# Demonstration of Hong-Ou-Mandel interference in an LNOI directional coupler

Silia Babel1  Laura Bollmers1  Marcello Massaro1  Kai Hong Luo1  Michael
Stefszky1  Federico Pegoraro1  Philip Held1  Harald Herrmann1  Christof
Eigner1  Benjamin Brecht1  Laura Padberg 1  and Christine Silberhorn1  1
Paderborn University, Integrated Quantum Optics, Institute for Photonic
Quantum Systems (PhoQS), Warburger Str. 100, 33098 Paderborn, Germany
<EMAIL_ADDRESS>
###### Abstract

Interference between single photons is key for many quantum optics experiments
and applications in quantum technologies, such as quantum communication or
computation. It is advantageous to operate the systems at telecommunication
wavelengths and to integrate the setups for these applications in order to
improve stability, compactness and scalability. A new promising material
platform for integrated quantum optics is lithium niobate on insulator (LNOI).
Here, we realise Hong-Ou-Mandel (HOM) interference between telecom photons
from an engineered parametric down-conversion source in an LNOI directional
coupler. The coupler has been designed and fabricated in house and provides
close to perfect balanced beam splitting. We obtain a raw HOM visibility of
($93.5\pm 0.7$) $\%$, limited mainly by the source performance and in good
agreement with off-chip measurements. This lays the foundation for more
sophisticated quantum experiments in LNOI.

††journal: oe††articletype: Research Article

##  1 Introduction

Linear optical networks are the basis of quantum technologies. Many important
applications of quantum technologies such as quantum key distribution [1] and
boson sampling [2] were demonstrated using linear optics. At the heart of
these linear networks lie optical beam splitters (BS) and phase shifters,
since it was shown that with these two components any unitary matrix operation
can be implemented [3].
A particular interest has been devoted to the study and realisation of
integrated optical circuits, since they offer high stability, the possibility
of compact devices and high efficiencies, and thus enable scalability in
contrast to their free space or bulk counterparts. The history of integrated
linear networks is summarised in many review articles see, e.g., [4].
A new promising material for the implementation of future quantum circuits is
lithium niobate on insulator (LNOI) since it combines a high integration
density, as known from silicon photonics, with its unique functionalities of
electro-optical modulation and a second order nonlinearity [5]. It consists of
a thin-film of lithium niobate (LN) (300 nm to 1000 nm) bonded on a SiO2
layer. The light is guided in the LN thin-film and thus it inherits the
diverse property portfolio of LN, e.g., its wide transparency window, its
large second order non-linearity and electro-optical coefficients [6]. Due to
its structure with a SiO2 layer underneath and the low dimension of the LN
thin-film a strong confinement for rib waveguides can be realised, giving rise
not only to the high integration density, but also to a highly improved
effective non-linearity and the possibility of dispersion engineering.
LNOI has already been employed for the realisation of a variety of devices,
such as electro-optic modulators [7, 8, 9], directional couplers [10, 11], as
well as for second harmonic generation [12, 13, 14] and spontaneous parametric
down-conversion [15, 16, 17]. All of these functionalities combined with the
possibility to achieve very low loss waveguides [18] illustrate the high
potential of LNOI for future photonic quantum technologies. A key building
block of quantum networks is two-photon Hong-Ou-Mandel (HOM) interference.
Here, we demonstrate HOM interference of photons at telecommunication
wavelengths from a parametric down-conversion source in an LNOI directional
coupler. We observe an interference visibility of ($93.5\pm 0.7$) $\%$,
limited by the source performance and similar to an off-chip measurement. This
constitutes an important building block for realising optical quantum networks
on LNOI and paves the way towards future applications.
This paper is structured as follows. In section 2, we briefly discuss the
basics of HOM interference. Section 3 details the theory and the modeling of
the LNOI directional coupler and we describe the fabrication process in
section 4. In section 5 and section 6, we present the characterisation of the
coupler and the HOM measurements, respectively. Section 7 concludes the paper.

##  2 Fundamentals

HOM interference describes the bunching of two indistinguishable photons at a
balanced, non-polarising beam splitter (BS) where both photons enter from one
input port each, but leave them together in one of the outputs. Typically, the
degree of distinguishability between the two photons is continuously changed
from completely distinguishable to fully indistinguishable, by introducing a
time delay between the photons. Monitoring the number of coincidence events
between the output ports of the BS as function of delay then yields the famous
HOM dip [19]. The visibility of this dip is ideally 100 $\%$ but typically
limited by the imbalance of the BS and the indistinguishability of the
impinging photons. Here, we compare the dip visibility, which we obtain with
our LNOI device, against the visibility, which we observe with an off-chip
reference measurement. This allows us to gauge the impact of the LNOI
directional coupler on the photon indistinguishability and hence the
applicability of LNOI in integrated optical quantum networks.
Perfect HOM interference is only observed with a perfectly balanced BS.
Unavoidable fabrication intolerances will, however, lead to deviations from a
50:50 splitting. The impact of these on the HOM visibility can be summarised
as [20]

$V_{max}=\frac{2\cdot\eta\cdot(1-\eta)}{1-2\cdot\eta+2\cdot\eta^{2}},$ (1)

where $\eta$ is the reflectivity of the BS. To fabricate a 50:50 directional
coupler, we first carried out numerical simulations to estimate the coupling
behaviour. These are discussed in the next section.

##  3 Simulations of Waveguide and Directional Coupler Structures

The aim of our simulations is to find a suitable waveguide and directional
coupler geometry that meets our needs such as single modeness, short coupling
length, a gap width that is compatible with our fabrication capabilities, and
a large bandwidth. For the numerical evaluation, we briefly recap the
operation principle of a directional coupler.
A directional coupler consists of two waveguides that are brought close to one
another for a given length, $L_{I}$, with a small gap given by $G$. Along the
length $L_{I}$ the coupling takes place via the evanescent fields of the two
waveguides. Mathematically, this is explained by coupled mode theory which
yields that the incoupled power oscillates sinusoidally between the two
waveguides [21]. The coupling depends on the wavelength, the waveguide
geometry, the interaction length and the gap and is summarised by the coupling
coefficient. A user-chosen splitting ratio can be realised by adapting $L_{I}$
appropriately.
We first investigated the single modeness of a single waveguide using the MODE
solver of Lumerical [22].

(a) Single waveguide mode.

(b) Symmetric mode.

(c) Anti-symmetric mode.

Figure 1: Simulations of waveguide structures in LNOI. The three images show
the model of a single waveguide and of the directional coupler. The parameters
of the waveguide that can be changed are the thin-film thickness, the etching
depth, the top width, the angle and the thickness of the cladding layer and
for the directional coupler the gap can be varied. For more information, see
text.

For that, we fixed the thin-film thickness to 600 nm and used X-cut LNOI. The
sidewall angle of the waveguide is set to 60 ° mirroring our fabrication
process. Furthermore, the top width is adjusted to 1 µm and we applied a
cladding layer of SiO2. For these parameters we simulated different etching
depths to find a single mode waveguide geometry and chose 150 nm as an etching
depth. In 1(a) a single waveguide with the discussed geometry is shown. We can
see the fundamental mode guided in the LN thin-film, the SiO2 substrate
underneath the LN and the SiO2 cladding layer.
Afterwards, we simulated the coupling behaviour of a directional coupler and
we introduced a second waveguide to our model defining the center to center
distance between the two waveguides as a new parameter called gap $G$. The
calculated symmetric and anti-symmetric mode are shown in 1(b) and 1(c). The
resolution of the laser lithography used for defining the structures placed a
lower limit on our gap width, which set a minimum coupling length for us. A
gap width of 2.3 µm yielded a coupling length of 137 µm, leading to small
device footprint well suited for high-density integration.

Figure 2: The graph shows the normalised splitting ratio as a function of the
wavelength. The splitting ratio lies within 1 $\%$ of the desired 50:50 value
for wavelengths between 1540 nm and 1560 nm.

For characterising the bandwidth of the coupler we show in Figure 2 the
normalised splitting ratio as a function of the wavelength. In a range of
$\pm$ 10 nm the splitting ratio only varies by less than 1 $\%$ which shows
that the coupler is wavelength independent for a broad range. Since the
bandwidth of the single photons used for the HOM experiment is 1.8 nm at a
wavelength of 1542.22 nm, we found a stable geometry with a large enough
bandwidth for our experiment. In the next step, we fabricated this geometry to
test the coupling behaviour.

##  4 Fabrication of Directional Couplers

For the analysis of our directional couplers, we produced a sample with
multiple waveguide groups consisting of a directional coupler and a straight
waveguide to estimate the propagation losses. We changed the interaction
length $L_{I}$ of the couplers from 30 µm to 580 µm to be able to determine
the coupling length as explained in the next section.
We fabricate the waveguides via a physical dry etching process utilizing SiO2
structured by a lift-off process as an etching mask. To realise the etching
mask, we spin coat photo resist onto a LNOI sample (NanoLN), which is
subsequently structured with a laser lithography system (Heidelberg
Instruments DWL 66+). After developing, SiO2 is deposited on the sample via
sputtering (Prevac Sputtering system 518) and is structured by a lift-off
process. Afterwards, the structured SiO2 is used as an etching mask to
transfer the structure in the lithium niobate thin-film layer. We use a dry
etching process with a pure Argon plasma (Oxford Instruments
PlasmalabSystem100). In a last step, the end-facet of the sample are chemo-
mechanically polished. For this purpose, a SiO2 cladding layer is evaporated
to protect the sample during polishing.

##  5 Linear Characterisation of Waveguides and Directional Couplers

We measured the propagation losses of eight straight reference waveguides at a
wavelength of 1550 nm using the so-called Fabry-Pérot method [23]. The light
was coupled into and out of the LNOI waveguides with aspheric lenses. From
these measurements, we estimate average propagation losses of ($4.85\pm 0.95$)
dB/cm and note that these do not impede the quality of HOM interference, but
do impact the overall measurement time.
Next, we investigated the coupling behaviour of the directional couplers. For
that, we coupled laser light with a wavelength of 1550 nm (Santec TSL-550)
into one input arm of the coupler and imaged the two output ports on a camera
(Xenics Wildcat 640).

(a) Interaction length: 58 µm.

(b) Interaction length: 257 µm.

Figure 3: Output modes of two directional coupler with different interaction
length.
(a) All light coupled from one waveguide to the other. (b) A nearly 50:50
splitting ratio is reached.

In Figure 3 two examples of measured output modes can be seen. In 3(a) all the
light from one input port coupled to the opposite output port. In contrast, in
3(b) the incoupled light is split nearly 50:50. We then calculated for each
output port the power ratio via this relation:

$P=\frac{P_{1}}{P_{1}+P_{2}}.$ (2)

Here, $P_{1,2}$ is the power of the output mode 1 and 2, respectively. In
Figure 4 the power ratio is plotted against the interaction length of the
directional coupler for incoupling into input $a$ and input $b$. The offset
for zero coupling length is due to coupling in the bendings leading to and
from the coupling region. This effectively reduces the coupling length
required for 50:50 splitting.
We extract the coupling length of the directional coupler from a sinusoidal
fit, which yields lengths of ($114.85\pm 0.04$) µm and ($110.87\pm 0.04$) µm
for inputs $a$ and $b$, respectively. The slight discrepancy can be caused by
inhomogeneities in the fabrication. The average coupling length is ($112.86\pm
2.82$) µm, which is in excellent agreement with the expected 120 µm predicted
from the model.
For the HOM experiment a 50:50 coupler is needed. For this we choose the
coupler that is nearest to this splitting ratio which is the one with the
interaction length of 257 µm. Note that this implied that we use a higher-
order coupling process (light couples to and fro once), which is expected to
have a reduced spectral bandwidth compared to a 0-th order coupling.

Figure 4: Power ratio as a function of the interaction length. The measurement
of the coupling behaviour shows that the light oscillates as expected between
the two waveguides. The coupling length is (112.86$\pm$2.82) µm.

##  6 Hong-Ou-Mandel Interference Experiment

For our final quantum experiment of the HOM interference, we realized a novel
set-up as depicted in Figure 5.

Figure 5: Setup for the quantum experiment. The setup consists of two main
parts. The first part is the preparation of the single photons and the second
part consists of the LNOI BS.

The setup consists of two main parts. The first one is the source of the
single photons generated by a spontaneous parameteric down-conversion (PDC)
process in a ppKTP waveguide, and the second part is the LNOI BS. The state
preparation of the photons is the same as in [24]. A type-II PDC process
generates two photons with orthogonal polarisations at a wavelength of 1542.22
nm. The pulsed pump laser has a wavelength of 772.5 nm with a bandwidth of
$\approx$0.3 nm. We place a narrow band spectral filter with a FWHM of 1.8 nm
after the ppKTP sample to suppress background radiation. We set the mean
photon number of the generated PDC to a regime below 0.01 pairs per pulse to
reduce the generation of higher photon components which would reduce the
visibility. After the ppKTP sample the two photons are separated with a
polarising BS and sent to delay lines. After the delay line each photon is
coupled into a polarisation maintaining fiber and those fibers route to the BS
setup where we couple the two photons from fibers to free space. The
polarisation of one of the photons is controlled with a half-wave and quarter-
wave plate to ensure indistinguishability.
The photons are coupled into the two inputs of the LNOI BS using a single
aspheric lens. After the BS, we collect the photons with another lens and
couple them to two standard SMF-28 single mode fibers. These fibers are
connected to two superconducting nanowire single photon detectors (SNSPD) with
a 70 ns dead time and efficiencies >95 $\%$ which are connected to a time
tagger. To measure the HOM dip, we scan the delay line in 1µm steps for 250
µm. We perform several scans and add up the coincidences, to ensure that
longterm intensity fluctuations are not affecting the HOM dip. We extract the
visibility from a Gaussian fit (see Figure 6) and obtain a visibility of
($93.5\pm 0.7$) $\%$, where we did not subtract any background counts. Please
note that the data shown in Figure 6 has been normalised such that the
baseline of the Gaussian fit is equal to 1 for better visualisation.

Figure 6: Measured Hong-Ou-Mandel interference of the signal and idler photons
interfered at the LNOI directional coupler. The Gaussian fit
$1-0.935\cdot\text{exp}(-0.143\cdot t^{2})$ yields a visibility of ($93.5\pm
0.7$) $\%$.

Finally, we calculate the expected maximum visibility. From our model (see
Figure 4), we extract an average splitting ratio of (54.6$\pm$3.8) $\%$,
slightly off from perfect balanced splitting. This limits the HOM visibility
to (98.32$\pm$2.8) $\%$. Further, we measured the HOM visibility of the source
by replacing the LNOI BS with a tunable fibre BS and find a HOM visibility of
(98.01$\pm$0.24) $\%$. Combining the source visibility with the imperfect
splitting ratio of the LNOI coupler, we obtain a maximum expected visibility
of (96.36$\pm$2.8) $\%$. Compared with our measured visibility of
(93.5$\pm$0.7) $\%$, we have a deviation of 3 $\%$. This could be due to
effects that occur in high confined waveguide structures such as hybrid modes
which have already been witnessed in LNOI [25]. These hybrid modes can have an
influence on the polarisation of the photons which would alter the splitting
ratio of the devices and thus reduce the visibility of the dip. However, this
effect is challenging to quantify. Note that even though we did not reach the
maximal visibility, the measured visibility agrees with the expected one
within the uncertainty range. This demonstrates that the LNOI BS does not
adversely effect HOM interference.

##  7 Conclusion

In this paper, we demonstrated HOM interference with telecom photons on LNOI,
the missing piece for establishing LNOI as quantum-ready platform for
integration. For this, we fabricated a directional coupler based on numerical
simulations. After that, we estimated the losses and the coupling behaviour.
Finally, we performed a HOM experiment with single photons generated via a PDC
process in ppKTP. With a measured visibility of (93.5$\pm$0.7) $\%$ we can
conclude that LNOI is a favourable platform for integrated quantum optics.
Connecting several of these directional couplers to a sophisticated network
and combining it with already shown single photon sources and electro-optical
modulators in LNOI opens the possibility for a quantum processor in LNOI.

Funding Deutsche Forschungsgemeinschaft (SFB-Geschäftszeichen TRR142/3-2022,
Projektnummer 231447078.); Max Planck School of Photonics.

Acknowledgments Funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – SFB-Geschäftszeichen TRR142/3-2022 – Projektnummer
231447078. Laura Padberg and Silia Babel are part of the Max Planck School of
Photonics supported by the German Federal Ministry of Education and Research
(BMBF), the Max Planck Society, and the Fraunhofer Society.

Disclosures The authors declare no conflicts of interest.

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# Note on some properties of generalized affine Grassmannian slices

Yehao Zhou  Perimeter Institute for Theoretical Physics<EMAIL_ADDRESS>
###### Abstract.

Braverman, Finkelberg and Nakajima introduced the generalized affine
Grassmannian slices $\overline{\mathcal{W}}^{\lambda}_{\mu}$ and showed that
they are Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories. We prove a
conjecture of theirs that a transversal slice of $\mathcal{W}^{\nu}_{\mu}$ in
$\overline{\mathcal{W}}^{\lambda}_{\mu}$ is isomorphic to
$\overline{\mathcal{W}}^{\lambda}_{\nu}$. In addition, they conjecture that
Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories have symplectic
singularities, and we confirm this conjecture for generalized affine
Grassmannian slices $\overline{\mathcal{W}}^{\lambda}_{\mu}$. Along the way we
give a new proof of the smoothness of $\mathcal{W}^{\nu}_{\mu}$, which has
been previously proven by Muthiah and Weekes using different method.

##  1\. Introduction

For a complex reductive group $G$, and a pair of dominant coweights
$\lambda,\mu$ such that $\mu\leq\lambda$, the affine Grassmannian slice
$\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$ is a transversal slice
of $G_{\mathcal{O}}$-orbit $\operatorname{Gr}^{\mu}$ in the
$G_{\mathcal{O}}$-orbit closure
$\operatorname{\overline{\mathrm{Gr}}}^{\lambda}$. As they are transversal
slices, they contain important information on singularities of
$\operatorname{\overline{\mathrm{Gr}}}^{\lambda}$. It is shown that
$\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$ has symplectic
singularities [KWWY14], and this implies that
$\operatorname{\overline{\mathrm{Gr}}}^{\lambda}$ has rational and Gorenstein
singularities. The also play significant role in representation theory, for
example the intersection cohomology of
$\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$ can be identified with
the the weight $\mu$ subspace $V^{\lambda}(\mu)$ of the irreducible
representation $V^{\lambda}$ of the Langlands dual group $G^{\vee}$ of highest
weight $\lambda$, by the geometric Satake correspondence.

Recently, Braverman, Finkelberg and Nakajima defined a generalization of
affine Grassmannian slices in their seminal paper [BFN16a], which is still
denoted by $\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$ but $\mu$
is not required to be dominant (we will recall its definition in Section 2).
They have shown that the generalized affine Grassmannian slices are Coulomb
branches of $3d$ $\mathcal{N}=4$ quiver gauge theories, when $G$ is of ADE-
type, and later work [NW19] extended this result to BCFG-type using quivers
with symmetrizers.

Physically, a Coulomb branch is an irreducible component of the moduli space
of vacua of a supersymmetric gauge theory [BDG15]. It is argued in the physics
literature that Coulomb branch of $3d$ $\mathcal{N}=4$ gauge theory is the
moduli space of singular $G_{c}$-monopoles on $\mathbb{R}^{3}$ [HW97, Ton99,
dBHO+97, dBHOO97, CK98], denoted by $\mathcal{M}_{C}(\mu,\lambda)$, where
$G_{c}$ is the compact form of $G$, $\mu,\lambda:S^{1}\to G_{c}$ are coweights
attached at $\infty,0$ of $\mathbb{R}^{3}$. Here $\mu$ is called the magnetic
charge, and $\lambda$ is called the monopole charge which is assumed to be
dominant.

The variety $\mathcal{M}_{C}(\mu,\lambda)$ is a remarkable object. It carries
a natural Poisson structure, together with a canonical quantization
$\mathcal{M}^{q}_{C}(\mu,\lambda)$ with respect to the natural Poisson
structure [BFN16b]. It is expected to be symplectic dual to the Higgs branch
$\mathcal{M}_{H}(\mu,\lambda)$ of the same gauge theory [BPW12, BLPW14], and
the symplectic duality has been worked out in many cases when the gauge group
is simply-laced [Web16, KTW+18]. These features make it an ideal test ground
for geometric representation theory.

In this note we will focus on some basic geometric properties of
$\mathcal{M}_{C}(\mu,\lambda)$. According to the “monopole bubbling” [IOT12,
GOP12], the moduli space of singular $G_{c}$-monopoles on $\mathbb{R}^{3}$ has
a decomposition

$\mathcal{M}_{C}(\mu,\lambda)=\coprod_{\begin{subarray}{c}\nu:\text{
dominant}\\\
\mu\leq\nu\leq\lambda\end{subarray}}\mathring{\mathcal{M}}_{C}(\mu,\nu)$

where $\mathring{\mathcal{M}}_{C}(\mu,\nu)$ denotes the moduli space of
genuine $G_{c}$-monopoles on $\mathbb{R}^{3}$ with magnetic charge $\mu$ and
monopole charge $\nu$. In [Nak15], Nakajima made the following conjecture:

######  Conjecture 1.1.

$\mathring{\mathcal{M}}_{C}(\mu,\nu)$ is a symplectic leaf of
$\mathcal{M}_{C}(\mu,\lambda)$, and its transversal slice in
$\mathcal{M}_{C}(\mu,\lambda)$ is isomorphic to $\mathcal{M}_{C}(\nu,\mu)$.

By the results in [BFN16a, NW19], generalized affine Grassmannian slices are
examples of Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories
(with appropriate symmetrizers if needed), and in this case we have
$\mathcal{M}_{C}(\mu,\lambda)=\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$
and
$\mathring{\mathcal{M}}_{C}(\mu,\nu)=\operatorname{\mathcal{W}}^{\nu}_{\mu}$,
in particular we have the following “monopole bubbling” decomposition

$\overline{\mathcal{W}}^{\lambda}_{\mu}=\coprod_{\begin{subarray}{c}\nu:\text{
dominant}\\\ \mu\leq\nu\leq\lambda\end{subarray}}\mathcal{W}^{\nu}_{\mu}.$

The main result of this note is that we confirm the Conjecture 1.1 for
$\overline{\mathcal{W}}^{\lambda}_{\mu}$. We will review the definitions of
$\overline{\mathcal{W}}^{\lambda}_{\mu}$ and $\mathcal{W}^{\nu}_{\mu}$, and
explain the nature of this decomposition in Section 2. It is explained in
[BFN16a, Remark 3.19] that to prove the first part of Conjecture 1.1 for
$\overline{\mathcal{W}}^{\lambda}_{\mu}$, it suffices to prove the following

######  Theorem 1.2.

$\mathcal{W}^{\nu}_{\mu}$ is smooth.

This theorem has been proven in [MW19]. In Section 2, we will use a different
method to reprove Theorem 1.2, which is inspired from the proof of the Cohen-
Macaulay property of $\overline{\mathcal{W}}^{\lambda}_{\mu}$ in [BFN16a,
Lemma 2.16]. In fact, we will prove a slightly stronger result, which holds
for the Beilinson-Drinfeld version of the generalized affine Grassmannian
slices, see Proposition 2.4. In Section 4, we prove the second part of
Conjecture 1.1 for $\overline{\mathcal{W}}^{\lambda}_{\mu}$:

######  Theorem 1.3.

A transversal slice of $\mathcal{W}^{\nu}_{\mu}$ in
$\overline{\mathcal{W}}^{\lambda}_{\mu}$ is isomorphic to
$\overline{\mathcal{W}}^{\lambda}_{\nu}$.

Note that this theorem is known in [BFN16a] for two special cases: dominant
$\mu$ and $\mu\leq w_{0}\lambda$.

In [BFN16b], Braverman et. al. made the following conjecture on the
singularities of the Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories.

######  Conjecture 1.4.

$\mathcal{M}_{C}(\mu,\lambda)$ has symplectic singularities.

For gauge theories defined by simple quivers, this conjecture has been proven
by Weekes using Coulomb branch techniques [Wee20]. In particular, this implies
that $\overline{\mathcal{W}}^{\lambda}_{\mu}$ has symplectic singularities
when $G$ is of ADE-type. Using techniques developed in [NW19], his result
extends to BCFG-type as well. We reprove the special case of this conjecture
for $\overline{\mathcal{W}}^{\lambda}_{\mu}$ in Section 4 without Coulomb
branch machinery. Finally, we end this note with generalization of [KWWY14,
Theorem 2.7, 2.9] to not necessarily dominant $\mu$.

### Acknowledgement

The author would like to thank Alex Weekes and Vasily Krylov for discussions
and comments on earlier version of this note. This research is supported by
Perimeter Institute for Theoretical Physics. Research at Perimeter Institute
is supported by the Government of Canada through Industry Canada and by the
Province of Ontario through the Ministry of Research and Innovation.

##  2\. Smoothness of $\mathcal{W}^{\lambda}_{\mu}$

Let $G$ be a split reductive group scheme over $\mathbb{Z}$, let’s recall the
definition of the generalized affine Grassmannian slice
$\overline{\mathcal{W}}^{\lambda}_{\mu}$ for a dominant coweight $\lambda$,
and a coweight $\mu$ such that $\lambda\geq\mu$ [BFN16a]. First, we have a
functor $\overline{\mathsf{W}}^{{\lambda}}_{\mu}$ sending a scheme $S$ to the
data

  * (1)

a $G$-bundle $\mathcal{F}_{G}$ on $\mathbb{P}^{1}_{S}$;

  * (2)

a trivialization $\sigma$ of $\mathcal{F}_{G}$ on $\mathbb{P}^{1}_{S}-0$ with
pole of order $\leq\lambda$;

  * (3)

a $B$-structure $\phi$ on $\mathcal{F}_{G}$ of degree $w_{0}\mu$ having fiber
$B\times S$ at $\infty\times S\subset\mathbb{P}^{1}_{S}$.

It’s easy to see that $\overline{\mathsf{W}}^{{\lambda}}_{\mu}$ is represented
by a finite type scheme over $\mathbb{Z}$, and we define
$\overline{\mathcal{W}}^{\lambda}_{\mu}$ to be the reduced scheme
$\overline{\mathsf{W}}^{{\lambda}}_{\mu,\mathrm{red}}$.

From the definition, we read out that there is a Cartesian diagram

${\overline{\mathcal{W}}^{{\lambda}}_{\mu}}$${\operatorname{Bun}_{B}^{w_{0}\mu}(\mathbb{P}^{1},\infty)}$${\operatorname{\overline{\mathrm{Gr}}}^{\lambda}}$${\operatorname{Bun}_{G}(\mathbb{P}^{1},\infty)}$$\scriptstyle{\mathbf{p}}$$\scriptstyle{\mathbf{r}}$$\scriptstyle{p}$$\scriptstyle{h^{\rightarrow}_{G}}$

where $\operatorname{Bun}_{G}(\mathbb{P}^{1},\infty)$ is the stack of
$G$-bundles on $\mathbb{P}^{1}$ with a trivialization at $\infty$, and
$\operatorname{Bun}_{B}^{w_{0}\mu}(\mathbb{P}^{1},\infty)$ is the degree
$w_{0}\mu$ component. In this diagram, the map $p$ sends a $B$-bundle to its
induced $G$-bundle, and the map $h^{\rightarrow}_{G}$ sends a pair
$(\mathcal{F}_{G},\sigma)$ in
$\operatorname{\overline{\mathrm{Gr}}}^{\lambda}$ to the $G$-bundle
$\mathcal{F}_{G}$ together with the canonical trivialization induced from
$\sigma$ at $\infty$.

Recall that there is a decomposition
$\operatorname{\overline{\mathrm{Gr}}}^{\lambda}=\coprod_{\nu\leq\lambda}\operatorname{Gr}^{\nu}$
for dominant $\nu$, thus we have corresponding decomposition for generalized
affine Grassmannian slice:

$\overline{\mathcal{W}}^{\lambda}_{\mu}=\coprod_{\begin{subarray}{c}\nu:\text{
dominant}\\\ \mu\leq\nu\leq\lambda\end{subarray}}\mathcal{W}^{\nu}_{\mu}.$

This is known as the “monopole bubbling” in the physics literature.

###  2.1. Beilinson-Drinfeld generalized affine Grassmannian slice

If moreover there is a decomposition $\lambda=\sum_{i=1}^{N}\lambda_{i}$, we
can define the Beilinson-Drinfeld version of the generalized affine
Grassmannian slice $\mathcal{W}^{\underline{\lambda}}_{\mu}$, where
$\underline{\lambda}=(\lambda_{1},\cdots,\lambda_{N})$ is the defect type.
First, we define a functor $\mathsf{W}^{\underline{\lambda}}_{\mu}$ sending a
scheme $S$ to the data

  * (1)

a collection of $S$-points
$\underline{z}=(z_{1},\cdots,z_{N})\in\mathbb{A}^{N}(S)$;

  * (2)

a $G$-bundle $\mathcal{F}_{G}$ on $\mathbb{P}^{1}_{S}$;

  * (3)

a trivialization $\sigma$ of $\mathcal{F}_{G}$ on
$\mathbb{P}^{1}_{S}-\\{z_{1},\cdots,z_{N}\\}$ with pole exactly of order
$\sum_{i=1}^{N}\lambda_{i}\cdot z_{i}$;

  * (4)

a $B$-structure $\phi$ on $\mathcal{F}_{G}$ of degree $w_{0}\mu$ having fiber
$B\times S$ at $\infty\times S\subset\mathbb{P}^{1}_{S}$.

It’s easy to see that $\mathsf{W}^{\underline{\lambda}}_{\mu}$ is represented
by a finite type scheme over $\mathbb{Z}$, and then we define
$\mathcal{W}^{\underline{\lambda}}_{\mu}=\mathsf{W}^{\underline{\lambda}}_{\mu,\mathrm{red}}$.

######  Proposition 2.2.

The natural morphism
$\mathcal{W}^{\underline{\lambda}}_{\mu}\to\mathbb{A}^{N}$ is smooth.

###### Proof.

We can read out from the definition that
$\mathsf{W}^{\underline{\lambda}}_{\mu}$ fits into the commutative diagram

${\operatorname{Spec}\mathbb{Z}}$${\mathsf{Gr}^{\underline{\lambda}}_{\mathrm{BD}}}$${\mathsf{W}^{\underline{\lambda}}_{\mu}}$${\operatorname{Bun}_{G}(\mathbb{P}^{1},\infty)}$${\mathsf{H}^{\underline{\lambda}}_{\mathrm{BD}}}$${\mathsf{C}^{\underline{\lambda}}_{\mu}}$${\operatorname{Bun}_{G}(\mathbb{P}^{1},\infty)}$${\operatorname{Bun}_{B}^{w_{0}\mu}(\mathbb{P}^{1},\infty)}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{h^{\leftarrow}_{G}}$$\scriptstyle{h^{\rightarrow}_{G}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{q}$$\scriptstyle{p}$

where all squares are Cartesian. In this diagram,
$\operatorname{Bun}_{G}(\mathbb{P}^{1},\infty)\overset{h^{\leftarrow}_{G}}{\longleftarrow}\mathsf{H}^{\underline{\lambda}}_{\mathrm{BD}}\overset{h^{\rightarrow}_{G}}{\longrightarrow}\operatorname{Bun}_{G}(\mathbb{P}^{1},\infty)$
is the Beilinson-Drinfeld Hecke stack, defined by sending a scheme $S$ to the
groupoid of

  * (1)

a collection of $S$-points
$\underline{z}=(z_{1},\cdots,z_{N})\in\mathbb{A}^{N}(S)$;

  * (2)

a $G$-bundle $\mathcal{F}^{0}_{G}$ on $\mathbb{P}^{1}_{S}$ with a
trivialization $\sigma^{0}_{\infty}$ at $\infty$;

  * (3)

a $G$-bundle $\mathcal{F}^{1}_{G}$ on $\mathbb{P}^{1}_{S}$ with a
trivialization $\sigma^{1}_{\infty}$ at $\infty$;

  * (4)

an isomorphism
$\varphi:\mathcal{F}^{0}_{G}|_{\mathbb{P}^{1}_{S}-\\{z_{1},\cdots,z_{N}\\}}\cong\mathcal{F}^{1}_{G}|_{\mathbb{P}^{1}_{S}-\\{z_{1},\cdots,z_{N}\\}}$
with pole exactly of order $\sum_{i=1}^{N}\lambda_{i}\cdot z_{i}$, and
$\varphi$ maps $\sigma^{0}_{\infty}$ to $\sigma^{1}_{\infty}$;

$h^{\leftarrow}_{G}$ (resp. $h^{\rightarrow}_{G}$) maps the above data to
$(\mathcal{F}^{0}_{G},\sigma^{0}_{\infty})$ (resp.
$(\mathcal{F}^{1}_{G},\sigma^{1}_{\infty})$).
$i:\operatorname{Spec}\mathbb{Z}\to\operatorname{Bun}_{G}(\mathbb{P}^{1},\infty)$
is the trivial $G$-bundle with the canonical trivialization at $\infty$. Note
that $\mathsf{C}^{\underline{\lambda}}_{\mu}$ is nothing but a variation of
$\mathsf{W}^{\underline{\lambda}}_{\mu}$ by allowing the domain of the
isomorphism
$\sigma:\mathcal{F}^{\mathrm{triv}}_{G}|_{\mathbb{P}^{1}_{S}-\\{z_{1},\cdots,z_{N}\\}}\cong\mathcal{F}_{G}|_{\mathbb{P}^{1}_{S}-\\{z_{1},\cdots,z_{N}\\}}$
to vary.

Since $i$ induces an isomorphism between $\operatorname{Spec}\mathbb{Z}$ and
the open substack
$\operatorname{Bun}_{G}^{\mathrm{triv}}(\mathbb{P}^{1},\infty)\subset\operatorname{Bun}_{G}(\mathbb{P}^{1},\infty)$
of trivial $G$-bundles, we see that $j$ is an open immersion, thus the
proposition follows from the following

  * (•)

the natural morphism
$\mathsf{C}^{\underline{\lambda}}_{\mu,\mathrm{red}}\to\mathbb{A}^{N}$ is
smooth.

To prove this assertion, we notice that $h^{\rightarrow}_{G}$ is a locally
trivial fibration in the smooth topology with fiber
$\mathsf{Gr}^{\underline{\lambda}^{*}}_{\mathrm{BD}}$, hence $q$ has the same
property. Since $\operatorname{Bun}_{H}(\mathbb{P}^{1},\infty)$ is a smooth
stack on $\mathbb{Z}$ for any smooth group scheme $H$, the assertion (•)
follows from next lemma. ∎

######  Lemma 2.3.

The natural morphism
$\mathrm{Gr}^{\underline{\lambda}}_{\mathrm{BD}}:=\mathsf{Gr}^{\underline{\lambda}^{*}}_{\mathrm{BD},\mathrm{red}}\to\mathbb{A}^{N}$
is smooth.

###### Proof.

Recall the well-known fact that
$\overline{\mathrm{Gr}}^{\underline{\lambda}}_{\mathrm{BD}}\to\mathbb{A}^{N}$
is flat with geometrically reduced fibers. Now
$\mathrm{Gr}^{\underline{\lambda}}_{\mathrm{BD}}\subset\overline{\mathrm{Gr}}^{\underline{\lambda}}_{\mathrm{BD}}$
and its geometric fibers along the projection to $\mathbb{A}^{N}$ are products
of $G_{\mathcal{O}}$-orbits with reduced scheme structures, so the fibers are
smooth, hence
$\mathrm{Gr}^{\underline{\lambda}}_{\mathrm{BD}}\to\mathbb{A}^{N}$ is smooth.
∎

In fact, we can formulate a similar commutative diagram to the diagram that is
used in the proof of Proposition 2.2, but allow the Hecke stack to have poles
of order less than or equal to $\lambda$, then the up-right corner is
$\overline{\mathcal{W}}^{\lambda}_{\mu}$, which maps smoothly to
$\overline{\mathsf{C}}^{\lambda}_{\mu}$ (defined as the pullback of the right
projection of the Hecke stack). Now the projection
$\overline{\mathsf{C}}^{\lambda}_{\mu}\to\operatorname{Bun}^{w_{0}\mu}_{B}(\mathbb{P}^{1},\infty)$
is a fibration with fiber $\operatorname{\overline{\mathrm{Gr}}}^{\lambda}$.
It is known that the smooth locus of
$\operatorname{\overline{\mathrm{Gr}}}^{\lambda}$ is exactly
$\operatorname{Gr}^{\lambda}$, thus we have the following sharper result:

######  Proposition 2.4.

The natural morphism
$\operatorname{\overline{\mathcal{W}}}^{\underline{\lambda}}_{\mu}\to\mathbb{A}^{N}$
is flat and the smooth locus is
$\operatorname{\mathcal{W}}^{\underline{\lambda}}_{\mu}$.

##  3\. Transversal Slice of $\mathcal{W}^{\nu}_{\mu}$ in
$\overline{\mathcal{W}}^{\lambda}_{\mu}$

In this section we will prove Theorem 1.3. For simplicity we will assume that
$G$ is a complex reductive group, but the proof holds for reductive groups
over arbitrary algebraically closed field. We begin with proving a variant of
this theorem.

###  3.1. Infinite $\boldsymbol{\lambda}$

By sending $\lambda$ to infinity, we mean considering
$\operatorname{\mathcal{W}}_{\mu}$ (resp. $\operatorname{\mathcal{W}}_{\nu}$)
instead of $\overline{\mathcal{W}}^{\lambda}_{\mu}$ (resp.
$\overline{\mathcal{W}}^{\lambda}_{\nu}$). First of all, we notice that there
exists a natural $G_{1}[\\![z^{-1}]\\!]$ action on $\mathcal{W}_{\mu}$ which
makes $\mathbf{p}:\mathcal{W}_{\mu}\to\mathrm{Gr}$ equivariant. In fact, we
have [BFN16a, 2(xi)]

$\mathcal{W}_{\mu}=B_{1}[\\![z^{-1}]\\!]z^{\mu}B_{1}^{-}[\\![z^{-1}]\\!]=G_{1}[\\![z^{-1}]\\!]z^{\mu}B^{-}(\\!(z^{-1})\\!)/B^{-}[z]$

and the natural projection $\mathbf{p}:\mathcal{W}_{\mu}\to\mathrm{Gr}$ is a
further step quotient by $G[z]$. So the left multiplication by
$G_{1}[\\![z^{-1}]\\!]$ is an action on $\mathcal{W}_{\mu}$ and $\mathbf{p}$
is equivariant with respect to this action and the natural left multiplication
of $G_{1}[\\![z^{-1}]\\!]$ on $\mathrm{Gr}$.

Next, recall that $\mathcal{W}_{\nu}$ is a transversal slice of
$\mathrm{Gr}^{\nu}$ in $\mathrm{Gr}$, in fact there is an isomorphism:

$\mathcal{W}_{\nu}\cong G_{1}[\\![z^{-1}]\\!]\cap
z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu},\;U[z]z^{\nu}G[z]/G[z]\cong G[z]\cap
z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}.$

Let’s call the latter space $\mathcal{U}^{\nu}$, it’s an open subvariety of
$\mathrm{Gr}^{\nu}$. In this way we identify $\mathcal{W}_{\nu}$ and
$\mathcal{U}^{\nu}$ with subgroups of $z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}$
and the multiplication maps

$\displaystyle m_{l}:G_{1}[\\![z^{-1}]\\!]\cap
z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}\times G[z]\cap
z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}\to z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}$
$\displaystyle m_{r}:G[z]\cap z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}\times
G_{1}[\\![z^{-1}]\\!]\cap z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}\to
z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}$

are isomorphisms of schemes. We focus on $m_{l}$ for now. Since
$z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}\to
z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}\cdot z^{\nu}G[z]/G[z]$ is open immersion,
we obtain an open immersion

$m_{l}:\mathcal{W}_{\nu}\times\mathcal{U}^{\nu}\to\mathrm{Gr},$

such that $1\times\mathcal{U}^{\nu}$ (resp. $\mathcal{W}_{\nu}\times 1$) are
mapped identically to $\mathcal{U}^{\nu}$ (resp. $\mathcal{W}_{\nu}$).

The result we state below is a special case of the scheme-theoretic version of
[CG09, Corollary 3.2.21], we reprove it for completeness.

######  Proposition 3.2.

A transversal slice of
$\mathbf{p}^{-1}(\mathrm{Gr}^{\nu})=\mathcal{W}^{\nu}_{\mu}$ in
$\mathcal{W}_{\mu}$ is isomorphic to $\mathcal{W}_{\nu}$.

###### Proof.

Consider the Cartesian diagram

${X}$${\mathcal{W}_{\mu}}$${\mathcal{W}_{\nu}\times\mathcal{U}^{\nu}}$${\mathrm{Gr}}$$\scriptstyle{m_{l}^{\prime}}$$\scriptstyle{\mathbf{p}}$$\scriptstyle{m_{l}}$

so $m_{l}^{\prime}$ is open immersion as well. We claim that
$X\cong\mathcal{W}_{\nu}\times\mathbf{p}^{-1}(\mathcal{U}^{\nu})\subset\mathcal{W}_{\nu}\times\mathcal{W}^{\nu}_{\mu}$.
Let $S$ be a $\mathbb{C}$-scheme, given an $S$-point $x$ in $X$, we can write
$\mathbf{p}(x)=g\cdot y$ for a unique pair
$(g,y)\in\mathcal{W}_{\nu}(S)\times\mathcal{U}^{\nu}(S)$, so $g^{-1}\cdot
x\in\mathbf{p}^{-1}(\mathcal{U}^{\nu})(S)$, and we can write
$x=g\cdot(g^{-1}\cdot x)$, this gives rise to a morphism
$f:X\to\mathcal{W}_{\nu}\times\mathbf{p}^{-1}(\mathcal{U}^{\nu})$. Conversely,
the action map
$\mathcal{W}_{\nu}\times\mathbf{p}^{-1}(\mathcal{U}^{\nu})\to\mathcal{W}_{\mu}$
factors through $X$, and it’s easy to see that this is inverse to $f$.

Since Weyl conjugations of ${\mathcal{U}}^{\nu}$ is a covering of
$\operatorname{Gr}^{\nu}$, and same method can be applied to
$\mathcal{W}_{\nu}\times w\mathcal{U}^{\nu}w^{-1}$ for Weyl group element $w$,
hence we conclude that a transversal slice of
$\mathbf{p}^{-1}(\mathrm{Gr}^{\nu})=\mathcal{W}^{\nu}_{\mu}$ in
$\mathcal{W}_{\mu}$ is isomorphic to $\mathcal{W}_{\nu}$. ∎

One might wonder if this result, after truncation to finite $\lambda$,
directly proves the Theorem 1.3. The answer is “No”, because the image of
$\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\nu}\times\mathbf{p}^{-1}(\mathcal{U}^{\nu})$
under $m_{l}$ is not contained in
$\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$. For example, consider
$G=\operatorname{SL}_{2}$, $\mu=0,\nu=\alpha,\lambda=2\alpha$, where $\alpha$
is the unique positive coroot, in this case generalized transversal slices are
usual transversal slices so we can embed them as subvarieties of
$\operatorname{Gr}_{G}$. Then we take two elements

$\displaystyle\begin{bmatrix}1&0\\\ z^{-3}&1\\\ \end{bmatrix}\cdot
z^{\alpha}\in\operatorname{\overline{\mathcal{W}}}^{2\alpha}_{\alpha},\;\begin{bmatrix}1&z\\\
0&1\\\ \end{bmatrix}\cdot
z^{\alpha}\in\operatorname{\mathcal{W}}_{0}\cap\mathcal{U}^{\alpha}.$

The map $m_{l}$ sends this pair to

$\displaystyle\begin{bmatrix}1&0\\\ z^{-3}&1\\\
\end{bmatrix}\cdot\begin{bmatrix}1&z\\\ 0&1\\\ \end{bmatrix}\cdot
z^{\alpha}=\begin{bmatrix}z&1\\\ z^{-2}&z^{-1}+z^{-3}\\\ \end{bmatrix},$

which is not in $\operatorname{\overline{\mathrm{Gr}}}^{2\alpha}$. So we have
to do something more to go from infinite $\lambda$ to finite $\lambda$.

###  3.3. Finite $\boldsymbol{\lambda}$

First of all, we have shown above that the map

$m_{r}:\mathcal{U}^{\nu}\times\mathcal{W}_{\nu}\to\mathrm{Gr},$

is an open immersion such that $\mathcal{U}^{\nu}\times 1$ (resp.
$1\times\mathcal{W}_{\nu}$) are mapped identically to $\mathcal{U}^{\nu}$
(resp. $\mathcal{W}_{\nu}$). Next lemma show that by considering $m_{r}$
instead of $m_{l}$, the truncation to finite $\lambda$ gives rise to a
transversal slice of $\mathcal{U}^{\nu}$ in
$\operatorname{\overline{\mathrm{Gr}}}^{\lambda}$.

######  Lemma 3.4.

By identifying $\mathcal{U}^{\nu}\times\mathcal{W}_{\nu}$ with the image of
$m_{r}$, we have

$(\mathcal{U}^{\nu}\times\mathcal{W}_{\nu})\cap\overline{\mathrm{Gr}}^{\lambda}=\mathcal{U}^{\nu}\times\overline{\mathcal{W}}^{\lambda}_{\nu}.$

Hence the restriction of $m_{r}$ to
$\mathcal{U}^{\nu}\times\overline{\mathcal{W}}^{\lambda}_{\nu}$ gives an open
immersion to $\overline{\mathrm{Gr}}^{\lambda}$.

###### Proof.

Let $S$ be a $\mathbb{C}$-scheme, given an $S$-point $x$ in
$(\mathcal{U}^{\nu}\times\mathcal{W}_{\nu})\cap\overline{\mathrm{Gr}}^{\lambda}$,
we can write $x=g\cdot h$ for a unique pair
$(g,h)\in\mathcal{U}^{\nu}(S)\times\mathcal{W}_{\nu}(S)$. Since
$\mathcal{U}^{\nu}\subset G[z]$ and $\overline{\mathrm{Gr}}^{\lambda}$ is
stable under the action of $G[z]$, we see that
$h=g^{-1}x\in\overline{\mathcal{W}}^{\lambda}_{\nu}(S)$. Conversely, for each
pair $(g,h)\in\mathcal{U}^{\nu}(S)\times\mathcal{W}_{\nu}(S)$, we have
$gh\in(\mathcal{U}^{\nu}\times\mathcal{W}_{\nu})\cap\overline{\mathrm{Gr}}^{\lambda}(S)$.
We establish an equality for $S$-points, whence the lemma follows. ∎

Next we lift these results to $G(\\!(z^{-1})\\!)$. Denote by
$\widetilde{\mathcal{U}}^{\nu}$ the ind-scheme $\mathcal{U}^{\nu}z^{\nu}G[z]$
which is open in $G[z]z^{\mu}G[z]$, here we also identify $\mathcal{U}^{\nu}$
with subgroup of $z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}$. Recall that
$G[z]z^{\lambda}G[z]$ (resp. $\overline{G[z]z^{\lambda}G[z]}$) is denoted by
$\mathcal{X}^{\lambda}$ (resp. $\overline{\mathcal{X}}^{\lambda}$) in [MW19].
Then we have open immersion

$m_{r}:\widetilde{\mathcal{U}}^{\nu}\times\overline{\mathcal{W}}^{\lambda}_{\nu}\hookrightarrow\overline{\mathcal{X}}^{\lambda},\;(g_{1}z^{\nu}g_{2},h)\mapsto
g_{1}hz^{\nu}g_{2}$

To state the next result, let’s slightly generalize $\mathcal{W}_{\mu}$ and
define

$\prescript{\lambda_{1}}{\lambda_{2}}{\mathcal{X}}_{\mu}=z^{\lambda_{1}}U_{1}[\\![z^{-1}]\\!]z^{-\lambda_{1}}\cdot
z^{\mu}\cdot z^{-\lambda_{2}}B^{-}_{1}[\\![z^{-1}]\\!]z^{\lambda_{2}},$

for dominant coweights $\lambda_{1},\lambda_{2}$ and a general coweight $\mu$.
Note that $\prescript{\lambda_{1}}{\lambda_{2}}{\mathcal{X}}_{\mu}$ is a
scheme and in fact it’s a closed subscheme of the ind-scheme
${\mathcal{X}}_{\mu}$ defined in [MW19]. The same argument in loc. cit. shows
that

######  Lemma 3.5.

The scheme $\prescript{\lambda_{1}}{\lambda_{2}}{\mathcal{X}}_{\mu}$ is
isomorphic to
$\mathcal{U}^{\lambda_{1}}\times\mathcal{W}_{\mu}\times\mathcal{U}^{\lambda_{2}^{*}}$.
The scheme
$\prescript{\lambda_{1}}{\lambda_{2}}{\mathcal{X}}_{\mu}\cap\overline{\mathcal{X}}^{\lambda}$
is isomorphic to
$\mathcal{U}^{\lambda_{1}}\times\overline{\mathcal{W}}^{\lambda}_{\mu}\times\mathcal{U}^{\lambda_{2}^{*}}$.

######  Remark 3.6.

Here we identify $z^{\lambda_{1}}U_{1}[\\![z^{-1}]\\!]z^{-\lambda_{1}}\cap
G[z]$ (resp. $z^{-\lambda_{2}}U^{-}_{1}[\\![z^{-1}]\\!]z^{\lambda_{2}}\cap
G[z]$) with $\mathcal{U}^{\lambda_{1}}$ (resp.
$\mathcal{U}^{\lambda_{2}^{*}}$).

######  Lemma 3.7.

For every dominant coweight $\nu$, the left multiplication of
$z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}$ stabilizes
$\prescript{\nu}{\lambda}{\mathcal{X}}_{\mu}$ for sufficiently dominant
$\lambda$.

###### Proof.

By a conjugation of $z^{-\nu}$, it’s enough to prove the lemma for $\nu=0$.
Then we have

$\displaystyle G_{1}[\\![z^{-1}]\\!]\prescript{0}{\lambda}{\mathcal{X}}_{\mu}$
$\displaystyle=G_{1}[\\![z^{-1}]\\!]z^{\mu}\cdot
z^{-\lambda}B^{-}_{1}[\\![z^{-1}]\\!]z^{\lambda}$
$\displaystyle=U_{1}[\\![z^{-1}]\\!]z^{\mu}\cdot T_{1}[\\![z^{-1}]\\!]\cdot
z^{-\mu}U^{-}_{1}[\\![z^{-1}]\\!]z^{\mu}\cdot
z^{-\lambda}B^{-}_{1}[\\![z^{-1}]\\!]z^{\lambda}$

Then for sufficiently dominant $\lambda$,
$z^{-\mu}U^{-}_{1}[\\![z^{-1}]\\!]z^{\mu}$ is a subgroup of
$z^{-\lambda}B^{-}_{1}[\\![z^{-1}]\\!]z^{\lambda}$, whence the lemma follows.
∎

######  Lemma 3.8.

Identify
$\widetilde{\mathcal{U}}^{\nu}\times\overline{\mathcal{W}}^{\lambda}_{\nu}$
with the image of $m_{r}$ in $\overline{\mathcal{X}}^{\lambda}$, then for
sufficiently dominant $\lambda^{\prime}$, we have

$m_{r}\left((\prescript{\nu}{\lambda^{\prime}}{\mathcal{X}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu})\times\overline{\mathcal{W}}^{\lambda}_{\nu}\right)\subset\prescript{\nu}{\lambda^{\prime}}{\mathcal{X}}_{\mu}\cap\overline{\mathcal{X}}^{\lambda}.$

###### Proof.

Let’s take
$g_{1}z^{\nu}g_{2}\in\prescript{\nu}{\lambda^{\prime}}{\mathcal{X}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu}(S)$
for an $\mathbb{C}$-scheme $S$, and for arbitrary
$h\in\overline{\mathcal{W}}^{\lambda}_{\nu}(S)$, we have

$m_{r}(g_{1}z^{\nu}g_{2},h)=g_{1}hz^{\nu}g_{2}=g_{1}hg_{1}^{-1}\cdot
g_{1}z^{\nu}g_{2}$

and since both $g_{1}$ and $h$ are elements in
$z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}(S)$, we see that
$m_{r}(g_{1}z^{\nu}g_{2},h)$ is in the $z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}$
orbit of $g_{1}hz^{\nu}g_{2}$, hence it lies inside
$\prescript{\nu}{\lambda^{\prime}}{\mathcal{X}}_{\mu}(S)$ for sufficiently
dominant $\lambda^{\prime}$ by Lemma 3.7. ∎

Next, we embed
$(\operatorname{\mathcal{W}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu})\times\overline{\mathcal{W}}^{\lambda}_{\nu}$
into
$(\prescript{\nu}{\lambda^{\prime}}{\mathcal{X}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu})\times\overline{\mathcal{W}}^{\lambda}_{\nu}$
using Lemma 3.5, compose it with $m_{r}$ so that the image is in
$\prescript{\nu}{\lambda^{\prime}}{\mathcal{X}}_{\mu}\cap\overline{\mathcal{X}}^{\lambda}$,
and then compose with the projection to
$\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$ using Lemma 3.5 again.
Denote this map by $\psi$, and we have the following

######  Lemma 3.9.

$\psi:(\operatorname{\mathcal{W}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu})\times\overline{\mathcal{W}}^{\lambda}_{\nu}\to\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$
is a monomorphism of schemes.

###### Proof.

We need to show that $\psi$ is injective on $S$-points for all
$\mathbb{C}$-scheme $S$. Let’s take
$g_{1}z^{\nu}g_{2}\in\operatorname{\mathcal{W}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu}(S)$
and $h\in\overline{\mathcal{W}}^{\lambda}_{\nu}(S)$, where we use the
decomposition $\widetilde{\mathcal{U}}^{\nu}={\mathcal{U}}^{\nu}z^{\nu}G[z]$,
in other word $g_{1}\in\mathcal{U}^{\nu}(S)$ and $g_{2}\in G[z](S)$, then

$\psi(g_{1}z^{\nu}g_{2},h)=g_{1}hg_{1}^{-1}\cdot
g_{1}z^{\nu}g_{2}\mod\mathcal{U}^{\nu}\times\mathcal{U}^{\lambda^{\prime*}}$

where $\mathcal{U}^{\nu}$ acts from the left and
$\mathcal{U}^{\lambda^{\prime*}}$ acts from the right. Now
$g_{1}hg_{1}^{-1}\in
z^{\nu}G_{1}[\\![z^{-1}]\\!]z^{-\nu}(S)=\mathcal{U}^{\nu}(S)\times\mathcal{W}_{\nu}(S)$,
so there exists a unique pair
$(s,t)\in\mathcal{U}^{\nu}(S)\times\mathcal{W}_{\nu}(S)$ such that
$g_{1}hg_{1}^{-1}=st$. As a result, we have

$\psi(g_{1}z^{\nu}g_{2},h)=tg_{1}z^{\nu}g_{2}\mod\mathcal{U}^{\lambda^{\prime*}}.$

Suppose that there is another pair
$(g_{1}^{\prime}z^{\nu}g_{2}^{\prime},h^{\prime})\in\operatorname{\mathcal{W}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu}(S)\times\overline{\mathcal{W}}^{\lambda}_{\nu}(S)$
such that
$\psi(g_{1}z^{\nu}g_{2},h)=\psi(g_{1}^{\prime}z^{\nu}g_{2}^{\prime},h^{\prime})$,
then from above computation we see that
$tg_{1}z^{\nu}g_{2}=t^{\prime}g_{1}^{\prime}z^{\nu}g_{2}^{\prime}\mod\mathcal{U}^{\lambda^{\prime*}}$,
thus

$tg_{1}z^{\nu}=t^{\prime}g_{1}^{\prime}z^{\nu}\mod G[z]$

Since
$m_{l}:\operatorname{\mathcal{W}}_{\nu}\times\mathcal{U}^{\nu}\to\operatorname{Gr}$
is an open immersion, we conclude that $t=t^{\prime}$ and
$g_{1}=g_{1}^{\prime}$.

From these two equations and the definition of $(s,t)$, we arrive at
$g_{1}hg_{1}^{-1}=\tilde{s}g_{1}h^{\prime}g_{1}^{-1}$, where
$\tilde{s}=ss^{\prime-1}\in\mathcal{U}^{\nu}(S)$, and this implies that

$hh^{\prime-1}=g_{1}^{-1}\tilde{s}g_{1}\in\mathcal{U}^{\nu}(S),$

but by definition $hh^{\prime-1}\in\operatorname{\mathcal{W}}_{\nu}(S)$, hence
$h=h^{\prime}$.

Finally, plug three equations we already have into the condition
$\psi(g_{1}z^{\nu}g_{2},h)=\psi(g_{1}^{\prime}z^{\nu}g_{2}^{\prime},h^{\prime})$,
and we get
$g_{1}hz^{\nu}g_{2}=g_{1}hz^{\nu}g_{2}^{\prime}\mod\mathcal{U}^{\lambda^{\prime*}}$,
equivalently $g_{2}^{\prime}=g_{2}y$ where
$y\in\mathcal{U}^{\lambda^{\prime*}}(S)$. Then both $g_{1}z^{\nu}g_{2}$ and
$g_{1}z^{\nu}g_{2}y$ are elements in $\operatorname{\mathcal{W}}_{\nu}(S)$,
and this implies that

$y\in\operatorname{\mathcal{W}}_{\nu}(S)\cap\mathcal{U}^{\lambda^{\prime*}}(S)=1$

This concludes the proof of
$(g_{1}z^{\nu}g_{2},h)=(g_{1}^{\prime}z^{\nu}g_{2}^{\prime},h^{\prime})$. ∎

###### Proof of Theorem 1.3.

By Lemma 3.9,
$\psi:(\operatorname{\mathcal{W}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu})\times\overline{\mathcal{W}}^{\lambda}_{\nu}\to\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$
is a monomorphism between normal varieties of the same dimension, in
particular $\psi$ induces isomorphism between generic points, thus $\psi$ is
an open immersion by Zariski’s main theorem [Sta20, Tag 05K0]. Moreover it
induces identity map on
$(\operatorname{\mathcal{W}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu})\times 1$,
so $\overline{\mathcal{W}}^{\lambda}_{\nu}$ is a transversal slice of
$\operatorname{\mathcal{W}}_{\mu}\cap\widetilde{\mathcal{U}}^{\nu}$ in
$\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$. Since Weyl
conjugations of $\widetilde{\mathcal{U}}^{\nu}$ is a covering of
$\mathcal{X}^{\nu}$, and same method can be applied to
$\operatorname{\mathcal{W}}_{\mu}\cap w\widetilde{\mathcal{U}}^{\nu}w^{-1}$
for Weyl group element $w$, and see that there is an open immersion
$\psi_{w}:(\operatorname{\mathcal{W}}_{\mu}\cap
w\widetilde{\mathcal{U}}^{\nu}w^{-1})\times\overline{\mathcal{W}}^{\lambda}_{\nu}\to\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$
which induces identity map on $(\operatorname{\mathcal{W}}_{\mu}\cap
w\widetilde{\mathcal{U}}^{\nu}w^{-1})\times 1$. ∎

##  4\. Symplectic Singularities

In this section we prove Conjecture 1.4 for
$\overline{\mathcal{W}}^{\lambda}_{\mu}$. Note that our main result (Theorem
4.7) can be deduced using Coulomb branch techniques, see [Wee20] for ADE-type
and [NW19] for extension to non-simply laced cases. Here we provide an
elementary approach.

######  Corollary 4.1.

$\operatorname{\overline{\mathcal{W}}}^{{\lambda}}_{\mu}$ has rational
Gorenstein singularities.

###### Proof.

This is known for dominant $\mu$ [KWWY14]. The general case follows from the
dominant case and the transversal slice description of local singularities of
$\operatorname{\overline{\mathcal{W}}}^{{\lambda}}_{\mu}$ (Theorem 1.3). ∎

Before stating more results, let’s recall a theorem of Namikawa [Nam00,
Theorem 6], which is essential in the proof of our main result.

######  Theorem 4.2 (Namikawa).

A normal variety $X$ has symplectic singularities if and only if there is a
symplectic form $\Omega$ on the smooth locus $X^{\mathrm{reg}}$, and $X$ has
rational Gorenstein singularities.

This means that we only need to show that there exists a natural Poisson
structure on $\overline{\mathcal{W}}^{\lambda}_{\mu}$ and it’s regular on
${\mathcal{W}}^{\lambda}_{\mu}$. Recall that there is an open embedding
$\mathring{Z}^{\alpha^{*}}=\overline{\mathcal{W}}^{0}_{\mu-\lambda}\hookrightarrow\overline{\mathcal{W}}^{\lambda}_{\mu}$,
and in fact this is a section on $\mathring{Z}^{\alpha^{*}}$ of the natural
map $\mathbf{q}:\overline{\mathcal{W}}^{\lambda}_{\mu}\to Z^{\alpha^{*}}$
[BFN16a].

######  Proposition 4.3.

The natural Poisson structure on $\mathring{Z}^{\alpha^{*}}$ extends to a
Poisson structure on $\overline{\mathcal{W}}^{\lambda}_{\mu}$, and it’s
regular on ${\mathcal{W}}^{\lambda}_{\mu}$.

For simply laced $G$, this is [BFN16a, Proposition 3.18]. Here we present a
proof for general simple $G$. From now on, we will freely use notations from
[BFN16a]. Recall that there is an smooth open subvariety
$\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}$ of ${Z}^{\alpha^{*}}$, such that
$\mathbf{q}^{-1}({Z}^{\alpha^{*}}-\accentset{\mbox{\large.}}{Z}^{\alpha^{*}})$
has codimension $2$ in $\overline{\mathcal{W}}^{\lambda}_{\mu}$, so it’s
enough to prove the proposition for
$\accentset{\mbox{\large.}}{\mathcal{W}}^{\lambda}_{\mu}=\mathbf{q}^{-1}(\accentset{\mbox{\large.}}{Z}^{\alpha^{*}})$.

The following is a generalization of [BFN16a, Lemma 2.11, 2.12] to non-simply
laced case, and its proof is identical to that of loc. cit..

######  Lemma 4.4.

Let $I$ be the set of simple coroots, and let $d_{i}=1$ for short coroots and
$d_{i}=2$ or $3$ for long coroots, then:

  * (1)

$\mathrm{div}\left(F_{\alpha^{*}}\right)=\sum_{i\in
I}d_{i}\partial_{i}\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}$;

  * (2)

$\mathrm{div}\left(\mathbf{q}^{*}F_{\alpha^{*}}\right)=\sum_{i\in
I}d_{i}\mathbf{q}^{-1}_{*}\left(\partial_{i}\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}\right)+\sum_{i\in
I}\langle\lambda,d_{i}\check{\alpha}_{i^{*}}\rangle\accentset{\mbox{\large.}}{E}_{i}$;

  * (3)

$\mathbf{p}^{*}\mathcal{L}=\mathcal{O}_{\accentset{\mbox{\large.}}{\mathcal{GZ}}^{-\mu}_{\lambda}}\left(\sum_{i\in
I}d_{i}\mathbf{q}^{*}\left(\partial_{i}\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}\right)-\sum_{i\in
I}\langle\lambda,d_{i}\check{\alpha}_{i^{*}}\rangle\accentset{\mbox{\large.}}{E}_{i}\right)\cong\mathcal{O}_{\accentset{\mbox{\large.}}{\mathcal{GZ}}^{-\mu}_{\lambda}}\left(-\sum_{i\in
I}\langle\lambda,d_{i}\check{\alpha}_{i^{*}}\rangle\accentset{\mbox{\large.}}{E}_{i}\right)$;

  * (4)

$\mathrm{div}\left(\mathbf{q}^{*}\pi_{\alpha^{*}}^{*}\mathbb{A}^{\alpha^{*}}_{i}\right)=E_{i}+\mathbf{q}^{-1}_{*}\left(\pi_{\alpha^{*}}^{*}\mathbb{A}^{\alpha^{*}}_{i}\right)$.

Define the ideal sheaf
$\mathcal{K}_{i}=\mathcal{I}_{i}^{\langle\lambda,d_{i}\check{\alpha}_{i^{*}}\rangle}+\mathcal{J}_{i}^{d_{i}}$,
where $\mathcal{I}_{i}$ (resp. $\mathcal{J}_{i}$) is the ideal sheaf of
divisor $\partial_{i}\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}$ (resp.
$\pi_{\alpha^{*}}^{*}\mathbb{A}^{\alpha^{*}}_{i}$). Note that subvarieties
defined by different $\mathcal{K}_{i}$ are disjoint. Then define
$\mathcal{K}=\cap_{i\in I}\mathcal{K}_{i}$. The same argument in the proof of
[BFN16a, Proposition 2.10] shows that
$\mathbf{q}^{-1}\mathcal{K}\cdot\mathcal{O}_{\accentset{\mbox{\large.}}{\mathcal{GZ}}^{-\mu}_{\lambda}}\cong\mathbf{p}^{*}\mathcal{L}$.

The universal property of blow up implies that there is projective morphism
$\Upsilon:\accentset{\mbox{\large.}}{\mathcal{GZ}}^{-\mu}_{\lambda}\to\mathrm{Bl}_{\mathcal{K}}\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}$.
Moreover, the fiber of
$\mathbf{q}:\accentset{\mbox{\large.}}{\mathcal{GZ}}^{-\mu}_{\lambda}\to\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}$
is either a point or isomorphic to $\mathbb{P}^{1}$, and the fiber of
$\mathrm{Bl}_{\mathcal{K}}\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}\to\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}$
has dimension $\leq 1$ (since locally $\mathcal{K}$ is generated by $2$
elements). It follows that $\Upsilon$ is finite, hence it’s a normalization
map. In other words we have shown the following generalization of [BFN16a,
Proposition 2.10]

######  Proposition 4.5.

The identity isomorphism over
$\pi_{\alpha^{*}}^{-1}(\mathbb{G}_{m}^{\alpha^{*}})$ extends to a regular
isomorphism of varieties over $\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}$:

$\accentset{\mbox{\large.}}{\mathcal{GZ}}^{-\mu}_{\lambda}\overset{\sim}{\longrightarrow}\widetilde{\mathrm{Bl}}_{\mathcal{K}}\accentset{\mbox{\large.}}{Z}^{\alpha^{*}},\;\accentset{\mbox{\large.}}{\mathcal{W}}^{\lambda}_{\mu}\overset{\sim}{\longrightarrow}\widetilde{\mathrm{Bl}}^{\mathrm{aff}}_{\mathcal{K}}\accentset{\mbox{\large.}}{Z}^{\alpha^{*}}.$

Replacing ${\mathrm{Bl}}^{\mathrm{aff}}_{\mathcal{K}}$ by
$\widetilde{\mathrm{Bl}}^{\mathrm{aff}}_{\mathcal{K}}$ in [BFN16a, (2.13)], we
see that [BFN16a, Proposition 2.9] is true for all simple $G$, without
assumption of ADE-type (this is pointed out in Remark 2.14 of loc. cit.). We
record it here for completeness.

######  Proposition 4.6 (Factorization).

The birational isomorphism $\varphi$ extends to a regular isomorphism of
varieties over
$\left(\mathbb{G}_{m}^{\beta^{*}}\times\mathbb{A}^{1}\right)_{\mathrm{disj}}$:

$\left(\mathbb{G}_{m}^{\beta^{*}}\times\mathbb{A}^{1}\right)_{\mathrm{disj}}\times_{\mathbb{A}^{\alpha^{*}}}\overline{\mathcal{W}}^{\lambda}_{\mu}\overset{\sim}{\longrightarrow}\left(\mathbb{G}_{m}^{\beta^{*}}\times\mathbb{A}^{1}\right)_{\mathrm{disj}}\times_{\mathbb{A}^{\beta^{*}}\times\mathbb{A}^{1}}\left(\mathring{Z}^{\beta^{*}}\times\mathcal{S}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}\right)$

where $\beta=\alpha-\alpha_{i}$, and
$\mathcal{S}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}$ is the
$A_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle-1}$ singularity defined by
equation $xy=w^{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}$, and the
natural map $\mathcal{S}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}\to
Z^{\alpha_{i}^{*}}\cong\mathbb{A}^{2}$ is given by $(x,y,w)\mapsto(y,w)$.

###### Proof of Proposition 4.3.

For type $A_{1}$, this is a straightforward computation. For general $G$, we
deduce from Proposition 4.6 that the Poisson structure extends over
$\accentset{\mbox{\large.}}{E}_{i}$ for every $i\in I$, so it extends
globally. And it’s regular outside of a set of codimension $2$, hence it’s
regular on $\mathcal{W}^{\lambda}_{\mu}$. ∎

Combine Corollary 4.1, Theorem 4.2 and Proposition 4.3, we obtain the main
results of this section:

######  Theorem 4.7.

$\overline{\mathcal{W}}^{\lambda}_{\mu}$ has symplectic singularities.

###  4.8. $\mathbb{Q}$-Factorial Terminalization

Assume that $G$ is of adjoint type, and suppose that
$\lambda=\sum_{i=1}^{n}\lambda_{i}$, where $\lambda_{i}$ are fundamental
coweights, then consider the convolution

$m:\mathrm{Gr}^{\vec{\lambda}}=\mathrm{Gr}^{\lambda_{1}}\widetilde{\times}\cdots\widetilde{\times}\mathrm{Gr}^{\lambda_{n}}\subset\overline{\mathrm{Gr}}^{\vec{\lambda}}=\overline{\mathrm{Gr}}^{\lambda_{1}}\widetilde{\times}\cdots\widetilde{\times}\overline{\mathrm{Gr}}^{\lambda_{n}}\to\overline{\mathrm{Gr}}^{\lambda}$

and define

$\operatorname{\mathcal{W}}^{\vec{\lambda}}_{\mu}=\mathrm{Gr}^{\vec{\lambda}}\times_{\overline{\mathrm{Gr}}^{\lambda}}\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu},\;\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}=\overline{\mathrm{Gr}}^{\vec{\lambda}}\times_{\overline{\mathrm{Gr}}^{\lambda}}\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$

Here what we called
$\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}$ is denoted by
$\widetilde{\mathcal{W}}^{\underline{\lambda}}_{\mu}$ in [BFN18]. Note that
$m:\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}\to\operatorname{\overline{\mathcal{W}}}^{\underline{\lambda}}_{\mu}$
is projective and birational, in fact it’s an isomorphism on the open loucs
$\mathring{Z}^{\alpha^{*}}$, so
$\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}$ inherits a
Poisson structure on $\mathring{Z}^{\alpha^{*}}$. The same commutative diagram
in the proof of Proposition 2.2 with
$\mathsf{H}^{\underline{\lambda}}_{\mathrm{BD}}$ replaced by a convolution
version $\mathsf{H}^{\vec{\lambda}}_{\mathrm{BD}}$ shows the following:

######  Proposition 4.9.

$\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}$ is normal and
its smooth locus is $\operatorname{\mathcal{W}}^{\vec{\lambda}}_{\mu}$.

The factorization of
$\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}$ is also proven
in [BFN18], and it holds for general simple $G$ without assumption on ADE
type. Here we record it for completeness:

######  Proposition 4.10 (Factorization).

The birational isomorphism $\varphi$ extends to a regular isomorphism of
varieties over
$\left(\mathbb{G}_{m}^{\beta^{*}}\times\mathbb{A}^{1}\right)_{\mathrm{disj}}$:

$\left(\mathbb{G}_{m}^{\beta^{*}}\times\mathbb{A}^{1}\right)_{\mathrm{disj}}\times_{\mathbb{A}^{\alpha^{*}}}\overline{\mathcal{W}}^{\vec{\lambda}}_{\mu}\overset{\sim}{\longrightarrow}\left(\mathbb{G}_{m}^{\beta^{*}}\times\mathbb{A}^{1}\right)_{\mathrm{disj}}\times_{\mathbb{A}^{\beta^{*}}\times\mathbb{A}^{1}}\left(\mathring{Z}^{\beta^{*}}\times\widetilde{\mathcal{S}}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}\right)$

where $\beta=\alpha-\alpha_{i}$, and
$\widetilde{\mathcal{S}}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}$ is
the minimal resolution of the
$A_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle-1}$ singularity
$\mathcal{S}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}$.

######  Theorem 4.11.

$m:\overline{\mathcal{W}}^{\vec{\lambda}}_{\mu}\to\overline{\mathcal{W}}^{\lambda}_{\mu}$
is a $\mathbb{Q}$-factorial terminalization map.

###### Proof.

We observe that

  * (1)

$\forall i,j\in I$, $\lambda_{i}-\alpha_{j}$ is not dominant, thus every
stratum ${\mathcal{W}}^{\vec{\lambda}^{\prime}}_{\mu}$ in the singular locus
has codimension at least $4$, i.e.
$\overline{\mathcal{W}}^{\vec{\lambda}}_{\mu}$ is smooth at codimension $3$;

  * (2)

The pull back of the canonical Poisson structure along the resolution map
$\widetilde{\mathcal{S}}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}\to\mathcal{S}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}$
is a symplectic structure, so by the factorization we see that the Poisson
structure on the open locus
$\mathring{Z}^{\alpha^{*}}\subset\overline{\mathcal{W}}^{\vec{\lambda}}_{\mu}$
extends over the exceptional divisors $E_{i}$ for all $i\in I$, moreover it’s
symplectic outside of a subset of codimension $2$;

  * (3)

$\overline{\mathcal{W}}^{\vec{\lambda}}_{\mu}\to\operatorname{Bun}_{B}^{w_{0}\mu}(\mathbb{P}^{1},\infty)$
is an open substack of a locally trivial fibration with fibers isomorphic to
$\operatorname{\overline{\mathrm{Gr}}}^{\vec{\lambda}^{*}}$, and the latter is
known to be $\mathbb{Q}$-factorial [KWWY14, Theorem 2.7], thus
$\overline{\mathcal{W}}^{\vec{\lambda}}_{\mu}$ is $\mathbb{Q}$-factorial as
well.

The first two facts together with Flenner’s extendability theorem [Fle88]
shows that $\overline{\mathcal{W}}^{\vec{\lambda}}_{\mu}$ has symplectic
singularities. Moreover Namikawa shows that regularity at codimension $3$
implies that it’s terminal [Nam01].

Finally we only need to show that
$m:\overline{\mathcal{W}}^{\vec{\lambda}}_{\mu}\to\overline{\mathcal{W}}^{\lambda}_{\mu}$
has zero discrepancy. Again, this follows from the factorization. Note that
the minimal resolution
$\widetilde{\mathcal{S}}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}\to\mathcal{S}_{\langle\lambda,\check{\alpha}_{i^{*}}\rangle}$
is well-known to have zero discrepancy. ∎

We also have a partial generalization of [KWWY14, Theorem 2.9].

######  Proposition 4.12.

The following are equivalent:

  * (1)

$\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$ has a symplectic
resolution.

  * (2)

$\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}$ is smooth.

  * (3)

$\operatorname{\mathcal{W}}^{\vec{\lambda}}_{\mu}=\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}$.

  * (4)

If there are dominant coweights $\nu_{i},i=1,\cdots,n$, such that
$\nu_{i}\leq\lambda_{i}$ and $\mu\leq\sum_{i=1}^{n}\nu_{i}$, then we must have
$\nu_{i}=\lambda_{i}$.

###### Proof.

$(2)\Longrightarrow(1)$ is tautological. $(2)\Longleftrightarrow(3)$:
$\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}$ is smooth if and
only if it agrees with its smooth locus, which is equivalent to
$\operatorname{\mathcal{W}}^{\vec{\lambda}}_{\mu}=\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}$
by Proposition 4.9.

$(3)\Longleftrightarrow(4)$: Note that

$\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}=\operatorname{\mathcal{W}}^{\vec{\lambda}}_{\mu}\cup\coprod_{\vec{\nu}}\operatorname{\mathcal{W}}^{\vec{\nu}}_{\mu}$

where $\vec{\nu}$ satisfy condition $(3)$, then the equivalence is obvious.

$(1)\Longrightarrow(2)$: For dominant $\mu$, this is already proven in
[KWWY14, Theorem 2.9]. For general $\mu$, there is a subtle complication: loc.
cit. uses [Nam11, 5.6] where a conical $\mathbb{G}_{m}$ action is assumed, but
the natural $\mathbb{G}_{m}$ action on
$\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$ is not conical 111The
author thanks Vasily Krylov for pointing out this subtlety..

Assume that $(2)$ is false, then by what we have shown, $(4)$ is false, i.e.
there exists dominant coweights $\nu_{i}\leq\lambda_{i}$ such that
$\mu\leq\sum_{i}\nu_{i}$ and there is some $i$ that $\nu_{i}\neq\lambda_{i}$.
Define $\nu=\sum_{i}\nu_{i}$, then the implication $(1)\Longrightarrow(2)$ for
dominant $\nu$ implies that $\overline{\mathcal{W}}^{\lambda}_{\nu}$ has no
symplectic resolution. We shall use this to show that in this case, (1) must
be false.

Note that for symplectic singularity $Y$, a symplectic resolution $f:X\to Y$
is the same as a crepant resolution of $Y$, since nondegeneracy of
$f^{*}\omega_{Y}$ is equivalent to $f^{*}\omega_{Y}^{n}$ having no zero along
exceptional divisors, which in turn is equivalent to $f$ being crepant.

Suppose that (1) holds, then the restriction of a symplectic resolution to an
open locus gives rise to a crepant resolution of
$\overline{\mathcal{W}}^{\lambda}_{\nu}\times\mathcal{W}^{\nu}_{\mu}$, by
Theorem 1.3. Then we apply the following lemma to the crepant resolution of
$\overline{\mathcal{W}}^{\lambda}_{\nu}\times\mathcal{W}^{\nu}_{\mu}$, we get
a crepant resolution of $\overline{\mathcal{W}}^{\lambda}_{\nu}$, which is
automatically symplectic by previous discussion. However this contradicts with
what we have already deduced, so we conclude that $(1)$ must be false. ∎

######  Lemma 4.13.

Suppose that $X$ is a normal variety, and $Y$ is a smooth variety, and $f:Z\to
X\times Y$ is a crepant resolution. Then there exists a Zariski open subset
$U$ in $Y$ such that $\forall y\in U$, the fiber $f_{y}:Z_{y}\to X\times{y}$
is a crepant resolution.

###### Proof.

By generic smoothness theorem, there exists Zariski open subset $U$ in $Y$
such that the composition of $f$ and projection to $Y$ is smooth. Shrink $U$
if needed, we can assume that $\forall y\in U$, $f_{y}$ is birational and is
isomorphism on the smooth locus of $X$, thus $f_{y}:Z_{y}\to X\times{y}$ is a
resolution of singularity. Shrink $U$ again if needed, we assume that the
canonical divisor $K_{U}$ is trivial, so the Weil divisor $[K_{X\times U}]$
can be written as $[K_{X}]\times U$. Now by the assumption that $f$ is
crepant, we have $f^{*}([K_{X}]\times U)=[K_{Z_{U}}]$. Restrict the equation
to fiber, and we see that $f_{y}^{*}([K_{X}])=[K_{Z_{y}}]$, $\forall y\in U$.
This means that $f_{y}$ is a crepant resolution. ∎

######  Example 4.14.

In Type $A$, fundamental coweights are minuscule, so condition (4) is
automatically satisfied, thus
$m:\operatorname{\overline{\mathcal{W}}}^{\vec{\lambda}}_{\mu}\to\operatorname{\overline{\mathcal{W}}}^{\lambda}_{\mu}$
is a symplectic resolution.

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