# Multifractality Breaking from Bounded Random Measures

L. Moriconi  Instituto de Física, Universidade Federal do Rio de Janeiro,
C.P. 68528, CEP: 21945-970, Rio de Janeiro, RJ, Brazil

###### Abstract

Multifractal systems usually have singularity spectra defined on bounded sets
of Hölder exponents. As a consequence, their associated multifractal scaling
exponents are expected to depend linearly upon statistical moment orders at
high enough orders – a phenomenon referred to as the linearization effect.
Motivated by general ideas taken from models of turbulent intermittency and
focusing on the case of two-dimensional systems, we investigate the issue
within the framework of Gaussian multiplicative chaos. As verified by means of
Monte Carlo simulations, it turns out that the linearization effect can be
accounted for by Liouville-like random measures defined in terms of upper-
bounded scalar fields. The coarse-grained statistical properties of Gaussian
multiplicative chaos are furthermore found to be preserved in the linear
regime of the scaling exponents. As a related application, we look at the
problem of turbulent circulation statistics, and obtain a remarkably accurate
evaluation of circulation statistical moments, recently determined with the
help of massive numerical simulations.

##  I Introduction

Soon after the realization that strange attractors should be characterized by
a set of generalized dimensions rather than a single fractal dimension
hentschel-procaccia ; grassberger ; schertzer-lovejoy , related concepts were
further developed and applied to the problem of homogeneous and isotropic
turbulence benzi_etal ; frisch-parisi ; meneveau-sreenivasan1 ; chhabra_etal ;
meneveau-sreenivasan2 . In the latter context, domains of the fluid velocity
field which have prescribed singular Hölder exponents have been conjectured to
be fractal. This is the essential content of the multifractal approach to
turbulence, which allows one to recover, within inertial range scales, the
anomalous scaling properties of the turbulent energy cascade O62 ; K62 ;
anselmet_etal ; frisch .

The multifractal mindset has since then crossed the borders of its fluid
dynamical birth place and is by now a valuable tool for the investigation of
problems in fields as diverse as seismology, meteorology, ecology, condensed
matter physics, dynamical systems, etc. sism_met ; mat_cond ; sis_din ; eco ;
signals . In the particular case of turbulence, its worth emphasizing that
multifractality has been noted to be closely related to the Onsager’s long-
standing conjecture on flow singularities onsager ; eyink_sreenivasan ; eyink
and to the phenomenon of spontaneous stochasticity, a subject of growing
interest, as far as it leads to a breaking of the deterministic paradigm of
classical mechanics, in a sense which is even stronger than the one usually
implied by chaotic behavior bernard_etal ; chaves_etal ; eyink_drivas ;
thalabard_etal ; eyink_bandak .

Multifractal modeling, however, is expected to be broken by extreme events: at
enough high orders, statistical moments of the physical observables of
interest are found to depend linearly upon its moment orders, at variance with
the typical non-linear profiles predicted for multifractal systems molchan1 ;
molchan2 . While such a linearization effect can be actually explained in a
natural way within the multifractal formalism ossiander_etal ; Lahermes_etal1
; abry_etal ; lashermes_etal2 ; muzy_etal ; bacry_etal ; angeletti_etal , its
account from alternative perspectives on multifractality has been puzzingly
and, as a consequence, a point of concern in applications. We have in mind,
more specifically, the connection betweeen mutifractality and the theory of
Gaussian multiplicative chaos (GMC) kahane ; rhodes_vargas , which has been a
fruitful tool in the development of finance duchon-robert , turbulence
chevillard_etal1 ; pereira_etal ; chevillard_etal2 ; apol-mori  and even
quantum gravity models barral_etal . Our aim, in this work, is to address the
linearization effect in the framework of GMC and to illustrate, as a
meaningful case study, an application of the proposed solution to the problem
of turbulent circulation statistics Iyer_etal ; migdal ; apol_etal .

This paper is organized as follows. In the next section, we clarify the
problem we are interested to study, recalling some relevant technical details
of the multifractal formalism and the theory of GMC, with specific attention
to the case of two-dimensional modeling. In Sec. III, heuristic arguments are
introduced, based on phenomenological descriptions of turbulent cascades
benzi_etal ; O62 ; K62 , which motivate us to put forward, as a proposal, the
necessary ingredients for the realization of the linearization effect along
the lines of GMC. Our conjectures are fully confirmed in Sec. IV by means of
Monte Carlo simulations. We then carry out, in Sec. V, an application of the
freshly derived results to the problem of circulation fluctuations in
turbulence, obtaining excellent comparisons with evaluations obtained from
previous numerical experiments Iyer_etal . Finally, in Sec. VI, we summarize
our findings and point out directions of further research.

##  II Problem Setup

To set the stage for the issues we aim to address in this paper, let us
consider the example of a $d$-dimensional positive-definite multifractal
scalar field $\psi(x)$, described by some translation invariant probability
measure, in such a way that the statistical moments of the renormalized fields

$\psi_{a}(x)\equiv\frac{1}{a^{d}}\int_{{\cal{D}}_{a}}d^{d}x\psi(x)\ ,\ $ (1)

where ${\cal{D}}_{a}$ is the spatial domain $|x^{\prime}-x|\leq a$, behave as

$\mathbb{E}[(\psi_{a}(x))^{q}]\sim a^{\tau_{q}}\ .\ $ (2)

The scaling exponents $\tau_{q}$ can be derived within the multifractal
language as follows frisch-parisi . Let $D(h)$, referred to as the singularity
spectrum, be the fractal dimension of the set of points which have Hölder
exponent $h$ in the ensemble realizations of $\psi(x)$. Regarding $h$ as a
random variable to be sorted when an arbitrary region of size $a$ is probed,
the probability to find the local scaling behavior $\psi_{a}(x)\sim
a^{h^{\prime}}$ where $h^{\prime}\in[h,h+dh]$ is, thus, $\rho(h)dh\sim
a^{d-D(h)}dh$. We get, from these assumptions, that

$\mathbb{E}[(\psi_{a}(x))^{q}]\sim\int dh\rho(h)a^{qh}\sim\int
dha^{hq+d-D(h)}\ .\ $ (3)

At small enough scales, the dependence of the above expectation values upon
$a$ can be estimated with the help of the saddle-point method, which leads to
(2), with

$\tau_{q}={\hbox{inf}}_{h}[hq+d-D(h)]\ .\ $ (4)

The scaling exponent $\tau_{q}$, therefore, is nothing but the Legendre
transform of the fractal codimension $d-D(h)$.

General arguments frisch  tell us that $\tau_{q}$ is a concave function of the
moment order $q$. For the sake of clarity, we adopt here the convention that
multifractality refers to the case of scaling exponents which are strictly
concave, that is, $d^{2}\tau_{q}/dq^{2}<0$. The singularity spectrum $D(h)$
is, accordingly, a strictly concave function of $h$.

As already alluded in the introductory section, multifractality is expected to
be broken at high enough moment orders. More concretely, this stands for the
fact that for $q\geq q_{c}$, where $q_{c}$ is a model-dependent critical
moment order, $\tau_{q}$ becomes a linear function of $q$ molchan1 ; molchan2
; ossiander_etal ; Lahermes_etal1 ; abry_etal ; lashermes_etal2 ; muzy_etal ;
bacry_etal ; angeletti_etal . An essential explanation of the linearization
effect, apprehended from the aforementioned works, is that the domain of the
singularity spectrum function is usually bounded from below by a limiting
Hölder exponent $h_{\ast}$. Therefore, taking into account that as $q$ grows,
the value of $h$ which minimizes the RHS of (4) gets smaller, it turns out
that at some critical moment order $q_{c}$ the minimizer in (4) saturates to
$h_{\ast}$, leading to the monofractal relation

$\tau_{q}=h_{\ast}q+d-D(h_{\ast})\ ,\ $ (5)

for $q\geq q_{c}$.

The singularity spectrum can be well approximated in very many instances by a
parabolic function of $h$ over a broad range of Hölder exponents, so that
fluctuations of $\psi(x)$ can be effectively described by lognormal
probability distribution functions. In equivalent words, the scaling exponents
$\tau_{q}$ are given, in this approximation, by quadratic functions of $q$. In
this connection, one notes that the combined existence of pointwise lognormal
distributions for $\psi(x)$ and the scaling behavior of the coarse-grained
variables (2) can be reproduced with the help of the Liouville measures as
defined in the theory of GMC rhodes_vargas .

Centering our attention on two-dimensional modeling, the GMC approximation
means, in practice, that $\psi(x)$ may be expressed as the Liouville measure
density

$\psi(x)=\psi_{0}\exp\left\\{\gamma\phi(x)-\frac{\gamma^{2}}{2}\mathbb{E}[\phi^{2}]\right\\}\
,\ $ (6)

where $\psi_{0}>0$ and $\gamma$ are arbitrary parameters, and $\phi(x)$ is a
free scalar field zinn-justin  with fluctuations governed by the functional
probability measure

$d\mu[\phi]=D[\phi]\exp\\{-S[\phi]\\}\ ,\ $ (7)

where

$S[\phi]=\frac{1}{2}\int_{{\cal{D}}_{L}}d^{2}x(\partial_{i}\phi)^{2}\ .\ $ (8)

Periodic boundary conditions are assumed for the scalar field $\phi(x)$ in the
domain ${\cal{D}}_{L}$, which is furthermore discretized in a lattice of
lattice parameter $\eta$ (a necessary technical detail for the ultraviolet
regularization of the free field Green’s functions). Taking $\eta\ll a\ll L$,
it follows from (1) and (6-8) that (2) is satisfied, that is,

$\mathbb{E}[(\psi_{a}(x))^{q}]=c_{q}\psi_{0}^{q}\left(\frac{a}{L}\right)^{\tau_{q}}\
,\ $ (9)

where $c_{q}$ is an unimportant dimensionless constant (for our purposes) and

$\tau_{q}=\frac{\gamma^{2}}{4\pi}q(1-q)\ .\ $ (10)

The bare field $\psi(x)$ can be identified with the ultraviolet regularized
field $\psi_{\eta}(x)$. A scaling relation similar to (9),

$\mathbb{E}[(\psi(x))^{q}]=\left(\frac{\eta}{L}\right)^{\tau_{q}}\ ,\ $ (11)

can then be derived from (6-8) as well.

We wonder, thus, if it is possible to implement modifications in the GMC
computational scheme based on Eqs. (6-8) so as to get a crossover of the
scaling exponents $\tau_{q}$, as $q$ grows, from (10) to (5), while still
having power laws like (9) and (11). In the next section, we propose a
solution to this problem, relying on heuristic arguments inspired on well-
known phenomenological models of turbulent intermittency.

The apparent methodological restriction represented by the use of turbulence
phenomenology should not be a matter of concern at all, since multifractal
phenomena and techniques are usually traded without much difficulty among
models of completely different nature.

##  III Bounded Cascades

We briefly outline, in the following subsections A and B, two phenomenological
views on the turbulent cascade, which when placed vis a vis the theory of GMC
and the multifractal formalism, give relevant hints on how to establish the
linearization effect in the framework of the GMC, a task addressed in
subsection C.

###  III.1 The Obukhov-Kolmogorov lognormal model of turbulent intermittency

If $\psi(x)$ is used to model the turbulent dissipation field in homogeneous
and isotropic turbulence, commonly denoted by $\epsilon(x)$, relations (1) and
(2) yield a precise formulation of the Kolmogorov refined similarity
hypothesis, a central point in the Obukhov-Kolmogorov (OK62) modeling of
turbulent intermittency O62 ; K62 .

In the OK62 phenomenology, multiplicative cascade fluctuations of the energy
transfer rates per unit mass and unit time, $\epsilon_{a}$ and $\epsilon_{b}$,
across two different length scales $a$ and $b$, respectively, are related as

$\epsilon_{a}=\epsilon_{b}W_{1}W_{2}...W_{n}\ .\ $ (12)

The $W^{\prime}s$ are lognormally i.i.d. random variables, with unit mean, and
$n=\log_{2}(b/a)$, taken to be a positive integer, gives the number of modeled
steps in the turbulent energy cascade between the scales $a$ and $b$. They are
assumed to lay within the inertial range scales, that is, $\eta\leq a<b\leq
L$, where $L$ and $\eta$ define the integral and dissipative length scales,
respectively, of the turbulent flow. Eq. (12) is to be understood in the
probabilistic sense as an equality in law for $\epsilon_{a}$ and
$\epsilon_{b}$.

Considering $b=L$, that is, the scale where energy is injected into the flow
with non-fluctuating energy transfer rate $\epsilon_{L}$, then it is a
straightforward exercise to show, from (12), that

$\mathbb{E}[\epsilon_{a}^{q}]\sim\epsilon_{L}^{q}\left(\frac{a}{L}\right)^{\tau_{q}}$
(13)

holds for $\eta\leq a\leq L$, in the same fashion as (9) and (11), where
$\tau_{q}$ is given as in (10), with

$\gamma^{2}=2\pi\log_{2}\mathbb{E}[W^{2}]\ .\ $ (14)

As a relevant note for future use, we introduce the Gaussian random variable
$X_{p}$, through

$W_{p}\equiv\exp(\gamma X_{p})\ .\ $ (15)

The OK62 cascade argument (12) can in this way be recalled to suggest, taking
a look at (6), that pointwise fluctuations of $\psi(x)$ can be derived from
$\psi\sim\exp(\gamma\phi)$, where

$\phi=\sum_{p=1}^{\log_{2}(L/\eta)}X_{p}\ .\ $ (16)

###  III.2 The random $\beta$-model of turbulent intermittency

An alternative OK62-like cascade picture of the energy transfer rate
fluctuations across scales, as synthetized in Eq. (12), can be put forward in
order to render it closer to contemporary multifractal ideas and in compliance
with general physical principles like energy conservation.

In the random $\beta$-model benzi_etal , an arbitrary energy-containing eddy
defined at length scale $a$ produces, during its lifetime, a random number
$M_{a}\leq 2^{d}$ of descendent eddies (in $d$ dimensions), all of them
defined at length scale $a/2$. Energy conservation implies that the power
supplied by the mother-eddy to its descendents has to be same as the total
power supplied by the latter ones to their further descendents and, as a
consequence, $M_{a}\epsilon_{a/2}(a/2)^{d}=\epsilon_{a}a^{d}$, that is

$\epsilon_{a/2}=\beta_{a}^{-1}\epsilon_{a}\ ,\ $ (17)

where $\beta_{a}=M_{a}/2^{d}$ is the fraction of volume that the whole group
of descendent eddies (the “sibling-eddies”) occupy with the respect to the
volume of their mother-eddy.

Assuming that generation after generation the $\beta^{\prime}s$ are completely
independent and randomly distributed according to the same probability density
function $f(\beta)$, Eq. (12) still holds for the energy transfer rates of
each individual eddy, with (subindices suppressed)

$W=\beta^{-1}\ .\ $ (18)

We have, therefore,

$\epsilon_{a}=\left[\prod_{i=1}^{n}\beta_{i}^{-1}\right]\epsilon_{L}\ ,\ $
(19)

where $n=\log_{2}(L/a)$. Energy transfer rates have, now, statistical moments

$\displaystyle\mathbb{E}[\epsilon_{a}^{q}]\sim\epsilon_{L}^{q}\left[\int_{0}^{1}d\beta
f(\beta)\beta^{1-q}\right]^{n}$
$\displaystyle=\epsilon_{L}^{q}\mathbb{E}[\beta^{1-q}]^{n}\sim\epsilon_{L}^{q}\left(\frac{a}{L}\right)^{\tau_{q}}\
,\ $ (20)

where

$\tau_{q}=-\log_{2}\mathbb{E}[\beta^{1-q}]\ .\ $ (21)

Note that the expectation value (20) takes into account the fact that in the
random $\beta$-model eddies are not space-filling structures.

The particular modeling case where $\beta$ is fixed to some arbitrary value
$\beta_{0}$ frisch_beta , associated to the probability distribution function

$f(\beta)=\delta(\beta-\beta_{0})\ ,\ $ (22)

gives, in view of (21), the linear scaling exponents

$\tau_{q}=(q-1)\log_{2}\beta_{0}\ .\ $ (23)

Furthermore, still considering the situation of fixed $\beta$, we infer that a
mother-eddy at the integral scale $L$ (the “mother of all mothers”) is the
source, along the turbulent cascade, of a number

$N_{a}\sim(2^{d}\beta_{0})^{\log_{2}(L/a)}\sim\left(\frac{L}{a}\right)^{d+\log_{2}\beta_{0}}$
(24)

of descendent eddies at length scale $a$. The scaling law (24) indicates that
the fractal dimension of the energy-containing eddies is, here,

$d_{F}=d+\log_{2}\beta_{0}\ .\ $ (25)

###  III.3 The linearization effect in the theory of GMC

As discussed in Sec. II, the linearization effect takes place when statistical
moments get dominated by fluctuations associated to the most singular set of
configurations, which are the ones which have the minimum available Hölder
exponent, denoted in Eq. (5) by $h_{\ast}$. Due to the concavity properties of
the singularity spectrum, we expect the fractal dimension of the most singular
set, $D(h_{\ast})$, to be the smallest allowed one (for the evaluation of
positive order moments).

We also note that Eq. (25) can be used to establish a mapping between values
of $\beta$, from the side of the random $\beta$-model, to the fractal
dimensions encompassed by the singularity spectrum $D(h)$, from the side of
the multifractal formalism. In the language of the random $\beta$-model, the
linearization effect follows from the existence of a minimum value of $\beta$,
say $\beta_{\ast}$, obtained from

$D(h_{\ast})=d+\log_{2}\beta_{\ast}\ .\ $ (26)

Taking (18) into account, we conclude that the cascade factors $W^{\prime}s$
are, under these conditions, upper bounded random variables, viz., $W\leq
1/\beta_{\ast}$. Correspondingly, we see, from the context of the OK62
phenomenology, that bounded $W^{\prime}s$ should be related to bounded scalar
fields $\phi(x)$ in the GMC setup, as indicated by (15) and (16).

Relying upon the above heuristic considerations, we are, now, ready to propose
a modified version of the two-dimensional GMC, as given by Eqs. (6-8), in
order to accommodate in its formal structure the linearization effect. To do
so, we actually keep the definition of the functional probability measure (7),
but

(i) replace the Liouville measure (6) by the more general expression

$\psi(x)=\frac{\psi_{0}}{\mathbb{E}[\tilde{\psi}(x)]}\tilde{\psi}(x)\ ,\ $
(27)

where

$\tilde{\psi}(x)=\exp[\gamma\phi(x)]\ ;\ $ (28)

(ii) replace the Euclidean action, Eq. (8), by

$S[\phi]=\int d^{2}x\left[\frac{1}{2}(\partial_{i}\phi)^{2}+V(\phi)\right]\ ,\
$ (29)

where

$V(\phi)=\begin{cases}0\ ,\ \text{if $\phi<\phi_{0}$}\ ,\ \\\ V_{0}\ ,\
\text{if $\phi\geq\phi_{0}$}\ ,\ \\\ \end{cases}$ (30)

with $V_{0}\rightarrow\infty$ and

$\phi_{0}=C\ln(L/\eta)\ ,\ $ (31)

where $C$ is an adjustable positive constant (observe that (31) follows from
(16) by taking $X_{p}=C\ln 2$).

In short words, we have just postulated that the Liouville measure (27) gets
upper bounded due to the existence of a scalar field threshold $\phi_{0}$, and
that it fluctuates as usually determined by the free field action (8), if
$\phi(x)<\phi_{0}$ in an arbitrary neighborhood of $x$.

An analytical treatment of the modified GMC scenario, as defined by Eqs.
(27-31), is challenging. However, it is possible to proceed along with Monte
Carlo numerical validations, as detailed next.

##  IV Monte Carlo Simulations

We have performed Monte Carlo simulations to study the fluctuations of the
non-normalized Liouville measure (28), with $\gamma=1$, using (29-31), for the
pure GMC ($\phi_{0}=\infty$) and modified GMC ($\phi_{0}<\infty$) cases.

Statistical ensembles with configurations of $\tilde{\psi}(x)$ have been
produced for systems of three different sizes: $L/\eta=30,50,$ and $100$,
through the application of the standard Metropolis algorithm binder-heermann .
An educated guess for the value of $C$ in (31) gives

$\phi_{0}>\sqrt{\mathbb{E}_{0}[\phi^{2}]}\ ,\ $ (32)

where $\mathbb{E}_{0}[\cdot]$ stands for expectation values taken in the pure
GMC scheme. The rationale for (32) is that at low enough orders, statistical
moments of $\tilde{\psi}(x)$ are expected to be approximately described by
quadratic scaling exponents like the ones of the pure GMC case, since the
scalar field $\phi(x)$ will very rarely fluctuate beyond the standard
deviation range, $\sqrt{\mathbb{E}_{0}[\phi^{2}]}$. On the other hand, as the
moment order grows, larger fluctuations of $\phi(x)$ come into play, reaching
more frequently the upper bound $\phi_{0}$, thus opening the way to the onset
of the linearization effect. A direct computation yields

$\mathbb{E}_{0}[\phi^{2}]=\frac{1}{2\pi}\ln\left(\frac{L}{\eta}\right)\ .\ $
(33)

Taking, $C\equiv 2/\ln(30)$, one can then easily check that the inequality
(32) holds in fact for all the studied system sizes.

Each Monte Carlo run consisted of $10^{7}$ iterations, sampled at every other
10 steps, which evolved from the initial state $\phi(x)=0$. The field
derivatives in the action (29) were evaluated by means of central differences.
Monte Carlo variations of $\phi(x)$ (defined at lattice sites) were given by
independent pseudorandom numbers uniformly distributed in the interval
$[-1,1]$.

Statistical moments of the bare and the coarse-gained non-renormalized
Liouville measures, $\tilde{\psi}(x)$ and

$\tilde{\psi}_{a}(x)\equiv\frac{1}{a^{d}}\int_{{\cal{D}}_{a}}d^{d}x\tilde{\psi}(x)\
,\ $ (34)

respectively, are reported in Figs. 1 and 2. As evidenced from Fig. 1, the
linearization effect is well reproduced in the modified GMC framework for the
moment order range $5\leq q\leq 11$ ($q=11$ is the largest analysed order).
Fig. 1 also shows the excellent collapse of data for the investigated systems,
which supports the finite-size dependent definition of the upper bound (31).

Figure 1: Solid lines/symbols and open symbols refer, respectively, to the
pure and modifed GMC cases. Symbols yield scaling exponents obtained from
Monte Carlo simulations. The solid line (either in the main plot or the inset)
is the predicted parabolic scaling exponent $q^{2}/4\pi$, for the moments of
(28) in the pure GMC case, as it can be derived from Eq. (10), with
$\gamma=1$. Dashed lines (in the main plot and inset) are linear and parabolic
fits.

Figure 2: Scaling regimes for the moments of the coarse-grained non-normalized
Liouville measures $\tilde{\psi}_{a}(x)$, as evaluated from Monte Carlo
simulations (symbols) performed on a system of size $L/\eta=100$. The solid
straight lines have slopes which are the scaling exponents obtained from the
linear fit shown in Fig. 1, for the moment orders $q=5,7,$ and $9$.

The Monte Carlo results depicted in Fig. 2 indicate that $\tilde{\psi}_{a}(x)$
scales with the same scaling exponent as $\tilde{\psi}(x)\equiv\psi_{\eta}(x)$
at small length scales ($a/L<0.1$), even for moment orders where the
linearization effect is observed.

The linearization effect for the coarse-grained bounded Liouville measures is
a remarkable phenomenon, which has an immediate impact in turbulence modeling,
since it bridges the linearization effect for scaling quantities like the
velocity structure functions to the linearization effect for the turbulent
dissipation field, if one assumes, of course, that the Kolmogorov refined
similarity hypothesis is still valid. We examine, in the following, this
interesting phenomenological point in connection with a recently discussed
model for the turbulent fluctuations of the circulation variable apol_etal .

##  V Turbulent Circulation Statistics

The relevance of the circulation variable acheson  as a multiscale
“mathematical probe” of turbulent vortical structures, pointed for the first
time some 25 years ago migdal2 , has recently found renewed interest with the
advent of high performance computing and improved data storage capability
Iyer_etal . Novel modeling ideas have been put forward Iyer_etal ; migdal ;
apol_etal , including possible connections between the statistics of
circulation in classical and quantum turbulent flows muller_etal .

Let us center our attention on the particular definition of circulation as

$\Gamma_{R}\equiv\int_{\mathcal{C}}d^{2}{\bf{r}}\,\omega({\bf{r}})\ ,\ $ (35)

where $\mathcal{C}$ is a disk of radius $R$ and $\omega({\bf{r}})$ is the
component of vorticity which is normal (with arbitrary orientation) to the
plane that contains $\mathcal{C}$. The scaling form for the circulation
moments,

$\mathbb{E}[|\Gamma_{R}|^{q}]\sim R^{\lambda_{q}}\ ,\ $ (36)

is observed to hold for the inertial range of scales $\eta\ll R\ll L$
Iyer_etal . We are here mainly interested to model the scaling exponents
$\lambda_{q}$ in (36). The Kolmogorov phenomenological description of
turbulence (K41) frisch  yields $\lambda_{q}=4q/3$, which has been noted to be
a very good approximation only for $q\leq 4$ Iyer_etal .

Tracing back circulation fluctuations to the presence of vortex tubes, it was
proposed, in Ref. apol_etal , that the vorticity field in (35) can be
effectively represented, for the purpose of evaluating the statistical moments
(36), as

$\omega({\bf{r}})\sim\xi_{R}\tilde{\omega}({\bf{r}})\ ,\ $ (37)

where

$\xi_{R}\equiv\frac{1}{\pi
R^{2}}\int_{\mathcal{C}}d^{2}{\bf{r}}\,\sqrt{\epsilon({\bf{r}})}$ (38)

is a functional of the dissipation field $\epsilon({\bf{r}})$, modeled as a
Liouville measure density, and $\tilde{\omega}({\bf{r}})$ is an independent
Gaussian random field, with vanishing mean and correlator

$\mathbb{E}[\tilde{\omega}({\bf{r}})\tilde{\omega}({\bf{r}}^{\prime})]\sim|{\bf{r}}-{\bf{r}}^{\prime}|^{-\alpha}\
.\ $ (39)

The scaling exponent $\alpha$ in (39) can be determined, as we will see in a
moment, from the imposition of general phenomenological constraints comment .

Figure 3: Scaling exponents for the circulation moments. Symbols give the
values obtained through direct numerical simulations Iyer_etal . The dashed
line is the K41 linear profile, $\lambda_{q}=4q/3$, while the solid line is
the prediction of the present model, as given in Eq. (44).

Since powers of Liouville measures are Liouville measures as well, as it can
be clearly seen from the definition (28), we are able to obtain (36) by
putting together (35), (37), and the coarse-grained Liouville measure (38),
with

$\lambda_{q}=\tau_{q/2}+(4-\alpha)\frac{q}{2}\ ,\ $ (40)

where $\tau_{q}$ is the energy transfer rate exponent formally introduced in
(13). We determine, now, the crossover moment order $q_{c}$ that defines the
onset of the linearization effect. For $q<q_{c}$ the OK62 lognormal model
gives O62 ; K62 ; frisch

$\tau_{q/2}=\frac{\mu}{8}q(2-q)\ ,\ $ (41)

where $\mu=0.17\pm 0.01$ tang_etal . The relation between $q_{c}$ and the
Hölder exponent minimizer $h_{\ast}$ (associated to singularities of the
dissipation field) can be worked out without much difficulty; we get

$q_{c}=1-2\frac{h_{\ast}}{\mu}\ .\ $ (42)

Determinations of the singularity spectrum of the energy dissipation field
from high Reynolds number experiments was accomplished in Ref. meneveau-
sreenivasan1 ; chhabra_etal . It turns out, from a careful analysis of the
reported data, that $h_{\ast}\simeq-0.5$. This leads us, from (42), to
$q_{c}\simeq 6.88$.

It remains to discuss the yet undetermined exponent $\alpha$. Considering that
there is no anomalous scaling for the third order velocity structure
functions, as signalized in Kolmogorov’s 4/5 law frisch , we postulate that
$\lambda_{3}=4$, exactly as in K41 phenomenology comment2 . Using (40) and
(41) with $q=3$, we obtain, thus,

$\alpha=\frac{4}{3}-\frac{\mu}{4}\ .\ $ (43)

Collecting all the above pieces of information, we write down the circulation
scaling exponent as

$\lambda_{q}=\begin{cases}\bar{\lambda}_{q}\equiv\frac{4}{3}q+\frac{\mu}{8}q(3-q)\
,\ \text{if $q<q_{c}$}\ ,\ \\\
\frac{1}{2}\left(h_{\ast}+\frac{8}{3}+\frac{\mu}{4}\right)(q-q_{c})+\bar{\lambda}_{q_{c}}\
,\ \text{if $q\geq q_{c}$}\ .\ \end{cases}$ (44)

The comparison of the predicted values of $\lambda_{q}$ with the results of
massive numerical simulations Iyer_etal  is excellent, as shown in Fig. 3. The
transition in behavior of the statistical moments of circulation as their
moment orders are varied was actually observed for the first time in Ref.
Iyer_etal . We see, therefore, that it can be consistently explained as a
manifestation of multifractality breaking, or, in other words, the
linearization effect, within the modeling arena of GMC.

##  VI Conclusions

We have been able to address a variation of GMC, as described from relations
(27-31), which gives room for the linearization effect, a phenomenon commonly
observed in multifractal systems. The key technical point in the definition of
the modified GMC setting is the introduction of upper-bounded Liouville
measures.

Our line of reasoning has been closely motivated by cascade models of
turbulent intermittency and their connections with the mulifractal language
and the theory of GMC. We validated the modified picture of GMC by means of
straightforward Monte Carlo simulations and applied it to the problem of
turbulent circulation statistics. We developed, in this way, accurate
evaluations of the scaling exponents for the statistical moments of
circulation, previously established only through extensive numerical
simulations Iyer_etal .

Further work is in order. It would be very interesting to devise a
mathematically rigorous analysis of the linearization effect in the GMC, as
discussed in this work, and to extend it to general space dimensions.
Additional Monte Carlo simulations are also welcome to explore the validity
range (as the field bound $\phi_{0}$, the system size, and moment orders are
changed) of the modified GMC picture. The empirical (numerical) implementation
of bounded Liouville measures to models based on the theory of GMC should not
present any relevant technical or conceptual difficulty.

Acknowledgments

The author thanks G.B. Apolinário, R.M. Pereira, and V.J. Valadão for several
enlightening discussions. This work was partially supported by CNPq.

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# duplex Hecke Algebras of type $B$

Yu Xie, An Zhang and Bin Shu  School of Mathematical Sciences, East China
Normal University, No. 500 Dongchuan Rd., Shanghai 200241, China
<EMAIL_ADDRESS>School of Mathematical Sciences, East China Normal
University, No. 500 Dongchuan Rd., Shanghai 200241, China
<EMAIL_ADDRESS>School of Mathematical Sciences, Ministry of
Education Key Laboratory of Mathematics and Engineering Applications &
Shanghai Key Laboratory of PMMP, East China Normal University, No. 500
Dongchuan Rd., Shanghai 200241, China<EMAIL_ADDRESS>
###### Abstract.

As a sequel to [14], in this article we first introduce a so-called duplex
Hecke algebras of type $\mathrm{\mathsf{B}}$ which is a
$\mathbb{Q}(q)$-algebra associated with the Weyl group
$\mathscr{W}(\mathrm{\mathsf{B}})$ of type $\mathrm{\mathsf{B}}$, and
symmetric groups $\mathfrak{S}_{l}$ for $l=0,1,\ldots,m$, satisfying some
Hecke relations (see Definition 2.1). This notion originates from the
degenerate duplex Hecke algebra arising from the course of study of a kind of
Schur-Weyl duality of Levi-type (see [12]), extending the duplex Hecke algebra
of type A arising from the related $q$-Schur-Weyl duality of Levi- type (see
[14]). A duplex Hecke algebra of type $\mathrm{\mathsf{B}}$ admits natural
representations on certain tensor spaces. We then establish a Levi-type
$q$-Schur-Weyl duality of type $\mathrm{\mathsf{B}}$, which reveals the double
centralizer property between such duplex Hecke algebras and $\imath$quantum
groups studied by Bao-Wang in [1].

###### Key words and phrases:

duplex Hecke algebras, $\imath$quantum groups, $q$-Schur-Weyl duality

###### 2020 Mathematics Subject Classification:

20G05; 17B20;17B45; 17B50

This work is supported by the National Natural Science Foundation of China
(Grant No. 12071136 and 12271345), and by Science and Technology Commission of
Shanghai Municipality (No. 22DZ2229014).

## Introduction

###  0.1. Background

An algebraic group $G$ is called a semi-reductive group if $G$ is a semi-
direct product of a reductive closed subgroup $G_{0}$ and the unipotent
radical $U$. It is important to study semi-reductive algebraic groups and
their Lie algebras in lots of cases when the underground filed is of
characteristic $p>0$ (see [11]). Set $\underline{G}=G\times_{\nu}V$, where
$G=\text{GL}(V)$ and $\nu$ be the natural representation of $G$ on $V$. Then
this enhanced group $\underline{G}$ naturally becomes a semi-reductive group
(called an enhanced reductive group by Shu-Xue-Yao [12]). In order to study
polynomial representations of general linear groups, classical Schur algebras
are produced. By analogy of this, the $m$th tensor representations of an
enhanced group $\underline{G}$ naturally produce the so-called enhanced Schur
algebra $\mathcal{E}(n,m)$. In the course of study in representations of
enhanced Schur algebras, an algebraic model of degenerate duplex Hecke
algebras is introduced in [12], where it is denoted by $\mathcal{H}_{m}$.
Roughly speaking, for $\underline{V}$ a one-dimensional extension of $V$
associated with $\text{GL}(V)$, Shu-Xue-Yao established the following duality
in [12]:

$\Xi(\mathcal{H}_{m})={\mathrm{End}}_{\mathbb{C}\Phi(\text{GL}_{n}\times\textsf{G}_{\text{m}})}(\underline{V}^{\otimes
m});$ ${\mathrm{End}}_{\Xi(\mathcal{H}_{m})}(\underline{V}^{\otimes
m})=\mathbb{C}\Phi(\text{GL}_{n}\times\textsf{G}_{\text{m}}),$

where $\text{GL}_{n}$ and $\textsf{G}_{\text{m}}$ are the general linear group
and the one-dimensional multiplication group, respectively, and
$\text{GL}_{n}\times\textsf{G}_{\text{m}}$ is naturally regarded as a
connected closed subgroup of $\underline{G}$, and $\Phi$ and $\Xi$ denote the
natural representation of $\underline{G}$ and $\mathcal{H}_{m}$ on
$\underline{V}^{\otimes m}$ respectively. This duality is called a Levi-type
Schur-Weyl duality.

Recall that soon after quantum groups arising, Jimbo [7] established a duality
between quantum groups and Hecke algebras of type A, which is a
$q$-deformation of classical Schur-Weyl duality. This duality is called
$q$-Schur duality or Jimbo-Schur duality nowadays.

Combining Jimbo-Schur duality and Levi-type Schur-Weyl duality mentioned
above, Xue-Zhang [14] introduced duplex Hecke algebras (of type A) and
established the $q$-deformation of Levi-type Schur-Weyl duality111In [14], it
is called the doubled Hecke algebra. Here that notion is modified into the
current one “duplex Hecke algebras,” the latter of which can reflect more what
it looks like. More explanation about this notion can be refereed to [12]..
There is a natural question which seems nontrivial:

######  Question 0.1.

How to extend Xue-Zhang’s work to the case of type $\mathrm{\mathsf{B}}$, i.e.
how to define doubled Hecke algebras of type $\mathrm{\mathsf{B}}$, and how to
establish a $q$-deformation of Levi-type Schur-Weyl duality?

###  0.2. Our purpose

In this paper, our purpose is to answer the above question. We formulate a so-
called duplex Hecke algebra of type $\mathrm{\mathsf{B}}$, and establish a
related $q$-deformation of Levi Schur-Weyl duality of type
$\mathrm{\mathsf{B}}$ by exploiting $\imath$quantum groups.

Recall that a symmetric pair $(\mathfrak{g},\mathfrak{g}^{\theta})$ consists
of a semisimple Lie algebra $\mathfrak{g}$ and its fixed-point subalgebra
under $\theta$ which is an involution on $\mathfrak{g}$. The classification of
irreducible symmetric pairs is equivalent to the classification of real forms
of complex simple Lie algebras. Let $\textbf{U}=\textbf{U}_{q}(\mathfrak{g})$
be a quantum group associated with $\mathfrak{g}$. According to Gail Letzter
(cf. [8]), a quantum symmetric pair $(\textbf{U},\textbf{U}_{q}^{\imath})$
consists of a quantum group U and its right coideal subalgebra
$\textbf{U}_{q}^{\imath}$ which specializes at $q\rightarrow 1$ to
$\mathbf{U}(\mathfrak{g})$ and $\mathbf{U}(\mathfrak{g^{\theta}})$. We call
$\textbf{U}_{q}^{\imath}$ an $\imath$quantum group. Bao-Wang (and Bao-Wang-
Watanabe) establish a kind of Schur duality (called $\imath$Schur duality)
between an $\imath$quantum group of type A and a Hecke algebra of type
$\mathrm{\mathsf{B}}$ (see [1, 2]).

###  0.3. Main results

#### 0.3.1.

We establish a duplex Hecke algebra
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ of type $\mathrm{\mathsf{B}}$,
which is a $\mathbb{Q}(q)$-algebra associated with the Weyl group
$\mathscr{W}(\mathrm{\mathsf{B}})$ of type $\mathrm{\mathsf{B}}$ and symmetric
groups $\mathfrak{S}_{l}$ for $l=0,1,\ldots,m$, satisfying some Hecke
relations (see Definition 2.1).

#### 0.3.2.

Based on Bao-Wang’s work along theirs jointly with Watanabe in [1, 2], we
introduce a Levi-type subalgebra ${L}_{q}^{\imath}(\mathfrak{sl}_{2r+4})$ of
$\imath$quantum group $\textbf{U}_{q}^{\imath}(\mathfrak{sl}_{2r+4})$ (see
§3.1). This becomes another important ingredient in our argument solving
Question 0.1.

#### 0.3.3.

Let $V$ be the $\mathbb{Q}(q)$-vector space of dimension $2r+2$, and let
$\underline{V}$ be a $2$-dimensional extension of $V$. Then there are both
natural actions $\Phi$ and $\Xi$ of $L_{q}^{\imath}(\mathfrak{sl}_{2r+4})$ and
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ on $\underline{V}^{\otimes m}$
respectively (see §1.2.1 and §2.2.2). We prove the following double
centralizer theorem (see Theorems 3.6 and 3.9).

$\displaystyle\Phi(L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4}))$
$\displaystyle={\mathrm{End}}_{\mathfrak{H}\kern-4.20004pt\mathfrak{H}_{m}}(\underline{V}^{\otimes
m});$ $\displaystyle\Xi(\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m})$
$\displaystyle={\mathrm{End}}_{L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})}(\underline{V}^{\otimes
m})^{\textsf{op}}.$

###  0.4. The structure of the paper

The article contains three sections. In the first section, we recall some
basic concepts, including quantum groups, $\imath$quantum group, Hecke
algebras and $\imath$Schur duality. In the second section, we introduce the
duplex Hecke algebra $\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ of type
$\mathrm{\mathsf{B}}$ and define its representation on $\underline{V}^{\otimes
m}$, and show that this representation is well-defined. In the third section,
we define a Levi-type subalgebra $L_{q}^{\imath}(\mathfrak{sl}_{2r+4})$ of
$\imath$quantum group
$\textbf{U}_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})$. We then establish
the $\imath$Schur duality of Levi-type.

Throughout the paper, $q$ is a fixed parameter which is assumed to be
transcendental over $\mathbb{Q}$. It is worth mentioning that the proof of
Theorem 3.6 and Lemma 3.7 are dependent on the condition of $q$ being
transcendental over $\mathbb{Q}$ (see [3]).

##  1\. Preliminaries

In this section, we recall some basic notions and introduce notations used in
this article, particularly involving $\imath$quantum groups. For details, one
can be referred to [1], [2] or [10].

###  1.1. Quantum groups

######  Definition 1.1.

The quantum group $\textbf{U}_{q}(\mathfrak{g}\mathfrak{l}_{n})$ associated
with $\mathfrak{gl}_{n}$ is the associative algebra generated by all $E_{i}$,
$F_{i}$, $D_{k}$, $D_{k}^{-1}$ with $1\leqslant i\leqslant n-1$ and
$1\leqslant k\leqslant n$ over ${\mathbb{Q}}(q)$, which satisfies the
following relations

$\displaystyle
D_{k}D_{k}^{-1}=1=D_{k}^{-1}D_{k};\;D_{k}D_{s}=D_{s}D_{k}\;(1\leq k,s\leq n);$
$\displaystyle
D_{i}E_{i}D_{i}^{-1}=qE_{i};\;D_{i}F_{i}D_{i}^{-1}=q^{-1}F_{i};$
$\displaystyle
D_{i+1}E_{i}D_{i+1}^{-1}=q^{-1}E_{i};\;D_{i+1}F_{i}D_{i+1}^{-1}=qF_{i};$
$\displaystyle D_{k}E_{i}D_{k}^{-1}=E_{i}\;(k\neq
i,i+1);\;D_{k}F_{i}D_{k}^{-1}=F_{i}\;(k\neq i,i+1);$ $\displaystyle
E_{i}F_{j}-F_{i}E_{j}=\delta_{ij}\frac{D_{i}D_{i+1}^{-1}-D_{i}^{-1}D_{i+1}}{q-q^{-1}};$
$\displaystyle\begin{cases}E_{i}E_{j}=E_{j}E_{i}\cr
F_{i}F_{j}=F_{j}F_{i}\end{cases}\text{ if }c_{ij}=0;$

and

$\displaystyle\begin{cases}E_{i}^{2}E_{j}-(q+q^{-1})E_{i}E_{j}E_{i}+E_{j}E_{i}^{2}=0\cr
F_{i}^{2}F_{j}-(q+q^{-1})F_{i}F_{j}F_{i}+F_{j}F_{i}^{2}=0\end{cases}\text{ if
}c_{ij}=-1,$

where $C=(c_{ij})_{(n-1)\times(n-1)}$ is the Cartan matrix of
$\mathfrak{s}\mathfrak{l}{}_{n}$.

######  Definition 1.2.

The quantum group ${\textbf{U}}_{q}(\mathfrak{sl}_{n})$ associated with
$\mathfrak{sl}_{n}$ is the $\mathbb{Q}(q)$-subalgebra of
$\textbf{U}_{q}(\mathfrak{g}\mathfrak{l}_{n})$ generated by all
$E_{i},F_{i},K_{i}:=D_{i}D_{i+1}^{-1}$ with $1\leq i\leq n-1.$

######  Remark 1.3.

The quantum group ${\textbf{U}}_{q}(\mathfrak{gl}_{n})$ is a Hopf algebra with
comultiplication
$\Delta:{\textbf{U}}_{q}(\mathfrak{gl}_{n})\rightarrow{\textbf{U}}_{q}(\mathfrak{gl}_{n})\otimes{\textbf{U}}_{q}(\mathfrak{gl}_{n})$
given by

$\displaystyle\Delta(D_{i})=D_{i}\otimes D_{i},\Delta(E_{i})=1\otimes
E_{i}+E_{i}\otimes K_{i}^{-1},\Delta(F_{i})=F_{i}\otimes 1+K_{i}\otimes
F_{i}.$

###  1.2. $\imath$Quantum groups

Let $r$ be a given positive integer. Set

$\displaystyle\mathbb{I}_{2r+2}$
$\displaystyle:=\\{-r-\frac{1}{2},-r+\frac{1}{2},\ldots,-\frac{1}{2},\frac{1}{2},\ldots,r-\frac{1}{2},r+\frac{1}{2}\\},$
$\displaystyle\mathbb{I}_{2r+1}$
$\displaystyle:=\\{-r,-r+1,\ldots,0,\ldots,r-1,r\\}.$

######  Definition 1.4.

Index the generators (or the simple roots) for
${\textbf{U}}_{q}(\mathfrak{s}\mathfrak{l}_{2r+2})$ by $\mathbb{I}_{2r+1}$.
Then the $\imath$quantum group
${\textbf{U}}_{q}^{\imath}(\mathfrak{sl}_{2r+2})$ is defined as the
$\mathbb{Q}(q)$-subalgebra of ${\textbf{U}}_{q}(\mathfrak{gl}_{2r+2})$
generated by all

$\displaystyle k_{i}$ $\displaystyle:=K_{i}K_{-i}^{-1}\;$
$\displaystyle(i\in\mathbb{I}_{2r+1}),$ $\displaystyle B_{i}$
$\displaystyle:=E_{i}+F_{-i}K_{i}^{-1}\;$ $\displaystyle(0\neq
i\in\mathbb{I}_{2r+1}),$ $\displaystyle B_{0}$
$\displaystyle:=E_{0}+qF_{0}K_{0}^{-1}+K_{0}^{-1}.$

####  1.2.1. Natural modules

Let $V$ be the $\mathbb{Q}(q)$-vector space spanned by $\\{\eta_{i}\mid
i\in\mathbb{I}_{2r+2}\\}$, and $\underline{V}$ be the $\mathbb{Q}(q)$-vector
space spanned by $\\{\eta_{j}\mid j\in\mathbb{I}_{2r+4}\\}$, the latter of
which is a two-dimensional extension of the former. Furthermore, there is a
natural representation of ${\textbf{U}}_{q}(\mathfrak{sl}_{2r+4})$ on
$\underline{V}$ via

$\displaystyle F_{i}\eta_{i-\frac{1}{2}}=\eta_{i+\frac{1}{2}},$ $\displaystyle
E_{i}\eta_{i+\frac{1}{2}}=\eta_{i-\frac{1}{2}},$ (1.1) $\displaystyle
F_{i}\eta_{j}=0,$ $\displaystyle E_{i}\eta_{j+1}=0,$ $\displaystyle(j\neq
i-\frac{1}{2}),$ $\displaystyle
K_{i}\eta_{i-\frac{1}{2}}=q\eta_{i-\frac{1}{2}},$ $\displaystyle
K_{i}\eta_{i+\frac{1}{2}}=q^{-1}\eta_{i+\frac{1}{2}},$ $\displaystyle
K_{i}\eta_{j}=\eta_{j}$ $\displaystyle(j\neq i\pm\frac{1}{2}).$

Consequently,

$\displaystyle k_{i}\eta_{i-\frac{1}{2}}=q\eta_{i-\frac{1}{2}},$
$\displaystyle k_{i}\eta_{-(i-\frac{1}{2})}=q\eta_{-(i-\frac{1}{2})},$ (1.2)
$\displaystyle k_{i}\eta_{i+\frac{1}{2}}=q^{-1}\eta_{i+\frac{1}{2}},$
$\displaystyle k_{i}\eta_{-(i+\frac{1}{2})}=q^{-1}\eta_{-(i+\frac{1}{2})},$
$\displaystyle k_{i}\eta_{j}=\eta_{j}\quad\text{for
}j\neq\pm(i\pm\frac{1}{2}).$

Hence, there is a ${\textbf{U}}_{q}^{\imath}(\mathfrak{sl}_{2r+4})$-module
structure on $\underline{V}$. Let $m$ be a fixed integer not less than 2. It
is clear that $\underline{V}^{\otimes m}$ is also a
$\textbf{U}_{q}(\mathfrak{s}\mathfrak{l}_{2r+4})$-module in a usual way.
Precisely, the action of $u\in\textbf{U}_{q}(\mathfrak{s}\mathfrak{l}_{2r+4})$
on $\underline{V}^{\otimes m}$ is defined as

$\Delta^{m}(u):=(\Delta\otimes\mathsf{id}^{\otimes
m-2})\circ\cdots\circ(\Delta\otimes\mathsf{id})\circ\Delta(u).$

Hereinafter we denote
$\overset{s}{\overbrace{\square\otimes\cdots\otimes\square}}$ by
$\square^{\otimes s}$ for an operator or a vector $\square$. Then we have

$\displaystyle\Delta^{m}(K_{i})=K_{i}^{\otimes m};$ (1.3)
$\displaystyle\Delta^{m}(E_{i})=\sum_{j=1}^{m}\mathsf{id}^{\otimes(j-1)}\otimes
E_{i}\otimes K_{i}^{\otimes(j-m)};$
$\displaystyle\Delta^{m}(F_{i})=\sum_{j=1}^{m}{K_{i}}^{\otimes(j-1)}\otimes
F_{i}\otimes\mathsf{id}^{\otimes(m-j)},$

where $K_{i}^{\otimes{(j-m)}}$ means $(K_{i}^{-1})^{\otimes(m-j)}$.

###  1.3. Hecke algebras of type $\mathrm{\mathsf{B}}$

Denote by $\mathbb{I}_{2r+4}^{m}$ the set of $m$-tuples from
$\mathbb{I}_{2r+4}$. This means

$\mathbb{I}_{2r+4}^{m}:=\\{(a_{1,}a_{2},\ldots,a_{m})\mid
a_{i}\in\mathbb{I}_{2r+4}\text{ with }i=1,2,\ldots,m\\}.$

View $f\in\mathbb{I}_{2r+4}^{m}$ as a function
$f:\\{1,\ldots,m\\}\rightarrow\mathbb{I}_{2r+4}$. For any
$f\in\mathbb{I}_{2r+4}^{m}$, we can write $f=(f(1),\ldots,f(m))$, and define
$M_{f}:=\eta_{f(1)}\otimes\eta_{f(2)}\otimes\cdots\otimes\eta_{f(m)}$. Then
the set $\\{M_{f}\mid f\in\mathbb{I}_{2r+4}^{m}\\}$ forms a basis for
$\underline{V}^{\otimes m}$. Let $\mathscr{W}(\mathrm{\mathsf{B}}_{m})$ be the
Coxeter group of type $\mathrm{\mathsf{B}}_{m}$ with simple reflections
$s_{j}$, $0\leq j\leq m-1$, while the subgroup generated by all $s_{i}$ with
$1\leq i\leq m-1$ is exactly $\mathscr{W}(\textsf{A}_{m-1})$ coinciding with
$\mathfrak{S}_{m}$.

Let $\mathcal{H}(\mathrm{\mathsf{B}}_{m})$ be the Iwahori-Hecke algebra of
type $\mathrm{\mathsf{B}}_{m}$ over $\mathbb{Q}(q)$. By definition,
$\mathcal{H}(\mathrm{\mathsf{B}}_{m})$ is generated by $H_{0}$, $H_{1}$, …,
$H_{m-1}$ as a $\mathbb{Q}(q)$-algebra, subject to the following relations,

$\displaystyle(H_{i}-q^{-1})(H_{i}+q)=0,$ for $\displaystyle i\geq 0;$
$\displaystyle H_{i}H_{i+1}H_{i}=H_{i+1}H_{i}H_{i+1},$ for $\displaystyle
0\leq i\leq m-2;$ $\displaystyle H_{i}H_{j}=H_{j}H_{i},$ for
$\displaystyle|i-j|>1;$ $\displaystyle
H_{0}H_{1}H_{0}H_{1}=H_{1}H_{0}H_{1}H_{0}.$

For $\sigma\in\mathscr{W}(\mathrm{\mathsf{B}}_{m})$ with a reduced expression
$\sigma=s_{i_{1}}\cdot\cdot\cdot s_{i_{k}}$, the corresponding element
$H_{\sigma}:=H_{i_{1}}\cdot\cdot\cdot
H_{i_{k}}\in\mathcal{H}(\mathrm{\mathsf{B}}_{m})$ is well-defined, due to
Matsumoto’s theorem (see for example [4]).

#### 1.3.1.

The group $\mathscr{W}(\mathrm{\mathsf{B}}_{m})$ and its subgroup
$\mathfrak{S}_{m}$ naturally act by transposition on $\mathbb{I}_{2r+4}^{m}$
(from the right side): for $0\leq j\leq m-1$ and any
$f\in\mathbb{I}_{2r+4}^{m}$,

$\displaystyle f\cdot s_{j}=\begin{cases}(\ldots,f(j+1),f(j),\ldots),&\text{
if }j>0,\cr(-f(1),f(2),\ldots,f(m)),&\text{ if }j=0.\end{cases}$

#### 1.3.2.

By a straightforward computation, there is correspondingly a right action of
the Hecke algebra $\mathcal{H}(\mathrm{\mathsf{B}}_{m})$ on the
$\mathbb{Q}(q)$-vector space $\underline{V}^{\otimes m}$ as below:

$\displaystyle M_{f}H_{i}=\begin{cases}q^{-1}M_{f},&\text{ if }i\neq 0\text{
and }f(i)=f(i+1);\cr M_{fs_{i}},&\text{ if }i\neq 0\text{ and }f(i)<f(i+1);\cr
M_{fs_{i}}+(q^{-1}-q)M_{f},&\text{ if }i\neq 0\text{ and }f(i)>f(i+1);\\\
M_{fs_{0}},&\text{ if }i=0\text{ and }f(1)>0;\cr
M_{fs_{0}}+(q^{-1}-q)M_{f},&\text{ if }i=0\text{ and }f(1)<0.\end{cases}$
(1.4)

### 1.4.

Recall the arguments in §1.2.1. The left action of
$\textbf{U}_{q}(\mathfrak{s}\mathfrak{l}_{2r+4})$ restricts to
$\textbf{U}_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})$. Therefore
$\textbf{U}_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})$ and
$\mathcal{H}(\mathrm{\mathsf{B}}_{m})$ have left and right action on
$\underline{V}^{\otimes m}$ respectively, which we denote by $\Phi$ and $\Psi$
respectively. On the other hand, it is worthwhile reminding that
$\mathcal{H}(\mathrm{\mathsf{B}}_{l})$ has a right action on $V^{\otimes l}$.
We will denote this action of $\mathcal{H}(\mathrm{\mathsf{B}}_{l})$ on
$V^{\otimes l}$ by $\Psi_{l}^{V}$.

The following is the remarkable $\imath$Schur duality established by Bao-Wang.

######  Theorem 1.5.

([1, §5.4]) Both
$\Phi(\mathrm{\mathbf{U}}_{q}^{\imath}(\mathfrak{sl}_{2r+4}))$ and
$\Psi(\mathcal{H}(\mathrm{\mathsf{B}}_{m}))$ commute with each other, and they
form double centralizers, i.e.,

${\mathrm{End}}_{\mathrm{\mathbf{U}}_{q}^{\imath}(\mathfrak{sl}_{2r+4})}(\underline{V}^{\otimes
m})^{\mathrm{op}}=\Psi(\mathcal{H}(\mathrm{\mathsf{B}}_{m})),$
${\mathrm{End}}_{\mathcal{H}(\mathrm{\mathsf{B}}_{m})}(\underline{V}^{\otimes
m})=\Phi(\mathrm{\mathbf{U}}_{q}^{\imath}(\mathfrak{sl}_{2r+4})).$

##  2\. Duplex Hecke algebras

In this section, we will introduce duplex Hecke algebras (of type
$\mathrm{\mathsf{B}}$). This concept originates from degenerate duplex Hecke
algebras in [12], extending the notion of duplex Hecke algebras of type A in
[14].

Recall that the symmetric group $\mathfrak{S}_{l}$ of $l$ letters ($l\geq 2$)
has a set of canonical generators $\\{s_{1},\ldots,s_{l-1}\\}$ with relations
$s_{i}^{2}=\mathsf{id}$ for $i=1,\ldots,l-1$, $s_{i}s_{j}=s_{j}s_{i}$ if
$i,j=1,\ldots,l-1$ satisfying $|i-j|>1$, and
$s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}$ for $i=1,\ldots,l-2$. Moreover, we
appoint that $\mathfrak{S}_{1}=\mathfrak{S}_{0}=\\{\mathsf{id}\\}$, the
identity group, for convenience.

###  2.1. Basic Definition

######  Definition 2.1.

A duplex Hecke algebra $\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ is a
$\mathbb{Q}(q)$-algebra generated by all $T_{i}$, with $i=0,1,\ldots\ ,m-1$,
and all $x_{\sigma}^{(l)}$, with $\sigma\in\mathfrak{S}_{l}$ for $0\leq l\leq
m$, subject to the follow relations:

$\displaystyle(T_{i}+q)(T_{i}-q^{-1})=0,\quad$ for $\displaystyle 0\leq i\leq
m-1;$ (2.1) $\displaystyle T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1},$ for
$\displaystyle 0\leq i\leq m-2;$ $\displaystyle T_{i}T_{j}=T_{j}T_{i},$ for
$\displaystyle|i-j|>1;$ $\displaystyle
T_{0}T_{1}T_{0}T_{1}=T_{1}T_{0}T_{1}T_{0}.$

and

$\displaystyle x_{\sigma}^{(l)}x_{s_{i}}^{(l)}$
$\displaystyle=\begin{cases}x_{\sigma s_{i}}^{(l)},&l(\sigma
s_{i})=l(\sigma)+1;\cr x_{\sigma
s_{i}}^{(l)}+(q^{-1}-q)x_{\sigma}^{(l)},&l(\sigma
s_{i})=l(\sigma)-1;\end{cases}$ (2.2) $\displaystyle
x_{s_{i}}^{(l)}x_{\sigma}^{(l)}$
$\displaystyle=\begin{cases}x_{s_{i}\sigma}^{(l)},&l(s_{i}\sigma)=l(\sigma)+1;\\\
x_{s_{i}\sigma}^{(l)}+(q^{-1}-q)x_{\sigma}^{(l)},&l(s_{i}\sigma)=l(\sigma)-1;\end{cases}$
$\displaystyle x_{\sigma}^{(l)}x_{\gamma}^{(k)}$ $\displaystyle=0,\quad l\neq
k,$

along with

$\displaystyle T_{i}x_{\sigma}^{(l)}$
$\displaystyle=x_{s_{i}}^{(l)}x_{\sigma}^{(l)},$ $\displaystyle\;0<i<l;$ (2.3)
$\displaystyle x_{\sigma}^{(l)}T_{i}$
$\displaystyle=x_{\sigma}^{(l)}x_{s_{i}}^{(l)},$ $\displaystyle\;0<i<l;$
$\displaystyle T_{i}x_{\sigma}^{(l)}$
$\displaystyle=q^{-1}x_{\sigma}^{(l)}=x_{\sigma}^{(l)}T_{i},$ $\displaystyle
i>l.$

######  Remark 2.2.

From the formula (LABEL:dha_def1), $\mathcal{H}(\mathrm{\mathsf{B}}_{m})$ can
be imbedded into $\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ as a
subalgebra, and we can regard $T_{i}$ as an element in
$\mathcal{H}(\mathrm{\mathsf{B}}_{m})$. In particular, all $T_{i}$ with
$i=0,1,\ldots,m-1$ are invertible.

###  2.2. Canonical representation of
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ on ${\underline{V}}^{\otimes m}$

Keep the notations as before. In particular, we keep the notations in §1.2.1.
In order to endow an $\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-module
structure on ${\underline{V}}^{\otimes m}$, we first introduce a decomposition
of ${\underline{V}}^{\otimes m}$. Let $n$ be a nonnegative integer. Set
$\underline{n}:=\\{1,2,\cdots,n-1,n\\}$ for $n>0$ and
$\underline{0}=\emptyset$, where $\emptyset$ denotes the empty set.

For $I\subset\underline{m}$ with $\\#I=l$, and
$J\subset\underline{m}\backslash I$, we set $\underline{V}_{I,J}^{\otimes m}$
to be the subspace of ${\underline{V}}^{\otimes m}$ spanned by all
$M_{f}:=\eta_{f(1)}\otimes\eta_{f(2)}\otimes\cdots\otimes\eta_{f(m)}$ with
$f(k)\in\mathbb{I}_{2r+2}$ for $k\in I$, $f(k)=-r-\frac{3}{2}$ for $k\in J$,
and $f(k)=r+\frac{3}{2}$ for $k\in\underline{m}\backslash(I\cup J)$, and let

$\underline{V}_{l}^{\otimes
m}=\underset{\\#I=l,J\subset\underline{m}\backslash
I}{\bigoplus}\underline{V}_{I,J}^{\otimes m}.$

In the set-up above, ${\underline{V}}^{\otimes m}$ can be decomposed into a
direct sum of subspaces

${\underline{V}}^{\otimes m}=\bigoplus_{l=0}^{m}\underline{V}_{l}^{\otimes
m}=\bigoplus_{I,J\subset\underline{m},I\cap
J=\emptyset}\underline{V}_{I,J}^{\otimes m}.$

From now on, we suppose the integer $l\in\\{0,1,\ldots,m\\}$. Notice that we
have $\underline{V}_{\underline{l},\emptyset}^{\otimes m}=V^{\otimes
l}\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l}$.

#### 2.2.1.

Keep the notations from §1.4. For $I\subset\underline{m}$ with $\\#I=l,$
$J\subset\underline{m}\backslash I$, there exists a linear transformation
$\omega_{I,J}$ mapping $\underline{V}_{I,J}^{\otimes m}$ onto
$\underline{V}_{\underline{l},\emptyset}^{\otimes m}$. Actually, this
$\omega_{I,J}$ can be expressed as a composition of $T_{i}^{\pm 1}$ with
$0\leq i\leq m-1$ and acts on $\underline{V}_{I,J}^{\otimes m}$ by natural
representation $\Psi$ of Hecke algebra $\mathcal{H}(\mathrm{\mathsf{B}}_{m})$.
We can regard it as an element in
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ (see Remark 2.2). We explain
this operation in the following lemma.

######  Lemma 2.3.

For $I\subset\underline{m}$ with $\\#I=l,$ $J\subset\underline{m}\backslash
I$, there exists an element $\omega_{I,J}$ in
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ generated by some $T_{i}^{\pm
1}$ with $i\in\\{0,1,\ldots,m-1\\}$ such that
$\underline{V}_{\underline{l},\emptyset}^{\otimes
m}=\Psi(\omega_{I,J})(\underline{V}_{I,J}^{\otimes m})$.

###### Proof.

We construct the desired element by induction on $\\#J$.

In the case when $\\#J=0$, i.e. $J=\emptyset$, we take $r$ to be the minimal
element of $\underline{m}\backslash I$. If further, $I=\underline{l}$, then we
have nothing to do. Now we suppose $I\neq\underline{l}$. Take the minimal
element $i$ in $I$ with $i>r$. For any
$\eta_{f(1)}\otimes\eta_{f(2)}\otimes\cdots\otimes\eta_{f(m)}\in\underline{V}_{I,\emptyset}^{\otimes
m}$, we have $f(k)>f(i)$ for any $r\leq k<i$ . Then by applying $\Psi$, we
have

$\Psi(T_{i-1}^{-1}T_{i-2}^{-1}\cdots T_{r}^{-1})(\underline{V}^{\otimes
m}_{I,\emptyset})=\Psi(T_{r}^{-1})\Psi(T_{r+1}^{-1})\cdots\Psi(T_{i-1}^{-1})(\underline{V}^{\otimes
m}_{I,\emptyset})=\underline{V}^{\otimes m}_{I^{\prime},\emptyset},$

where $I^{\prime}=(I\backslash\\{i\\})\cup\\{r\\}$. Applying operation
repeatedly in the case $I^{\prime}$, we can change index $I$ into
$\underline{l}$, which shows $\underline{V}^{\otimes
m}_{\underline{l},\emptyset}=\Psi(\omega_{I,\emptyset})(\underline{V}^{\otimes
m}_{I,J})$ for some $\omega_{I,\emptyset}$ as desired.

Next, consider the case when $\\#J>0$. Then there exists a minimal element
$j\in J$, i.e, all
$\eta_{f(1)}\otimes\eta_{f(2)}\otimes\cdots\otimes\eta_{f(m)}\in\underline{V}_{I,J}^{\otimes
m}$ satisfy $f(i)>f(j)$ for any $i<j$. Then by applying the action $\Psi$, we
have

$\Psi(T_{j-1}^{-1}T_{j-2}^{-1}\cdots
T_{1}^{-1}T_{0})(\underline{V}_{I,J}^{\otimes
m})=\Psi(T_{0})\Psi(T_{1}^{-1})\cdots\Psi(T_{j-1}^{-1})(\underline{V}_{I,J}^{\otimes
m})=\underline{V}_{I^{\prime},J^{\prime}}^{\otimes m},$

where $I^{\prime}$ and $J^{\prime}$ is some subset of $\underline{m}$ with
$\\#J^{\prime}=\\#J-1$. We can now apply induction to
$\underline{V}_{I^{\prime},J^{\prime}}^{\otimes m}$ such that
$\underline{V}_{\underline{l},\emptyset}^{\otimes
m}=\Psi(\omega_{I^{\prime},J^{\prime}})(\underline{V}_{I^{\prime},J^{\prime}}^{\otimes
m})$ for some $\omega_{I^{\prime},J^{\prime}}$. Thus
$\omega_{I,J}=T_{j-1}^{-1}T_{j-2}^{-1}\cdots
T_{1}^{-1}T_{0}\omega_{I^{\prime},J^{\prime}}$ as desired. ∎

#### 2.2.2.

Note that ${\underline{V}}^{\otimes
m}=\bigoplus_{l=0}^{m}\underline{V}_{l}^{\otimes
m}=\bigoplus_{I,J\subset\underline{m},I\cap
J=\emptyset}\underline{V}_{I,J}^{\otimes m}.$ By using Lemma 2.3, we can only
consider the actions on $\underline{V}_{\underline{l},\emptyset}$ and then
extend $\Psi$. We can obtain an
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-module structure on
$\underline{V}^{\otimes m}$. Such an extension seems natural by the structure
of duplex Hecke algebras. Recall that $\Psi_{l}^{V}$ in $\S\ref{sec: 1.4}$
denotes the natural representation of $\mathcal{H}(\mathrm{\mathsf{B}}_{l})$
on $V^{\otimes l}$.

######  Proposition 2.4.

There exists a unique $\mathbb{Q}(q)$-algebra anti-homomorphism
$\Xi:\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}\rightarrow{\mathrm{End}}(\underline{V}^{\otimes
m})$ such that

$\displaystyle\Xi(T_{i})=\Psi(H_{i}),$ $\displaystyle\text{for
}i=0,1,2,\ldots\ ,m-1;$ (2.4)
$\displaystyle\Xi(x_{\sigma}^{(l)})|_{\underline{V}_{\underline{l},\emptyset}^{\otimes
m}}=\Psi_{l}^{V}(H_{\sigma})\otimes\mathsf{id}^{\otimes m-l},$
$\displaystyle\text{for }\sigma\in\mathfrak{S}_{l},0\leq l\leq m;$
$\displaystyle\Xi(x_{\sigma}^{(l)})(M_{f})=0,$ $\displaystyle\text{for
}M_{f}\notin\underline{V}_{\underline{l},\emptyset}^{\otimes m}.$

###### Proof.

Since the action of generators of
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ are determined by formula
(LABEL:eq:_defining_Xi), the uniqueness is obvious if this homomorphism
exists. It is enough to show that the right action $\Xi$ of
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ on $\underline{V}^{\otimes m}$
keeps relations (LABEL:dha_def1)-(2.3).

With the start point of extending $\Psi$ to $\Xi$, $\Xi|_{\langle
T_{i},i=0,1,...,m-1\rangle}$ is exactly the representation $\Psi$ of
$\mathcal{H}(\mathrm{\mathsf{B}}_{m})$. So $\Xi$ keeps relations
(LABEL:dha_def1).

For $0<i<l$. Recall that in this case, the action of $H_{i}$ on

$M_{f}=\eta_{f(1)}\otimes\eta_{f(2)}\otimes\cdots\otimes\eta_{f(m)}$

only changes the first $l$ factors of $M_{f}$ which belongs to
$\underline{V}_{\underline{l},\emptyset}^{\otimes m}$. If
$M_{f}\in\underline{V}_{\underline{l},\emptyset}^{\otimes m}$, we then have

$\Xi(T_{i})(M_{f})=\Psi(H_{i})(M_{f})=\Psi_{l}^{V}(H_{i})\otimes\mathsf{id}^{\otimes
m-l}(M_{f})=\Xi(x_{s_{i}}^{(l)})(M_{f}),$

and hence

$\Xi(T_{i}x_{\sigma}^{(l)})(M_{f})=\Xi(x_{\sigma}^{(l)})\Xi(T_{i})(M_{f})=\Xi(x_{\sigma}^{(l)})\Xi(x_{s_{i}}^{(l)})(M_{f})=\Xi(x_{s_{i}}^{(l)}x_{\sigma}^{(l)})(M_{f}),$
$\Xi(x_{\sigma}^{(l)}T_{i})(M_{f})=\Xi(T_{i})\Xi(x_{\sigma}^{(l)})(M_{f})=\Xi(x_{s_{i}}^{(l)})\Xi(x_{\sigma}^{(l)})(M_{f})=\Xi(x_{\sigma}^{(l)}x_{s_{i}}^{(l)})(M_{f}).$

If $M_{f}\notin\underline{V}_{\underline{l},\emptyset}^{\otimes m}$, the both
sides of equations are zero, since $\Xi(x_{\sigma}^{(l)})(M_{f})=0$ and
$\Xi(T_{i})(M_{f})\notin\underline{V}_{\underline{l},\emptyset}$.

For $i>l$, we have

$\Xi(T_{i}x_{\sigma}^{(l)})=q^{-1}\Xi(x_{\sigma}^{(l)})=\Xi(x_{\sigma}^{(l)}T_{i}),$

since $\Xi(T_{i})=\Psi(H_{i})$ acts as multiplication by $q^{-1}$ on vector of
$\underline{V}_{\underline{l},\emptyset}^{\otimes m}$ and annihilates vectors
which are outside of $\underline{V}_{\underline{l},\emptyset}^{\otimes m}$.
Thus, $\Xi$ keeps relation (2.3).

For $l\neq k$, it follows easily from definition that
$\Xi(x_{\sigma}^{(l)}x_{\gamma}^{(k)})=\Xi(x_{\gamma}^{(k)}x_{\sigma}^{(l)})=0$.
Regard $\mathfrak{S}_{l}$ as a subgroup of $\mathfrak{S}_{m}$, we get
similarly for $0<i<l$

$\Xi(x_{\sigma}^{(l)}x_{s_{i}}^{(l)})=\Psi({H_{\sigma}H_{i}}).$

The relation (2.2) holds by the composition formula in Hecke algebra.

We conclude the existence of $\Xi$, which is a representation of
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ on $\underline{V}^{\otimes m}$.

∎

##  3\. Levi-type $q$-Schur-Weyl duality of type $\mathrm{\mathsf{B}}$

In this section, we establish a duality between the Levi-type $\imath$quantum
group and the duplex Hecke algebra
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$.

###  3.1. $\imath$quantum group of Levi-type

######  Definition 3.1.

Denote by $L_{q}^{\imath}(\mathfrak{sl}_{2r+4})$ the subalgebra of
${\textbf{U}}_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})$ generated by
$B_{0}:=E_{0}+qF_{0}K_{0}^{-1}+K_{0}^{-1}$, all
$B_{i}:=E_{i}+F_{-i}K_{i}^{-1}$ with $0\neq i\in\mathbb{I}_{2r+1}$, and all
$k_{i}:=K_{i}K_{-i}^{-1}$ with $i\in\mathbb{I}_{2r+3}$. Call
$L_{q}^{\imath}(\mathfrak{sl}_{2r+4})$ an $\imath$quantum group of Levi-type.

It is easy to see
$L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})\cong\textbf{U}_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+2})\oplus\langle
k_{r+1},k_{r+1}^{-1}\rangle$ as a space.

### 3.2.

The action of ${\textbf{U}}_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})$ on
$\underline{V}^{\otimes m}$ restricts to the subalgebra
$L_{q}^{\imath}(\mathfrak{sl}_{2r+4})$ and denoted by $\Phi$ again. On the
other hand, we establish the action of
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ on $\underline{V}^{\otimes m}$
in Proposition 2.4. We have the following observation, which reveals the
relation between the Levi-type $\imath$quantum group
$L_{q}^{\imath}(\mathfrak{sl}_{2r+4})$ and the duplex Hecke algebra
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$.

######  Lemma 3.2.

The actions of $L_{q}^{\imath}(\mathfrak{sl}_{2r+4})$ and
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$ on $\underline{V}^{\otimes m}$
commute with each other, i.e.
$\Phi(L_{q}^{\imath}(\mathfrak{sl}_{2r+4}))\subset{\mathrm{End}}_{\mathfrak{H}\kern-4.20004pt\mathfrak{H}_{m}}(\underline{V}^{\otimes
m})$.

###### Proof.

We only need to show the actions of $k_{i}$ with $i\in\mathbb{I}_{2r+3}$, and
$B_{i}$ with $i\in\mathbb{I}_{2r+1}$ on $\underline{V}^{\otimes m}$ commute
with the action of $T_{i}$ with $i=0,1,\ldots\ ,m-1$, and $x_{\sigma}^{(l)}$
with $\sigma\in\mathfrak{S}_{l}$ for $0\leq l\leq m$, respectively.

The mutual commutant property of all $\Xi(T_{i})$ with
$\Phi(L_{q}^{\imath}(\mathfrak{sl}_{2r+4}))$ is ensured by $\imath$Schur
duality (Theorem 1.5) and Remark 2.2. We first check that all $\Psi(B_{i})$
with $0\neq i\in\mathbb{I}_{2r+1}$ commute with all $\Xi(x_{\sigma}^{(l)})$
with $\sigma\in\mathfrak{S}_{l}$. From (LABEL:action2), we have that

$\displaystyle\Delta^{m}(B_{i})$
$\displaystyle=\Delta^{m}(E_{i}+F_{-i}K_{i}^{-1})=\Delta^{m}(E_{i})+\Delta^{m}(F_{-i}K_{i}^{-1})$
$\displaystyle=\sum_{j=1}^{m}\mathsf{id}^{\otimes(j-1)}\otimes E_{i}\otimes
K_{i}^{\otimes(j-m)}+(\sum_{j=1}^{m}{K_{-i}}^{\otimes(j-1)}\otimes
F_{-i}\otimes\mathsf{id}^{\otimes(m-j)})(K_{i}^{\otimes-m})$
$\displaystyle=\sum_{j=1}^{m}\mathsf{id}^{\otimes(j-1)}\otimes E_{i}\otimes
K_{i}^{\otimes(j-m)}+k_{i}^{\otimes(1-j)}\otimes F_{-i}K_{i}^{-1}\otimes
K_{i}^{\otimes(j-m)}.$

Notice any element of $\underline{V}_{\underline{l},\emptyset}$ can be written
as $\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l}$ with some $\alpha\in
V^{\otimes l}$. For $0\neq i\in\mathbb{I}_{2r+1}$, we can get

$\displaystyle\Phi(B_{i})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})=\Delta^{m}(B_{i})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l})$
$\displaystyle=$
$\displaystyle(\sum_{j=1}^{m}\mathsf{id}^{\otimes(j-1)}\otimes E_{i}\otimes
K_{i}^{\otimes(j-m)}+k_{i}^{\otimes(1-j)}\otimes F_{-i}K_{i}^{-1}\otimes
K_{i}^{\otimes(j-m)})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l}).$
$\displaystyle=$
$\displaystyle(\sum_{j=1}^{l}\mathsf{id}^{\otimes(j-1)}\otimes E_{i}\otimes
K_{i}^{\otimes(j-m)}+k_{i}^{\otimes(1-j)}\otimes F_{-i}K_{i}^{-1}\otimes
K_{i}^{\otimes(j-m)})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l})$
$\displaystyle=$
$\displaystyle(\sum_{j=1}^{l}\mathsf{id}^{\otimes(j-1)}\otimes
E_{i}+k_{i}^{\otimes(1-j)}\otimes
F_{-i}K_{i}^{-1})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l}$
$\displaystyle=$
$\displaystyle\Delta^{l}(B_{i})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}.$

The third and fourth equalities follow from the fact

$\displaystyle
E_{i}\eta_{r+\frac{3}{2}}=F_{-i}K_{i}^{-1}\eta_{r+\frac{3}{2}}=0,\text{ and
}K_{i}^{-1}\eta_{r+\frac{3}{2}}=\eta_{r+\frac{3}{2}}\text{ for
}i\in\mathbb{I}_{2r+1}.$ (3.1)

Hence, from $\imath$Schur duality on $V^{\otimes l}$, we have

$\displaystyle\Xi(x_{\sigma}^{(l)})\Phi(B_{i})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})=$
$\displaystyle\Xi(x_{\sigma}^{(l)})(\Delta^{l}(B_{i})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})$ $\displaystyle=$
$\displaystyle\Psi_{l}^{V}(H_{\sigma})\Delta^{l}(B_{i})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}$ $\displaystyle=$
$\displaystyle\Delta^{l}(B_{i})\Psi_{l}^{V}(H_{\sigma})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}$ $\displaystyle=$
$\displaystyle\Phi(B_{i})\Xi(x_{\sigma}^{(l)})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}).$

Now we consider the action on the other parts outside of
$\underline{V}_{\underline{l},\emptyset}$. We already have
$\Phi(B_{i})\Xi(x_{\sigma}^{(l)})=0$. For any
$M_{f}=\eta_{f(1)}\otimes\eta_{f(2)}\otimes\cdots\otimes\eta_{f(m)}\notin\underline{V}_{\underline{l},\emptyset}^{\otimes
m}$, there are two possibilities:

  * (i)

There exists $k\in\\{1,2,\ldots,l\\}$ such that
$f(k)\in\\{-r-\frac{3}{2},r+\frac{3}{2}\\}$;

  * (ii)

There exists $k\in\\{l+1,l+2,\ldots,m\\}$ with $f(k)\neq r+\frac{3}{2}$.

In the first case, following from formula (3.1), the $k$th factor of each term
of $\Psi(B_{i})(M_{f})$ does not lie in
$\underline{V}_{\underline{l},\emptyset}$ for $j\neq k$ and is zero for $j=k$.
It shows $\Xi(x_{\sigma}^{(l)})\Phi(B_{i})=0$ as required.

For the second case, the action of $E_{i}$, $F_{-i}K_{i}^{-1}$, $K_{i}^{-1}$,
and $k_{i}^{-1}$ for $i\in\mathbb{I}_{2r+1}$ cannot change $\eta_{k}$ to
$\eta_{r+\frac{3}{2}}$. We still have $\Xi(x_{\sigma}^{(l)})\Phi(B_{i})=0$.

It remains to show $\Phi(B_{0})$ commutes with $\Xi(x_{\sigma}^{(l)})$. Notice
that

$\displaystyle\Delta^{m}(B_{0})=\Delta^{m}(E_{0})+\Delta^{m}(qF_{0}K_{0}^{-1})+\Delta^{m}({K_{0}^{-1}})$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{m}\mathsf{id}^{\otimes(j-1)}\otimes
E_{0}\otimes
K_{i}^{\otimes(j-m)}+q(\sum_{j=1}^{m}{K_{-i}}^{\otimes(j-1^{)}}\otimes
F_{-i}\otimes\mathsf{id}^{\otimes(m-j)})(K_{i}^{\otimes-m})+K_{0}^{\otimes-m}$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{m}\mathsf{id}^{\otimes(j-1)}\otimes
E_{i}\otimes K_{i}^{\otimes(j-m)}+qk_{i}^{\otimes(1-j)}\otimes
F_{-i}K_{i}^{-1}\otimes K_{i}^{\otimes(j-m)}+K_{0}^{\otimes-m}.$

Similarly, we still have

$\displaystyle\Xi(x_{\sigma}^{(l)})\Phi(B_{0})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})=$
$\displaystyle\Psi_{l}^{V}(H_{\sigma})\Delta^{l}(B_{0})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}$ $\displaystyle=$
$\displaystyle\Delta^{l}(B_{0})\Psi_{l}^{V}(H_{\sigma})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}$ $\displaystyle=$
$\displaystyle\Phi(B_{0})\Xi(x_{\sigma}^{(l)})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}),$

and

$\Xi(x_{\sigma}^{(l)})\Phi(B_{0})(M_{f})=\Phi(B_{0})\Xi(x_{\sigma}^{(l)})(M_{f})=0,$

for any $M_{f}$ outside of $\underline{V}_{\underline{l},\emptyset}$, since
$K_{0}\eta_{r+\frac{3}{2}}=\eta_{r+\frac{3}{2}}$. The Lemma follows.

∎

######  Remark 3.3.

Let $\Phi_{l}^{V}$ be the natural representation of
$\textbf{U}_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+2})$ on $V^{\otimes l}.$
As in the proof of Lemma 3.2, we have

$\Phi(B_{i})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})=\Delta^{m}(B_{i})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})=\Delta^{l}(B_{i})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}=\Phi_{l}^{V}(B_{i})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l},$
$\Phi(k_{i})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})=\Delta^{m}(k_{i})(\alpha\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})=\Delta^{l}(k_{i})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}=\Phi_{l}^{V}(k_{i})(\alpha)\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l},$

for all $i\in\mathbb{I}_{2r+1}$. Hence, we can regard
$\Phi_{l}^{V}(\textbf{U}_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+2}))$ as a
subspace of
$\Phi(L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+2}))|_{\underline{V}_{\underline{l},\emptyset}^{\otimes
m}}.$

###  3.3. $q$-Schur algebras of type $\mathrm{\mathsf{B}}$ and their
gradations

We recall some general facts about $q$-Schur algebras associated with Hecke
algebras of type $\mathrm{\mathsf{B}}$. For more details, one can consult [6]
or [9].

For $M_{f}=\eta_{f(1)}\otimes\eta_{f(2)}\otimes\cdots\otimes\eta_{f(m)}\in V$,
the weight $\lambda(M_{f})$ of $M_{f}$ is defined to be the $(r+1)$-tuple

$(\lambda_{1},\cdots,\lambda_{r+1}),$

where $\lambda_{i}$ is the total number of occurrences of the indices
$i-\frac{1}{2}$ and $-(i-\frac{1}{2})$ in $M_{f}$. The set of all weights will
be denoted by $\Lambda_{\mathrm{\mathsf{B}}}(2r+2,m)$.

The space $V_{\lambda}$ is the span of all the basis vectors $M_{f}$ with
weight $\lambda$, the vector $\mathfrak{M}_{\lambda}$ is the unique basis
vector $M_{f}\in V_{\lambda}$ whose indices satisfy $f(1)\leq
f(2)\leq\cdots\leq f(m)$.

The parabolic subgroup $P_{\lambda}$ of
$\mathscr{W}(\textsf{A}_{m-1})=\left\langle s_{1},\cdots,s_{m-1}\right\rangle$
is generated by some $s_{i}\in\mathscr{W}(\textsf{A}_{m-1})$, which satisfy
$s_{i}\in P_{\lambda}$ if and only if the $i$th and $(i+1)$th entries of
$\mathfrak{M}_{\lambda}$ are equal.

The elements $x_{\lambda}$ of $\mathcal{H}(\mathrm{\mathsf{B}}_{m})$ is
defined by

$x_{\lambda}=\sum_{w\in P_{\lambda}}H_{w}.$

Following [6], we give the following definition.

######  Definition 3.4.

The $q$-Schur algebra of type $\mathrm{\mathsf{B}}_{m}$ is given by

$S_{q}^{\mathrm{\mathsf{B}}}(2r+2,m)={\mathrm{End}}_{\mathcal{H}(\mathrm{\mathsf{B}}_{m})}(\bigoplus_{\lambda\in\Lambda_{\mathrm{\mathsf{B}}}(2r+2,m)}x_{\lambda}\mathcal{H}(\mathrm{\mathsf{B}}_{m})).$

######  Proposition 3.5.

Keep the notations as above. The following statements hold.

  * (1)

$V_{\lambda}\cong x_{\lambda}\mathcal{H}({\mathrm{\mathsf{B}}_{m}})$;

  * (2)

$V^{\otimes
m}\cong\bigoplus_{\lambda\in\Lambda_{\mathrm{\mathsf{B}}}(2r+2,m)}x_{\lambda}\mathcal{H}(\mathrm{\mathsf{B}}_{m})$;

  * (3)

$S_{q}^{\mathrm{\mathsf{B}}}(2r+2,m)$ is a free $\mathbb{Q}(q)$-module.
Moreover, there is a $\mathbb{Q}(q)$-basis in
$S_{q}^{\mathrm{\mathsf{B}}}(2r+2,m)$:
$\\{\phi^{d}_{\lambda,\mu}\mid\lambda,\mu\in\Lambda_{\mathrm{\mathsf{B}}}(2r+2,m)\\}$
with $d$ is a distinguished double $P_{\lambda}-P_{\mu}$ coset representative,
such that each $\phi^{d}_{\lambda,\mu}$ only has nonzero image on $V_{\mu}$ ;

  * (4)

If $\mu=(\mu_{1},\mu_{2},\cdots,\mu_{r+1})$, then $\phi^{d}_{\lambda,\mu}$
only has nonzero image on $V^{\otimes m}_{k}$ where $k=m-\mu_{r+1}$.

###### Proof.

Parts (1)-(3) follow from [6, §4]. As to (4), we remark that
$\phi^{d}_{\lambda,\mu}$ only has nonzero image on $V_{\mu}$ and
$V_{\mu}\subset V^{\otimes m}_{k}$, where $k=m-\mu_{r+1}$ (with the notation
from §2.2). Then (4) follows immediately. ∎

### 3.4.

Now we can establish one part of the property of double centralizers.

######  Theorem 3.6.

If $q$ is transcendental over $\mathbb{Q}$. We have
$\Phi(L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4}))={\mathrm{End}}_{\mathfrak{H}\kern-4.20004pt\mathfrak{H}_{m}}(\underline{V}^{\otimes
m}).$

###### Proof.

By Lemma 3.2, it is enough to show
${\mathrm{End}}_{\mathfrak{H}\kern-4.20004pt\mathfrak{H}_{m}}(\underline{V}^{\otimes
m})\subset\Phi(L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})).$

Note that
${\mathrm{End}}_{\mathfrak{H}\kern-4.20004pt\mathfrak{H}_{m}}(\underline{V}^{\otimes
m})\subset{\mathrm{End}}_{\mathcal{H}(\mathrm{\mathsf{B}}_{m})}(\underline{V}^{\otimes
m})=S_{q}^{\mathrm{\mathsf{B}}}(2r+4,m)$. Thanks to Proposition 3.5, for every
$\phi\in{\mathrm{End}}_{\mathfrak{H}\kern-4.20004pt\mathfrak{H}_{m}}(\underline{V}^{\otimes
m})$, we have
$\phi=\underset{\lambda,\mu\in\Lambda_{\mathrm{\mathsf{B}}}(2r+4,m)}{\sum}a_{\lambda\mu
d}\phi^{d}_{\lambda,\mu}$, where $a_{\lambda\mu d}\in\mathbb{Q}(q)$. More
precisely,

$\phi=\sum_{l=0}^{m}\phi_{l}=\displaystyle\sum_{l=0}^{m}\sum_{\lambda,\mu\in\Lambda_{\mathrm{\mathsf{B}}}(2r+4,m),\mu_{r+2}=m-l}a_{\lambda\mu
d}\phi^{d}_{\lambda,\mu}.$

We have $\phi|_{\underline{V}_{l}^{\otimes m}}=\phi_{l}$ by Proposition
3.5(4). Now we will proceed the arguments by several steps.

(a) We show that $\phi_{l}$ stabilizes $\underline{V}_{I,J}^{\otimes m}$ with
$\\#I=l$.

We consider $\phi_{l}$ stabilizes
$\underline{V}_{\underline{l},\emptyset}^{\otimes m}$ at first. Suppose
$\nu_{j}\in\underline{V}_{\underline{l},\emptyset}^{\otimes m}$, and
$\phi_{l}(\nu_{j})=\nu_{l}+\nu_{k}$ with
$\nu_{l}\in\underline{V}_{\underline{l},\emptyset}^{\otimes m}$,
$\nu_{k}\notin\underline{V}_{\underline{l},\emptyset}^{\otimes m}$. Denote the
identity element of $\mathfrak{S}_{l}$ by $e$, then

$\Xi(x_{e}^{(l)})(\nu_{l}+\nu_{k})=\Xi(x_{e}^{(l)})\phi_{l}(\nu_{j})=\Xi(x_{e}^{(l)})\phi(\nu_{j})=\phi\Xi(x_{e}^{(l)})(\nu_{j})=\nu_{l}+\nu_{k}.$

By the definition of $\Xi$, we have $\Xi(x_{e}^{(l)})(\nu_{l})=\nu_{l}$, and
$\Xi(x_{e}^{(l)})(\nu_{k})=0$. It yields that $\nu_{k}=0$, and hence
$\phi_{l}$ stabilizes $\underline{V}_{\underline{l},\emptyset}^{\otimes m}.$

Then we consider a general summand $\underline{V}_{I,J}^{\otimes m}$ by using
Lemma 2.3. Note that $\phi\Xi(\omega_{I,J}^{-1})=\Xi(\omega_{I,J}^{-1})\phi$,
and
$\Xi(\omega_{I,J}^{-1})\phi(\underline{V}_{\underline{l},\emptyset}^{\otimes
m})=\Xi(\omega_{I,J}^{-1})\phi_{l}(\underline{V}_{\underline{l},\emptyset}^{\otimes
m})\subset\underline{V}_{I,J}^{\otimes m}$. It yields that

$\phi_{l}(\underline{V}_{I,J}^{\otimes
m})=\phi_{l}\Xi(\omega_{I,J}^{-1})(\underline{V}_{\underline{l},\emptyset}^{\otimes
m})=\phi\Xi(\omega_{I,J}^{-1})(\underline{V}_{\underline{l},\emptyset}^{\otimes
m})=\Xi(\omega_{I,J}^{-1})\phi(\underline{V}_{\underline{l},\emptyset}^{\otimes
m})\subset\underline{V}_{I,J}^{\otimes m},$

as desired.

(b) We claim that there exists an element $g_{l}\in
L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})$ satisfying

$\phi_{l}|_{\underline{V}_{l}^{\otimes
m}}=\Phi(g_{l})|_{\underline{V}_{l}^{\otimes m}}.$

We already have that

$\displaystyle\Xi(T_{0})|_{\underline{V}_{\underline{l},\emptyset}^{\otimes
m}}$ $\displaystyle=\Psi_{l}^{V}(H_{0})\otimes\mathsf{id}^{\otimes m-l},$
$\displaystyle\Xi(x_{\sigma}^{(l)})|_{\underline{V}_{\underline{l},\emptyset}^{\otimes
m}}$ $\displaystyle=\Psi_{l}^{V}(H_{\sigma})\otimes\mathsf{id}^{\otimes
m-l}\text{ for any }\sigma\in\mathfrak{S}_{l}.$

On the other hand, consider the $\imath$Schur duality of $\Phi_{l}^{V}$ and
$\Psi_{l}^{V}$ on $\underline{V}_{\underline{l},\emptyset}^{\otimes m}$
(extended from $V^{\otimes l}$). For any
$\phi\in{\mathrm{End}}_{\mathfrak{H}\kern-4.20004pt\mathfrak{H}_{m}}(\underline{V}^{\otimes
m})$, we have

$\phi_{l}|_{\underline{V}_{\underline{l},\emptyset}^{\otimes
m}}\in{\mathrm{End}}_{\mathcal{H}(\mathrm{\mathsf{B}}_{l})}(\underline{V}_{\underline{l},\emptyset}^{\otimes
m})=\Phi_{l}^{V}(\textbf{U}_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+2}))\subset\Phi(L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4}))|_{\underline{V}_{\underline{l},\emptyset}}.$

(See Remark 3.3.) So there exists $g_{l}\in
L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})$ such that
$\phi_{l}|_{\underline{V}_{\underline{l},\emptyset}^{\otimes
m}}=\Phi(g_{l})|_{\underline{V}_{\underline{l},\emptyset}^{\otimes m}}$.
Consequently,

$\phi_{l}(\underline{V}_{\underline{l},\emptyset}^{\otimes
m})=\phi_{l}\Xi(\omega_{I,J})(\underline{V}_{I,J}^{\otimes
m})=\Phi(g_{l})\Xi(\omega_{I,J})(\underline{V}_{I,J}^{\otimes m}).$

Note that
$\Phi(L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4}))\subset{\mathrm{End}}_{\mathfrak{H}\kern-4.20004pt\mathfrak{H}_{m}}(\underline{V}^{\otimes
m})$. Hence $\Phi(g_{l})$ commutes with $\Xi(\omega_{I,J})$. Thus, we have

$\displaystyle\phi_{l}(\underline{V}_{I,J}^{\otimes m})$
$\displaystyle=\Xi(\omega_{I,J}^{-1})\phi_{l}\Xi(\omega_{I,J})(\underline{V}_{I,J}^{\otimes
m})$
$\displaystyle=\Xi(\omega_{I,J}^{-1})\Phi(g_{l})\Xi(\omega_{I,J})(\underline{V}_{I,J}^{\otimes
m})$ $\displaystyle=\Phi(g_{l})(\underline{V}_{I,J}^{\otimes m}).$

Since every direct summands of $\underline{V}_{l}^{\otimes m}$ are stabilized
by $\phi_{l}$, we have $\phi_{l}|_{\underline{V}_{l}^{\otimes
m}}=\Phi(g_{l})|_{\underline{V}_{l}^{\otimes m}}$.

(c) We construct an element of
$L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})$ such that its image under
$\Phi$ coincides with $\phi$, which consequently complete the proof.

For this purpose, we firstly set $X:=k_{r+1}^{r+1}k_{r}^{r}\cdots
k_{2}^{2}k_{1}\in L_{q}^{\imath}(\mathfrak{sl}_{2r+4})$. From formula
(LABEL:action1.2) we have

$\displaystyle\Phi(X)(\eta_{r+\frac{3}{2}})=q^{-(r+1)}\eta_{r+\frac{3}{2}},$
$\displaystyle\Phi(X)(\eta_{-r-\frac{3}{2}})=q^{-(r+1)}\eta_{-r-\frac{3}{2}},$
$\displaystyle\Phi(X)(\eta_{i})=q\eta_{i},\text{ for }i\in\mathbb{I}_{2r+2}.$

Hence

$\Phi(X)|_{\underline{V}_{l}^{\otimes
m}}=q^{l-(m-l)(r+1)}\cdot\mathsf{id}|_{\underline{V}_{l}^{\otimes
m}}=q^{l(r+2)-m(r+1)}\cdot\mathsf{id}|_{\underline{V}_{l}^{\otimes m}}.$

We set $F(l):=q^{l(r+2)-m(r+1)}$ (the integers $m$ and $r$ are fixed). Since
$q$ is transcendental over $\mathbb{Q}$, we have $F(l)\neq F(k)$, for $l\neq
k$.

With the above $X$, we secondly consider the element

$G_{l}:=\frac{(X-F(1))\cdots\widehat{(X-F(l))}\cdots(X-F(m))}{(F(l)-F(1))\cdots\widehat{(F(l)-F(l))}\cdots(F(l)-F(m))},$

where the hat indicates as usual that the term is dropped. Then we have

$\Phi(G_{l})|_{\underline{V}_{l}^{\otimes
m}}=(\frac{(F(l)-F(1))\cdots\widehat{(F(l)-F(l))}\cdots(F(l)-F(m))}{(F(l)-F(1))\cdots\widehat{(F(l)-F(l))}\cdots(F(l)-F(m))})\cdot\mathsf{id}|_{\underline{V}_{l}^{\otimes
m}}=\mathsf{id}|_{\underline{V}_{l}^{\otimes m}},$

and

$\Phi(G_{l})|_{\underline{V}_{k}^{\otimes
m}}=(\frac{(F(k)-F(1))\cdots\widehat{(F(k)-F(l))}\cdots(F(k)-F(m))}{(F(k)-F(1))\cdots\widehat{(F(k)-F(l))}\cdots(F(k)-F(m))})\cdot\mathsf{id}|_{\underline{V}_{l}^{\otimes
m}}=0$

for $k\neq l$. Thus

$\displaystyle\Phi(g_{l})\Phi(G_{l})\cdot{V}_{l}^{\otimes m}$
$\displaystyle=\phi_{l}\cdot\underline{V}_{l}^{\otimes m};$
$\displaystyle\Phi(g_{l})\Phi(G_{l})\cdot\underline{V}_{k}^{\otimes m}$
$\displaystyle=\phi_{l}\cdot\underline{V}_{k}^{\otimes m}=0\;\text{ for }l\neq
k.$

Hence $\phi_{l}=\Phi(g_{l}\cdot G_{l})$. So we finally succeed in finding a
desired element

$\sum_{l=0}^{m}g_{l}\cdot G_{l}\in L_{q}^{\imath}(\mathfrak{sl}_{2r+4}),$

which satisfies

$\phi=\sum_{l=0}^{m}\phi_{l}=\Phi(\sum_{l=0}^{m}g_{l}\cdot
G_{l})\in\Phi(L_{q}^{\imath}(\mathfrak{sl}_{2r+4})).$

The proof is completed. ∎

### 3.5.

Now we begin to deal with the other part of the double centraliser property.
Under our assumption of $q$ being transcendental over $\mathbb{Q}$, we have
that $V^{\otimes l}$ is a completely reducible
$\mathcal{H}(\mathrm{\mathsf{B}}_{l})$-module. Suppose $V^{\otimes
l}=\underset{\pi\in I_{l}}{\bigoplus}S^{\pi}$, where $S^{\pi}$ is irreducible
$\mathcal{H}(\mathrm{\mathsf{B}}_{l})$-module and $I_{l}$ is the index set.
(We are not going to introduce this index set for simplicity, see details in
[3, §5.5].) Combining with notations in §2.2, we have

$\underline{V}_{\underline{l},\emptyset}^{\otimes m}=V^{\otimes
l}\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l}=\underset{\pi\in
I_{l}}{\bigoplus}S^{\pi}\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l},$

and hence

$\underline{V}_{I,J}^{\otimes m}=\Xi(\omega_{I,J}^{-1})(\underset{\pi\in
I_{l}}{\bigoplus}S^{\pi}\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l}).$

So we get

$\displaystyle\underline{V}_{l}^{\otimes m}$
$\displaystyle=\underset{\\#I=l,J\subset\underline{m}\backslash
I}{\bigoplus}\underline{V}_{I,J}^{\otimes m}$
$\displaystyle=\underset{\\#I=l,J\subset\underline{m}\backslash
I}{\bigoplus}\Xi(\omega_{I,J}^{-1})(\underset{\pi\in
I_{l}}{\bigoplus}S^{\pi}\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l})$
$\displaystyle=\underset{\pi\in
I_{l}}{\bigoplus}\underset{\\#I=l,J\subset\underline{m}\backslash
I}{\bigoplus}\Xi(\omega_{I,J}^{-1})(S^{\pi}\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}).$

Under the isomorphism $\underline{V}_{I,J}^{\otimes m}\simeq V^{\otimes l}$,
all $\Xi(\omega_{I,J}^{-1})(S^{\pi}\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l})$
are isomorphic to $S^{\pi}$ as an $\mathcal{H}(B_{l})$-module and we denote it
as $S^{\pi}_{I,J}$. We set
$D^{\pi}_{l}:=\underset{\\#I=l,J\subset\underline{m}\backslash
I}{\bigoplus}S^{\pi}_{I,J}$. The following lemma shows the property of
$D^{\pi}_{l}$.

######  Lemma 3.7.

Suppose that q is transcendental over $\mathbb{Q}$. Then $D^{\pi}_{l}$ is an
irreducible $\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-module.

###### Proof.

We first prove $D^{\pi}_{l}$ is an
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-module. It is sufficient to show
that $D^{\pi}_{l}$ is stabilized by $T_{i}$, with $i=0,1,\cdots,m-1$, and
$x_{\sigma}^{(l)}$, with $\sigma\in\mathfrak{S}_{l}$ for $0\leq l\leq m$.

From the proof of Lemma 2.3, $\Xi(\omega_{I,J}^{-1})$ only changes the
positions of some tensor factors and takes some $\eta_{r+\frac{3}{2}}$ to
$\eta_{-r-\frac{3}{2}}$. The action of $T_{0}$ preserves $S^{\pi}_{I,J}$ if
$1\in I$ and takes $S^{\pi}_{I,J}$ to $S^{\pi}_{I,J^{\prime}}$ for some subset
$J^{\prime}$ if $1\notin I$. Now consider $T_{i}$ with $i>0$. Note that
$-r-\frac{3}{2}<k<r+\frac{3}{2}$ for any $k\in\mathbb{I}_{2r+2}$. If both $i$
and $i+1$ belong to $I,J$, or $\underline{m}\backslash(I\cap J)$, the action
of $T_{i}$ preserves $S^{\pi}_{I,J}$. Otherwise, the action of $T_{i}$ or
$T_{i}^{-1}$ on $S^{\pi}_{I,J}$ takes $S^{\pi}_{I,J}$ to
$S^{\pi}_{I^{\prime},J^{\prime}}$ for some subsets $I^{\prime}$ and
$J^{\prime}$ with $\\#I^{\prime}=\\#I=l$. In summary, $D^{\pi}_{l}$ is stable
under the actions of all $T_{i}$.

By definition, $\Xi(x_{\sigma}^{(k)})(D^{\pi}_{l})=0$ for $k\neq l$ and
$\Xi(x_{\sigma}^{(l)})$ only has nonzero image on
$S^{\pi}\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l}$. As in the proof of Lemma
3.2, $\Xi(x_{\sigma}^{(l)})(S^{\pi}\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})=(\Psi_{l}^{V}(H_{\sigma})S^{\pi})\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}$ for irreducible $\mathcal{H}(\mathrm{\mathsf{B}}_{l})$-module $S^{\pi}$.
It shows that $x_{\sigma}^{(l)}$ stabilizes $D^{\pi}_{l}$. We get therefore
$D^{\pi}_{l}$ is an $\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-module.

We then show that $D^{\pi}_{l}$ is irreducible as an
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-module. It is enough to show
that for any nonzero vector $\nu\in D^{\pi}_{l}$, the
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-submodule generated by $\nu$
coincides with $D^{\pi}_{l}$, the former of which is denoted by $N$ in the
following.

Firstly, we can write

$\nu=\underset{\\#I=l,J\subset\underline{m}\backslash
I}{\sum}\Xi(\omega_{I,J}^{-1})(\nu_{I,J}\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}),$

where $\nu_{I,J}\in S^{\pi}$. By definition,
$\Xi(x_{e}^{(l)})\Xi(\omega_{I,J})(\nu)=\nu_{I,J}\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}$ for the identity element $e\in\mathfrak{S}_{l}$. Recall that $S^{\pi}$
is an irreducible $\mathcal{H}(\mathrm{\mathsf{B}}_{l})$-module, and

$\displaystyle\Xi(x_{\sigma}^{(l)})(\nu_{I,J}\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l})$
$\displaystyle=(\Psi_{l}^{V}(H_{\sigma})\nu_{I,J})\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}\text{ for }\sigma\in\mathfrak{S}_{l},$
$\displaystyle\Xi(T_{0})(\nu_{I,J}\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l})$
$\displaystyle=(\Psi_{l}^{V}(H_{0})\nu_{I,J})\otimes\eta_{r+\frac{3}{2}}^{\otimes
m-l}.$

By the action of $\Xi(x_{\sigma}^{(l)})$ with $\sigma$ running over
$\mathfrak{S}_{l}$, along with $\Xi(T_{0})$, it is readily known that $N$
contains the subspace $S^{\pi}\otimes\eta_{r+\frac{3}{2}}^{\otimes m-l}$,
which is the generator of $D^{\pi}_{l}$. Hence $N$ coincides with
$D^{\pi}_{l}$. We conclude that $D^{\pi}_{l}$ is an irreducible
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-module. ∎

######  Corollary 3.8.

If $q$ is transcendental over $\mathbb{Q}$. Then $\underline{V}^{\otimes m}$
is a completely reducible
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-module.

###### Proof.

Recall that

$\underline{V}^{\otimes m}=\bigoplus^{m}_{l=0}\underline{V}_{l}^{\otimes
m}=\bigoplus_{l=0}^{m}\bigoplus_{\pi\in I_{l}}D^{\pi}_{l}.$

By Lemma 3.7, $\underline{V}^{\otimes m}$ is a completely reducible
$\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m}$-module. ∎

######  Theorem 3.9.

If $q$ is transcendental over $\mathbb{Q}$, then
$\Xi(\mathfrak{H}\kern-6.00006pt\mathfrak{H}_{m})={\mathrm{End}}_{L_{q}^{\imath}(\mathfrak{s}\mathfrak{l}_{2r+4})}(\underline{V}^{\otimes
m})^{\rm{op}}$.

###### Proof.

By Theorem 3.6 and Corollary 3.8, we can use classical double commutant
theorem (see [5, §4.1.13]) to reach this conclusion. ∎

######  Remark 3.10.

(1) Recently, Di Wang developed a “super” version of degenerate duplex Hecke
algebras and related Schur-Sergeev duality of Levi-type (see [13]).

(2) In the case of odd dimensional vector spaces, the duality still holds and
the constructions are analogously.

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