of $\chi$ less than one as gravitational wells, as it can be related to the
Newtonian gravitational potential for weak gravity cases by

$\chi=\big{(}\frac{1}{1-2\Psi}\big{)}^{\frac{1}{4}}.$ (A.11)

One separates the extrinsic curvature into its trace $K=\gamma^{ij}K_{ij}$ and
traceless part $A_{ij}=K_{ij}-K\gamma_{ij}/3$ and then conformally decomposes
the traceless part, $\bar{A}_{ij}=A_{ij}/\chi$. We also define the conformal
connection functions $\bar{\Gamma}^{i}$ as contractions of the Christoffel
symbols of the conformal spatial metric, i.e.

$\bar{\Gamma}^{i}_{jk}=\frac{1}{2}\bar{\gamma}^{il}\big{(}\bar{\gamma}_{lj,k}+\bar{\gamma}_{lk,j}-\bar{\gamma}_{kj,l}\big{)},$
(A.12)

where indices after a comma indicate partial differentiation with respect to a
coordinate, and

$\bar{\Gamma}^{i}\equiv\bar{\gamma}^{jk}\bar{\Gamma}^{i}_{jk}=-\partial_{j}\bar{\gamma}^{ij}.$
(A.13)

The evolution equations for the conformal connections functions can be found
by combining their definition with the evolution equation of the conformal
spatial metric, after which the momentum constraints is used to eliminate a
$\partial_{j}\bar{A}^{ij}$ term that appears. Finally, the $\bar{\Gamma}^{i}$
can be used to to calculate the three-dimensional Ricci tensor, which appears
in the evolution equation of $\bar{A}_{ij}$ (it appears in the evolution
equation for $K$ as well, but this is eliminated using the Hamiltonian
constraint). One splits the Ricci tensor as

$R_{ij}=R_{ij}^{\chi}+\bar{R}_{ij},$ (A.14)

where

$R_{ij}^{\chi}=\frac{\bar{\gamma}_{ij}}{2\chi}(\bar{\gamma}^{k}l\bar{D}_{k}\bar{D}_{l}\chi-\frac{3}{2\chi}\bar{\gamma}^{kl}\partial_{k}\chi\partial_{l}\chi)+\frac{1}{2\chi}\big{(}\bar{D}_{i}\bar{D}-\frac{1}{2\chi}\partial_{i}\chi\partial_{j}\chi\big{)}$
(A.15)

and

$\bar{R}_{ij}=-\frac{1}{2}\bar{\gamma}^{kl}\partial_{k}\partial_{l}\bar{\gamma}_{ij}-\bar{\gamma}_{k(i}\partial_{j)}\bar{\Gamma}^{k}+\bar{\Gamma}^{k}\bar{\Gamma}_{(ij)k}+\bar{\gamma}^{mn}\big{(}2\bar{\Gamma}^{k}_{m(i}\bar{\Gamma}_{j)kn}+\bar{\Gamma}^{k}_{im}\bar{\Gamma}_{kjn}\big{)}.$
(A.16)

In this form, it was shown that the BSSN formalism is strongly hyperbolic and
therefore well-posed Sarbach:2002bt.

It is sometimes useful to have an expression for $\Gamma^{i}_{jk}$ in terms of
the conformal quantities, e.g. when one adds matter terms to a simulation with
evolution equations include $\Gamma^{i}_{jk}$. For reference, if the conformal
spatial metric is defined as $\gamma_{ij}\equiv\chi^{-n}\bar{\gamma}_{ij}$,
then

$\Gamma^{i}_{jk}=\bar{\Gamma}^{i}_{jk}-\frac{n}{2\chi}\big{(}\bar{\gamma}^{il}\bar{\gamma}_{lj}\partial_{k}\chi+\bar{\gamma}^{il}\bar{\gamma}_{lk}\partial_{j}\chi-\chi\bar{\gamma}^{il}\bar{\gamma}_{kj}\partial_{l}\chi\big{)}.$
(A.17)

The full set of BSSN evolution equations is then given by

$\displaystyle\partial_{t}\chi=$
$\displaystyle\beta^{k}\partial_{k}\chi+\frac{2}{3}\chi(\alpha
K-\partial_{k}\beta^{k})\,,$ (A.18a)
$\displaystyle\partial_{t}\bar{\gamma}_{ij}=$
$\displaystyle\beta^{k}\partial_{k}\bar{\gamma}_{ij}-2\alpha\bar{A}_{ij}+\bar{\gamma}_{ik}\partial_{j}\beta^{k}+\bar{\gamma}_{jk}\partial_{i}\beta^{k}-\frac{2}{3}\bar{\gamma}_{ij}\partial_{k}\beta^{k}\,,$
(A.18b) $\displaystyle\partial_{t}\bar{A}_{ij}=$
$\displaystyle\beta^{k}\partial_{k}\bar{A}_{ij}\chi\left\\{-D_{i}D_{j}\alpha+\alpha\left(\leftidx{{}^{(3)}}R_{ij}-8\pi\alpha
S_{ij}\right)\right\\}^{TF}+\alpha\left(\bar{A}_{ij}K-2\bar{A}_{ik}\bar{A}^{k}_{j}$
$\displaystyle+\bar{A}_{ik}\partial_{j}\beta^{k}+\bar{A}_{jk}\partial_{i}\beta^{k}-\frac{2}{3}\bar{A}_{ij}\partial_{k}\beta^{k}\,,$
(A.18c) $\displaystyle\partial_{t}K=$
$\displaystyle\beta^{k}\partial_{k}K-D_{i}D^{i}\alpha+\alpha\left(\bar{A}_{ij}\bar{A}^{ij}+\frac{1}{3}K^{2}\right)+4\pi\alpha(\rho+S)\,,$
(A.18d) $\displaystyle\partial_{t}\bar{\Gamma}^{i}=$
$\displaystyle\beta^{j}\partial_{j}\bar{\Gamma}^{i}+\bar{\gamma}^{jk}+\bar{\gamma}^{jk}\partial_{j}\partial_{k}\beta^{i}+\frac{1}{3}\bar{\gamma}^{ij}\partial_{j}\partial_{k}\beta^{k}-\bar{\Gamma}^{j}\partial_{j}\beta^{i}+\frac{2}{3}\bar{\Gamma}^{i}\partial_{j}\beta^{j}$
(A.18e)
$\displaystyle-2\bar{A}^{ij}\partial_{j}\alpha+2\alpha\left(\bar{\Gamma}^{i}_{jk}\bar{A}^{jk}-\frac{3}{2}\bar{A}^{ij}\frac{\partial_{j}\chi}{\chi}-\frac{2}{3}\bar{\gamma}^{ij}\partial_{j}K-8\pi\chi
S^{i}\right).$

###  Appendix B Spinning primordial black holes in a matter-dominated universe

####  B.1 Apparent horizons in FLRW spacetimes

To find the AH of the formed PBH, we follow the procedure described in
Thornburg:2003sf. The AH is the outermost marginally outer-trapped surface on
which the expansion of outgoing null geodesics $\Theta_{+}$ vanishes, i.e.

$\Theta_{+}\equiv D_{i}s^{i}+K_{ij}s^{i}s^{j}-K=0~{}.$ (A.18a)

In FLRW spacetimes one often encounters cosmological horizons, for which a
local measure can be found in a similar manner using the expansion of ingoing
null vectors $k_{-}^{a}=n^{a}-s^{a}$, equivalent to

$\Theta_{-}\equiv-D_{i}s^{i}+K_{ij}s^{i}s^{j}-K=0~{}.$ (A.18b)

In the following, we will refer to $\Theta_{+}$ ($\Theta_{-}$) as outgoing
(ingoing) expansion, and we note that it measures the fractional change in the
area of an outward (inward) spherical flash of light baumgarte_shapiro.

In general, these equations must be solved numerically, using a nonlinear root
finder algorithm such as the Newton-Raphson method or quasi-Newton methods.
However, to build some intuition for these quantities, it is useful to
consider specific cases with a high degree of symmetry, which can be solved
analytically. It can be shown baumgarte_shapiro that for a spherically
symmetric line element of the form

$ds^{2}=-\alpha^{2}dt^{2}+\psi(t,r)^{2}\left[dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right)\right]~{},$

and with $s^{r}=1/\psi(t,r)$, $s^{\theta}=s^{\phi}=0$, the outgoing/ingoing
expansion simplifies to

$\Theta_{\pm}=\frac{2}{\psi}\left[\frac{\partial_{t}\psi}{\alpha}\pm\left(\frac{\partial_{r}\psi}{\psi}+\frac{1}{r}\right)\right].$
(A.18c)

We find it instructive to treat some concrete examples here and we discuss the
resulting expressions for a Schwarzschild, FLRW and McVittie spacetime below.
We will only consider positive radius solutions to Eqn. (A.18a) and Eqn.
(A.18b), regarding negative radius solutions as unphysical.

  * (i)

For a Schwarzschild black hole, the conformal factor in isotropic coordinates
is $\psi_{\mathrm{BH}}(r)=\left(1+GM/2r\right)^{2}$ and the lapse is
$\alpha_{\mathrm{BH}}(r)=(1-GM/2r)/(1+GM/2r)$. The expansions are then given
by

$\Theta_{\pm}=\mp\frac{8\left(GM-2r\right)r}{(GM+2r)^{3}}~{},$ (A.18d)

which both vanish at $r=0$ and at the black hole horizon $r=GM/2$.

In these coordinates, the areal radius for the Schwarzschild metric is given
by $r\psi_{\mathrm{BH}}(t,r)$ and therefore, the area of a spherical surface
is $A=4\pi(r\psi_{\mathrm{BH}}(t,r))^{2}$ 111We note that at $r=GM/2$ and with
units restored, this area equals $16\pi G^{2}M^{2}$, as it should for this
spacetime.. The outgoing (ingoing) expansion can be thought of as measuring
the fractional change in the area of an outward (inward) spherical flash of
light. From the expression for $A$, this area decreases (increases) inside the
apparent horizon and increases (decreases) outside the apparent horizon when
the coordinate radius is increased (decreased). This justifies the fact that
$r=GM/2$ solves both $\Theta_{\pm}=0$.

  * (ii)

In an FLRW spacetime sliced by cosmic time $t$, $\psi_{\mathrm{FLRW}}(t)=a(t)$
and $\alpha_{\mathrm{FLRW}}=1$, so that

$\Theta_{\pm}=2\left(H\pm\frac{1}{ar}\right)~{},$ (A.18e)

and $\Theta_{-}$ vanishes at the comoving Hubble horizon $r=a^{-1}H^{-1}$.
This is the point at which the universe begins to expand superluminally
relative to the origin, and ingoing rays, that converge to the origin for
smaller radii, end up receding from us.

  * (iii)

A black hole immersed in an FLRW background can be represented by the McVittie
line element McVittie:1933zz, for which
$\alpha=\alpha_{\mathrm{BH}}\left(ar\right)$ and
$\psi=\psi_{\mathrm{FLRW}}(t)\psi_{\mathrm{BH}}\left(ar\right)$, so that

$\Theta_{\pm}=2\left(H\mp\frac{4ar\left(GM-2ar\right)}{\left(GM+2ar\right)^{3}}\right)~{}.$
(A.18f)

When $GMH\ll 1$, as is the case in the situations we consider, $\Theta_{-}$
vanishes for

$r\approx\frac{1}{aH}-\frac{2GM}{a}\quad\text{and}\quad
r\approx\frac{GM}{2a}+\frac{2H(GM)^{2}}{a}\,,$

while $\Theta_{+}$ vanishes for

$r\approx\frac{GM}{2a}-\frac{2H(GM)^{2}}{a}\quad\text{and}\quad
r\approx\frac{H(GM)^{2}}{4a}\,.$

This means that an AH finder should in principle be able to find the PBH and
cosmological horizon, as well as a perturbed $r=0$ solution. This is confirmed
by our simulations. We are able to distinguish the two solutions for
$\Theta_{+}=0$ because the physical area of the latter shrinks, rendering it
unphysical.

####  B.2 Initial data

In section 4.2.1 of the main text, we allude to the fact that to solve the
Einstein constraints on the initial hyperslice, we make an assumption through
Eqn. (2.42) that the linear momentum density obeys $S_{i}=\partial_{i}F$ for
some scalar function $F$ (the original method). Since the linear momentum
densities of the initial data defined in section 4.2.1 do not obey this
relation for any $F$, we must check that solving the constraints without this
assumption, i.e. solving Eqn. (2.40) and Eqn. (2.41) as four coupled equations
(the corrected method), results in the same initial configurations for the
extrinsic curvature $K_{ij}$ and does not change our final results.

To show this, we firstly solve the Einstein constraints for a perturbation
shell with
$A=0.09,R_{0}=1.2/H_{0},\lambda=0.15/H_{0},B=0.5,k=2,\omega=10H_{0}$, using
both the original and the corrected method. The only initial configurations
that then differ are the trace of the extrinsic curvature $K$ and the
traceless part of the extrinsic curvature $A_{ij}$. We compare configurations
obtained using both methods in Fig. B.1, which shows that the normalised
difference between method 1 and 2 (i.e. the original and the corrected method
respectively) is smaller than 0.02%. Therefore, the initial conditions used in
our production runs are virtually identical to the ones we would have obtained
using the correct method.

Fig. B.1: Comparison between initial data obtained with method 1 (original)
and method 2 (corrected) method. Shown in the top row are $K$ and one
representative component of $A_{ij}$, along the $x$-direction through the
entire simulation box and through the centre of the perturbation shell. The
bottom row shows the normalised difference between the two configurations in
the top row.

Secondly, we have convergence tested both the original and corrected method.
We use the same shell parameters as above, in a cubic box of length
$L_{\textrm{box}}=5/H_{0}$ and with base grid resolutions
$N_{1}=256,N_{2}=384,N_{3}=576$, yielding resolutions $\Delta_{1},\Delta_{2}$
and $\Delta_{3}$, so that the resolution ratio
$r=\Delta_{1}/\Delta_{2}=\Delta_{2}/\Delta_{3}=1.5$. Computational constraints
keep us from going to higher resolutions. The results of these tests for the
original (corrected) method are shown in Fig. B.2 (B.3).

It is clear from the top rows of both figures that the configurations obtained
at different resolutions are very similar. In the middle rows of the two
figures, we show the difference between solutions at $\Delta_{1}$ and
$\Delta_{2}$ (solid blue line), as well as the difference between solutions at
$\Delta_{2}$ and $\Delta_{3}$ rescaled by $r^{n}$ (dashed orange line), where
$n$ is the desired order of convergence that we set to 2, given the stencils
used in the solver. If the solver achieves second order convergence, the
orange line stays on or below the blue line, which is the case for both
methods. This shows that in this case, even though it is a mistake to use Eqn.
(2.42), it does not significantly affect the performance or rate of
convergence achieved by the solver, suggesting that the total error is not
dominated by the assumption of a curl-free $S_{i}$, at least at these
resolutions. It would be interesting to see whether either method achieves
this rate of convergence at arbitrarily high resolution, but we cannot draw
conclusions concerning that question based on these results.

The bottom panels of Fig B.2 and Fig. B.3 show the estimated numerical
solution error at resolution $\Delta_{3}$, defined as
$\epsilon=(u_{\Delta_{2}}-u_{\Delta_{3}})/(r^{n}-1)$. These are identical
between both methods and consistently small compared to the absolute values in
the top rows.

Fig. B.2: Convergence test for original method. Shown in the top row are the
configurations for $K$ and $A_{22}$ obtained at different resolutions. The
middle row compares the difference between solutions at different resolutions,
rescaled with respect to $r=1.5$ and $n=2$, while the bottom row shows an
estimate of the remaining numerical solution error at the finest resolution.
Fig. B.3: Convergence test for original method, showing the same quantities as
Fig. B.2.

Lastly, we have rerun a simulation using the corrected initial conditions, for
the same initial conditions as above. In Fig. B.4, we compare the mass and
angular momentum of the PBH formed from this perturbation with either method,
and we show that the results are qualitatively identical and are
quantitatively the same within 2%. We note that this figure has code time on
the $x$-axis. Because the $K$ configurations are initially slightly different
and because our lapse depends dynamically on $K$, one cannot expect the same
code time to correspond exactly to the same physical time with the original
and corrected method. We suspect that this causes at least part of the
differences shown in Fig. B.4.

We have not run new convergence tests with the corrected method, because these
runs are computationally expensive. However, since the difference between the
initial configurations and PBH mass and spin data is small, we are confident
that new convergence tests would show the same level of convergence as the
ones already present in chapter 4. These tests show convergence of PBH mass
and spin, as well as convergence of the norms of the Hamiltonian and momentum
constraint violation integrated over the simulation box.

In summary, we show that in this setting, the original and corrected methods
generate very similar initial configurations, that both methods show second
order convergence and that using the corrected method does not significantly
change the observed PBH mass and angular momentum. The conclusions presented
in our paper do not change when using the corrected method for solving the
Einstein constraints.

Fig. B.4: Comparison between PBH mass and angular momentum, obtained with the
original and corrected method. Top row show these two quantities, while the
bottom row shows the normalised difference.


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In this diagram, the top and the left maps are already known to be bijective.
We have now shown that $\Phi$ is bijective too. Also, it’s clear by unwrapping
the defintions that $\Phi$ is a filtration preserving map, and so is its
inverse. Therefore, we conclude that $\mathrm{gr}(\Phi)$ must also be a
bijection. This forces the fourth map in the commutative diagram to be a
bijection. ∎

###  3.6. Quantum Hamiltonian Reduction functor

Let $(\mathcal{D}(\mathfrak{g})\otimes W_{2n})$-$mod$ denote the category of
finitely generated $(\mathcal{D}(\mathfrak{g})\otimes W_{2n})$-modules. The
algebra $\mathcal{D}(\mathfrak{g})\otimes W_{2n}$ contains a subalgebra
$Z=\operatorname{Sym}(\mathfrak{g})^{\mathfrak{g}}$ of invariant differential
operators on $\mathfrak{g}$ with constant coefficients. Let $Z_{+}\subseteq Z$
be the augmentation ideal, consisting of differential operators with zero
constant term.

Furthermore, we have the algebra homomorphism
$\Theta_{2}:\mathcal{U}\mathfrak{g}\rightarrow\mathcal{D}(\mathfrak{g})\otimes
W_{2n}$. Also, let
$eu\in\mathcal{D}(\mathfrak{g})\subseteq\mathcal{D}(\mathfrak{g})\otimes
W_{2n}$ denote the Euler vector field on $\mathfrak{g}$. Let $\mathcal{U}$ be
the subalgebra of $\mathcal{D}(\mathfrak{g})\otimes W_{2n}$ generated by the
image of $\Theta_{2}$ and $eu$.

######  Definition 3.1.

Let $\mathcal{C}$ be the full subcategory of
$(\mathcal{D}(\mathfrak{g})\otimes W_{2n})$-$mod$, whose objects are
$(\mathcal{D}(\mathfrak{g})\otimes W_{2n})$-modules $M$, such that the action
on $M$ of the subalgebra $Z_{+}$ is locally nilpotent and the action of
$\mathcal{U}$ is locally finite.

For any $M\in\mathcal{C}$, we have a $\mathfrak{g}$-action on $M$ via the map
$\Theta_{2}$. As the group $G=Sp(V)$ is simply connected, the action of
$\mathcal{U}$ on $M$ being locally finite implies that the
$\mathfrak{g}$-action on $M$ can be integrated to get a rational
representation of the group $G$ on $M$. Thus, the local finiteness condition
in the above definition implies that the modules $M$ are $G$-equivariant.

We can identify $\mathcal{D}(\mathfrak{g})\otimes W_{2n}$ with the ring of
differential operators $\mathcal{D}(\mathfrak{g}\times L)$. Taking the degree
filtration on this algebra, for any finitely generated
$\mathcal{D}(\mathfrak{g})\otimes W_{2n}$-module $M$, there exists a
characteristic variety
$\operatorname{Ch}(M)\subseteq\mathfrak{g}\times\mathfrak{g}\times V$.

######  Proposition 3.6.1.

For any $G$-equivariant module $M\in(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})$-$mod$, we have $M\in\mathcal{C}$ if and only if
$\operatorname{Ch}(M)\subseteq X^{nil}.$ Furthermore, all the objects in
$\mathcal{C}$ are holonomic $(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})$-modules.

###### Proof.

The proof of the fact that $M\in\mathcal{C}$ if and only if
$\operatorname{Ch}(M)\subseteq X^{nil}$ follows by essentially repeating the
proof of Proposition 5.3.2 of [GG06] replacing
$\mathfrak{gl}(V)\times\mathbb{P}$ by $\mathfrak{sp}(V)\times L$ everywhere.
The holonomicity of the objects in $\mathcal{C}$ follows from the fact that
their characteristic variety lies in $X^{nil}$, which is a Lagrangian
subvariety of $\mathfrak{g}\times\mathfrak{g}\times V$. ∎

We now define the quantum Hamiltonian reduction functor. Let $Q$ be the
quotient $(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/((\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g}))$. We have the quantum Hamiltonian
reduction of $\mathcal{D}(\mathfrak{g})\otimes W_{2n}$ with respect to the
$\mathfrak{g}$-action, given by
$\mathcal{A}:=Q^{\mathfrak{g}}=\Big{(}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g})\Big{)}^{\mathfrak{g}}$. By Theorem 1.3.1,
we have an isomorphism $\mathcal{A}\simeq eH_{c}e$, where $eH_{c}e$ is the
spherical Cherednik algebra with parameter $c=(-1/4.-1/2)$. Let
$eH_{c}e\text{-}mod$ be the category of finitely generated $eH_{c}e$-modules.
Then, by Proposition 7.2.2 and Corollary 7.2.4 of [GG06], we have:

######  Proposition 3.6.2.

  1. (1)

The space $Q$ is a finitely generated $(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})$-module.

  2. (2)

There is an exact functor $\mathbb{H}$:

$\mathbb{H}:(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\text{-mod}\longrightarrow eH_{c}e\text{-mod}$
$M\mapsto\operatorname{Hom}_{\mathcal{D}(\mathfrak{g})\otimes
W_{2n}}(Q,M)=M^{\mathfrak{g}}.$

  3. (3)

The functor $\mathbb{H}$ has a left adjoint $\prescript{T}{}{\mathbb{H}}$:

$\prescript{T}{}{\mathbb{H}}:eH_{c}e\text{-mod}\longrightarrow(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\text{-mod}$ $M\mapsto Q\otimes_{\mathcal{A}}M,$

such that the canonical adjunction morphism
$M\rightarrow\prescript{T}{}{\mathbb{H}}(\mathbb{H}(M))$ is an isomorphism for
all $M\in(\mathcal{D}(\mathfrak{g})\otimes W_{2n})\text{-mod}$.

  4. (4)

The full subcategory $\ker(\mathbb{H})$ is a Serre subcategory of
$(\mathcal{D}(\mathfrak{g})\otimes W_{2n})\text{-mod}$ and the functor
$\mathbb{H}$ induces an equivalence of categories:

$(\mathcal{D}(\mathfrak{g})\otimes W_{2n})\text{-mod}/\ker(\mathbb{H})\simeq
eH_{c}e\text{-mod}$

Next, we recall that the algebra $eH_{c}e$ is generated by the subalgebras
$\mathbb{C}[\mathfrak{h}]^{W}$ and $\operatorname{Sym}(\mathfrak{h})^{W}$,
defined in $\mathsection{\ref{sec:RCA}}$. Let
$\operatorname{Sym}(\mathfrak{h})^{W}_{+}$ denote the augmentation ideal of
$\operatorname{Sym}(\mathfrak{h})^{W}$. Let $\mathcal{O}(eH_{c}e)$ be the
category of $\mathcal{O}$ of the algebra $H_{c}$, which is defined as the full
subcategory of $eH_{c}e\text{-}mod$ whose objects are finitely generated
$eH_{c}e$-modules with locally nilpotent action of
$\operatorname{Sym}(\mathfrak{h})_{+}^{W}\subseteq eH_{c}e$ (see, for example,
[BEG03, $\mathsection{2}$]).

######  Proposition 3.6.3.

The functor $\mathbb{H}$ restricts to an exact functor
$\mathbb{H}:\mathcal{C}\rightarrow\mathcal{O}(eH_{c}e)$. This induces an
equivelence of categories
$\mathcal{C}/\ker(\mathbb{H})\simeq\mathcal{O}(eH_{c}e).$

###### Proof.

Under the isomorphism $\mathcal{A}\simeq eH_{c}e$, the subalgebra $Z_{+}$ of
$\mathcal{A}$ is mapped exactly to the subalgebra
$\operatorname{Sym}(\mathfrak{h})_{+}^{W}$ of $eH_{c}e$. Therefore, for any
$M\in\mathcal{C}$, we have $\mathbb{H}(M)\in\mathcal{O}(eH_{c}e)$.
Furthermore, by the corollary to Lemma 2.5 of [BEG03], there exists an element
$h\in H_{c}$ that acts locally finitely on every element of
$\mathcal{O}(H_{c})\simeq\mathcal{O}(eH_{c}e)$, such that the image of the
Euler vector field $eu$ in $eH_{c}e$ is equal to $h$ upto an additive scalar
factor (see the proof of Formula 6.7 in [BEG03]). Thus, the local finiteness
condition on the $eu$-action in the definition of $\mathcal{C}$ is
automatically true in $\mathcal{O}(eH_{c}e)$. ∎

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P. Goyal: Department of Mathematics, University of Chicago, Chicago, IL, 60637

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