines a map on the associated graded algebra

$\bar{d_{\omega}}:\mathrm{gr}\mathbb{H}\otimes\mathrm{gr}\mathcal{C}\to\mathrm{gr}\mathbb{H}\otimes\mathrm{gr}\mathcal{C}.$

######  Lemma 5.6.

The map $\bar{d_{\omega}}$ is equal to $\bar{d_{0}}$ for any choice of
$\omega\in\rho{\mathbb{C}\widetilde{G}}$. Hence, the proof in [11] holds for
the following statements.

###### Proof.

Because $\mathcal{D}_{\omega}=\mathcal{D}_{0}+l.o.t.$ then
$\mathrm{gr}\mathcal{D}_{\omega}=\mathrm{gr}\mathcal{D}_{0}$ and
$\bar{d}_{\omega}=\bar{d}_{0}$ for any $\omega\in\mathbb{C}\widetilde{G}_{-}$.
∎

######  Proposition 5.7.

(cf. [3, Proposition 4.14]). The space $\ker\bar{d_{0}}^{\operatorname{triv}}$
decomposes as

$\ker\bar{d_{0}}^{\operatorname{triv}}=\operatorname{im}\bar{d_{0}}^{\operatorname{sgn}}\oplus\rho(\bar{\mathbb{C}\widetilde{G}}^{\widetilde{G}}).$

######  Corollary 5.8.

We have the following decomposition of
$\ker\bar{d_{\omega}}^{\operatorname{triv}}$

$\ker\bar{d_{\omega}}^{\operatorname{triv}}=\operatorname{im}\bar{d_{\omega}}^{\operatorname{sgn}}\oplus\rho(\bar{\mathbb{C}\widetilde{G}^{\widetilde{G}}}).$

####  5.2.1 Induction

We have proved that
$\ker\bar{d_{\omega}}=\operatorname{im}\bar{d_{\omega}}\oplus\rho(\bar{\mathbb{C}\widetilde{G}^{\widetilde{G}}}).$
To prove the equivalent statement for $d_{\omega}$ (Theorem 5.4) an induction
on the filtration of $\mathbb{H}\otimes\mathcal{C}$. The proof is verbatim to
[11, 3.3] and we omit it here.

###  5.3 Vogan’s Dirac homomorphism

######  Theorem 5.9.

The projection $\zeta_{\omega}:\ker
d_{\omega}^{\operatorname{triv}}\to\mathbb{C}\widetilde{G}^{\widetilde{G}}$
defined by Theorem 3.5 is an algebra homomorphism.

###### Proof.

Let $z_{1},z_{2}\in\ker d_{\omega}^{\operatorname{triv}}$, then
$z_{i}=\zeta(z_{i})+d_{\omega}^{\operatorname{sgn}}(a_{i})$. Multiplying
$z_{1}$, with $z_{2}$, we find

$z_{1}z_{2}=\zeta(z_{1})\zeta(z_{2})+d_{\omega}^{\operatorname{sgn}}(a_{1})\zeta(z_{2})+\zeta(z_{1})d_{\omega}^{\operatorname{sgn}}(a_{2})+d_{\omega}^{\operatorname{sgn}}(a_{1})d_{\omega}^{\operatorname{sgn}}(a_{2}).$

However, $d_{\omega}^{\operatorname{sgn}}$ is a derivation and
$d_{\omega}^{\operatorname{sgn}}(\rho\mathbb{C}\widetilde{G})=0$, hence

$z_{1}z_{2}=\zeta(z_{1})\zeta(z_{2})+d_{\omega}^{\operatorname{sgn}}(a_{1}\zeta(z_{2})\zeta(z_{1})a_{2})+d_{\omega}^{\operatorname{sgn}}(a_{1})d_{\omega}^{\operatorname{sgn}}(a_{2}).$

To finish the proof, we remark that $(d_{\omega}^{\operatorname{sgn}})^{2}=0$.
Therefore
$d_{\omega}^{\operatorname{sgn}}(a_{1})d_{\omega}^{\operatorname{sgn}}(a_{2})=d_{\omega}^{\operatorname{sgn}}(a_{1}d_{\omega}^{\operatorname{sgn}}(a_{2}))$.
We can conclude that $z_{1}z_{2}$ is equal to $\zeta(z_{1})\zeta(z_{2})$ plus
an element in the image of $d_{\omega}^{sgn}$. ∎

######  Definition 5.10.

Let $\mathcal{B}$ be an abelian subalgebra of
$\mathbb{H}\otimes\mathcal{C}^{\widetilde{G}}$ containing
$\Omega_{\tilde{\mathbb{H}}}$.

######  Proposition 5.11.

The dual of $\zeta_{\omega}$ is a map between
$\operatorname{Irr}\widetilde{G}$ and $\operatorname{Spec}\mathcal{B}$,

$\zeta_{\omega}^{*}:\operatorname{Irr}\widetilde{G}\to\operatorname{Spec}\mathcal{B}.$

######  Remark 5.12.

If $\kappa_{1}=0$, then we can take $\mathcal{B}=Z(\mathbb{H})\otimes 1$. Then
$\zeta_{\omega}$ is a map between the characters of $\widetilde{G}$ and
infinitesimal characters for $\mathbb{H}$.

###  5.4 Vogan’s conjecture

######  Definition 5.13.

Let $\mathcal{V}$ be an operator in $\mathbb{H}\otimes\mathcal{C}$. We say
that Vogan’s conjecture for $\mathcal{V}$ holds if, for all $z\in
Z(\mathbb{H})$, there exists a unique $\zeta(z)\in
Z(\rho\mathbb{C}\widetilde{G})$ and $a,b\in\mathbb{H}\otimes\mathcal{C}$ such
that

$z\otimes 1=\zeta(z)+\mathcal{V}a+a\mathcal{V}.$

######  Theorem 5.14.

Vogan’s conjecture holds for the family of Vogan operators defined in
Definition 4.4.

###### Proof.

By Theorem 5.4, for every $z\in\ker d_{\omega}^{\operatorname{triv}}$, there
exists an $a\in\mathbb{H}\otimes\mathcal{C}$ such that
$z=d_{\omega}^{\operatorname{sgn}}(a)+\zeta_{\omega}(z)$. The claim follows by
observing that $Z(\mathbb{H})$ is in the kernel of
$d_{\omega}^{\operatorname{triv}}$ and
$d_{\omega}^{\operatorname{sgn}}(a)=\mathcal{D}_{\omega}a+a\mathcal{D}_{\omega}$.
∎

###  5.5 Dirac cohomology

Let $(X,\pi_{X})$ be a representation of $\mathbb{H}$, We say that $X$ is an
admissible module if the decomposition of $X$ into
$\Omega_{\mathbb{H}}$-generalized eigenspaces

$X=\bigoplus_{\lambda\in\mathbb{C}}X_{\lambda}$

is such that every $X_{\lambda}$ is finite dimensional.

######  Definition 5.15.

For an irreducible $\mathbb{H}$-representation $(X,\pi_{X})$, let
$\chi:Z(\mathbb{H})\to\mathbb{C}$ be the infinitesimal character
$\pi_{X}|_{Z(\mathbb{H})}$.

######  Definition 5.16.

Let $X$ be an admissible $\mathbb{H}$-module and let $S$ be a spinor for
$\mathcal{C}$, then $X\otimes S$ is a $\mathbb{H}\otimes\mathcal{C}$ module
and $\mathcal{D}_{\omega}\in\mathbb{H}\otimes\mathcal{C}$ acts

$(D_{\omega})_{X}:X\otimes S\to X\otimes S,$

The Dirac $\omega$-cohomology of $X$ (and $S$) is defined as

$H(X,\omega)=\ker(D_{\omega})_{X}/\ker(D_{\omega})_{X}\cap\operatorname{im}(D_{\omega})_{X}.$

Since $\mathcal{D}_{\omega}$ $\operatorname{sgn}$-commutes with
$\widetilde{G}$ the Dirac $\omega$-cohomology of an admissible $X$ is a finite
dimension $\widetilde{G}$ module, or zero.

######  Theorem 5.17.

Let $X$ be an admissible irreducible $\mathbb{H}$-module. Let $\mathcal{B}$ be
an algebra as in Definition 5.10 Suppose $H(X,\omega)\neq 0$. If there is a
nonzero $\sigma\otimes\chi$-isotypic $\mathbb{C}\widetilde{G}\otimes B$
component in $H(X,\omega)$ for $\sigma\in\operatorname{Irr}(\widetilde{G})$
and $\chi\in\operatorname{Spec}B$, then

$\chi_{X}=\zeta_{\omega}^{*}(\sigma).$

###### Proof.

Once one has proved a Vogan conjecture for $\mathcal{D}_{\omega}$ (that is
Theorem 5.14) then the proof is identical to [11, Theorem 3.14] given [11,
Theorem 3.8]. ∎

Recall Theorem 4.15 states that all $V$-admissible elements for
$\mathbb{C}\tilde{G}_{-}$ are even. Suppose that $\omega$ is $V$-admissible,
because every group element occurring in $\omega$ is even then $\omega$ and
$\mathcal{D}$ commute.

######  Proposition 5.18.

If $X$ is a $\bullet$-unitary $\mathbb{H}$-module, then
$H(X,\omega)=\ker(\mathcal{D}_{\omega})$.

###### Proof.

If $X$ is $\bullet$-Hermitian then the image and kernel of
$\mathcal{D}_{\omega}$ are orthogonal with respect to
$\langle\,,\,\rangle_{X\otimes S}$ and hence
$\ker(\pi\mathcal{D}_{\omega})\cap\operatorname{im}(\pi\mathcal{D}_{\omega})={0}$
∎

######  Proposition 5.19.

Let $(\pi_{X},X)$ be a $\bullet$-unitary module for $\mathbb{H}$. Suppose that
there exists an admissible element $\omega\in\mathbb{C}\widetilde{G}_{-}$ such
that $\pi_{X}(\omega)\neq 0$. Then, there exists an admissible
$\lambda\omega\in\mathbb{C}\widetilde{G}_{-}$ such that
$H(X,\lambda\omega)\neq 0$.

###### Proof.

Since $X$ is Hermitian then
$H(X,\omega)=\ker(\mathcal{D}_{C})=\ker(\mathcal{D}_{C}^{2})$. We study the
kernel of the operator
$(\mathcal{D}_{\omega}-2\rho(\omega))\mathcal{D}_{\omega}=\mathcal{D}^{2}-\rho(\omega)^{2}$.
The elements $\mathcal{D}^{2}$ and $\rho(\omega)$ commute, hence have
simultaneous eigenvalues. Given $\pi(\rho(\omega))^{2}\neq 0$ it has positive
real eigenvalues. Let $\pi_{X\otimes S}\mathcal{D}^{2}$ and $\pi_{X\otimes
S}(C)^{2}$ have positive simultaneous eigenvalues $d\geq 0$ and $c>0$
respectively. One can modify $\mathcal{D}_{\omega}$ to
$\mathcal{D}_{\lambda\omega}$, with $\lambda=\sqrt{\frac{c}{d}}$. This ensures
that

$\mathcal{D}_{0}^{2}-\lambda^{2}\rho(\omega)^{2}$

has a non-zero kernel on $X\otimes S$. Thus proving that there exists a non-
zero kernel for the operator
$(\mathcal{D}_{\omega}-2\rho(\omega))\mathcal{D}_{\omega}$. Since the
composition of injective functions is injective, we find that one of
$(\mathcal{D}_{\omega}-2\rho(\omega))$ or $\mathcal{D}_{\omega}$ has a non-
zero kernel. ∎

######  Remark 5.20.

When $G=S_{n}$, we have shown that the set of $V$-admissible elements in
$\mathbb{R}\tilde{W}_{-}$ is equal to conjugacy classes associated to even
partitions. From [17], $Z_{0}(\mathbb{R}\tilde{W}_{-})$ is equal to
$\\{\sigma(M_{1}^{2},\ldots,M_{n}^{2})\mid\sigma\textup{ real symmetric
polynomial}\\}$. The element $\sigma(M_{1}^{2},\ldots,M_{n}^{2})$ acts on an
irreducible representation by evaluating $\sigma$ at specific real values [6,
Corollary 6.3]. In particular, we can conclude that for every
$\mathbb{C}\tilde{W}$-module there is an admissible element that does not act
by zero.

######  Theorem 5.21.

Let $\mathbb{H}$ be a degenerate affine Hecke algebra associated to the
symmetric group $S_{n}$. Then for any unitary $\mathbb{H}$-module $X$, there
exists a Vogan operator $\mathcal{D}_{\omega}$ such that

$H(X,\omega)\neq 0.$

###  5.6 Vogan inequalities

Throughout this section we assume that $\kappa_{g}=0$ for all
$g\in\ker\pi_{V}$.

Recall Theorem 4.15 states that all $V$-admissible elements for
$\mathbb{C}\tilde{G}_{-}$ are even. Let $\omega=\sum c_{\tilde{g}}\tilde{g}$,
this then implies $\operatorname{sgn}(\tilde{g})=1$ for every
$c_{\tilde{g}}\neq 0$. Suppose that $\omega$ is $V$-admissible, because every
group element occurring in $\omega$ is even then $\omega$ and $\mathcal{D}$
commute. On a $\star$-unitary module $X\otimes S$, the operator,
$\mathcal{D}_{\omega}^{2}$ is positive.

######  Proposition 5.22.

Let $(X,\pi_{X})$ be a $\bullet$-unitary module, then $(X\otimes
S,\pi_{X\otimes S})$ is $\star$-unitary and
$\mathcal{D}_{\omega}^{2}=\mathcal{D}^{2}+2\mathcal{D}\rho(\omega)+\rho(\omega)^{2}\geq
0.$ Hence for every $V$-admissible $\omega$ the following inequality holds ,

$\pi_{X}(\Omega_{\mathbb{H}})\leq\pi_{X\otimes
S}\Omega_{\tilde{G}}+\pi_{X\otimes S}((\omega+2\mathcal{D})\omega).$

Specialising the above inequality to $\omega=0$ gives the Dirac inequality.
Proposition 5.19 states that if there exists an $\omega$ such that
$\pi_{X\otimes S}(\omega)\neq 0$ then there exists a $V$-admissible element
such that the above inequality becomes an equality. For the Hecke algebra
associated to the symmetric group there is always non-zero
$\omega$-cohomology, this implies that for a particular choice of $\omega$ it
is always possible to make the inequality above strict.

##  6 Parthasarathy and Vogan operators

######  Proposition 6.1.

Consider elements of the form $\mathcal{D}+\rho A_{\widetilde{G}}$, where
$A_{\widetilde{G}}\in\mathbb{C}\widetilde{G}_{-}$. The only operator of this
form that is both a Vogan and Parthasarathy operator is

$\mathcal{D}+0=\mathcal{D}.$

###### Proof.

Suppose that $\mathcal{D}+\rho A_{\widetilde{G}}$ is a Parthasarathy operator,
by construction $\operatorname{sgn}$ of every group element in $A_{\tilde{g}}$
is $-1$. Now suppose that $\mathcal{D}+\rho A_{\widetilde{G}}$ is a Vogan
operator, then by Theorem 4.15, $\operatorname{sgn}$ of every group element is
$1$. Therefore $A_{\widetilde{G}}$ has no nonzero coefficient of group
elements and hence must be zero.

∎

## Acknowledgements

This research was supported by the Heilbronn Institute for Mathematical
Research.

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Kieran Calvert<EMAIL_ADDRESS>
Department of Mathematics, University of Manchester, Alan Turing Building,
Oxford Road, Manchester, UK.