# Supporting Material

Dimitrios C. Tzarouchis,1†‡ Brian Edwards,1† Nader Engheta1∗

1Department of Electrical and Systems Engineering,
School of Engineering and Applied Sciences,
University of Pennsylvania, Philadelphia, 19104, U.S.A.
†These authors contributed equally to this work.
‡Present address: Meta Materials Inc. (Europe),
Ap. Pavlou 10A, Marousi, 15123, Greece.
∗To whom correspondence should be addressed; e-mail<EMAIL_ADDRESS>
##  1 Details for the root finding algorithm

In the following the lowercase quantities are vectors, the capitalized ones
are matrices while Greek letters denote scalars - the subscripts follow a
logical notation. The problem statement for the root finding procedure is

$f({\bf z})=0$ (1)

find ${\bf z}\in\mathbb{C}^{m\times 1}$ that satisfies the above equation
(roots) of $f\in\mathbb{C}^{m\times 1}$. The above problem is solved using
Newton’s method for finding the root of a vector polynomial function [1]. For
example we have

${f}({\bf z})=\begin{pmatrix}f_{1}(z_{1},z_{2},z_{3},z_{4},z_{5})\\\
f_{2}(z_{1},z_{2},z_{3},z_{4},z_{5})\\\
f_{3}(z_{1},z_{2},z_{3},z_{4},z_{5})\\\
f_{4}(z_{1},z_{2},z_{3},z_{4},z_{5})\\\
f_{5}(z_{1},z_{2},z_{3},z_{4},z_{5})\end{pmatrix}$ (2)

with $z_{1\cdot 5}\in\mathbb{C}$ or (equivalently)

$\displaystyle f_{1}({\bf z})$
$\displaystyle=(z_{1}-r_{1})(z_{2}-4.2i)(z_{3}+2)(z_{4}-5i)(z_{5}-3.5)$ (3)
$\displaystyle f_{2}({\bf z})$
$\displaystyle=(z_{1}-3.9)(z_{2}-r_{2})(z_{3}+2.5i)(z_{4}-3.2i)(z_{5}-4.2)$
(4) $\displaystyle f_{3}({\bf z})$
$\displaystyle=(z_{1}+5.2i)(z_{2}-4)(z_{3}-r_{3})(z_{4}-4i)(z_{5}-7.1)$ (5)
$\displaystyle f_{4}({\bf z})$
$\displaystyle=(z_{1}-3)(z_{2}-7i)(z_{3}+4)(z_{4}-r_{4})(z_{5}-5i)$ (6)
$\displaystyle f_{5}({\bf z})$
$\displaystyle=(z_{1}-5.2i)(z_{2}-4)(z_{3}+4.75i)(z_{4}-8)(z_{5}-r_{4})$ (7)

where

$r=\frac{1}{4}\left(s_{1}+c_{1}i,-s_{1}+c_{1}i,-s_{2}-c_{2}i,s_{2}-c_{2}i,1i\right)^{T}$
(8)

with $c_{1}=\cos(2\pi/5)$, $c_{2}=\cos(\pi/5)$, $s_{1}=\sin(2\pi/5)$, and
$s_{2}=\sin(4\pi/5)$. The point corresponds to the vertices of a regular
pentagon. For the evaluation of Newton’s method we need to calculate the
Jacobian matrix, i.e., $J_{ij}=\frac{\partial f_{i}}{\partial x_{j}}$ (here
$i$ and $j$ are indexes) or

${J}_{f}({\bf z})=\begin{pmatrix}\frac{\partial f_{1}}{\partial
z_{1}}&\frac{\partial f_{1}}{\partial z_{2}}&\cdots&\frac{\partial
f_{1}}{\partial z_{5}}\\\ \frac{\partial f_{2}}{\partial z_{1}}&\frac{\partial
f_{2}}{\partial z_{2}}&\cdots&\frac{\partial f_{2}}{\partial z_{5}}\\\
\vdots&\vdots&\ddots&\vdots\\\ \frac{\partial f_{5}}{\partial
z_{1}}&\frac{\partial f_{5}}{\partial z_{2}}&\cdots&\frac{\partial
f_{5}}{\partial z_{5}}\end{pmatrix}$ (9)

therefore the root can be found as

${\bf z}_{n+1}={\bf z}_{n}-\alpha J^{-1}_{f}({\bf z}_{n})f({\bf z}_{n})$ (10)

where $\alpha$ is a relaxation constant. Here we used $\alpha=0.2$. In terms
of an algorithm, we have the following routine

Algorithm 1 Root finding with Newton’s method

1:Initial guess for ${z}_{1}$

2:for $n=1,\dots,m$ do

3: ${J}_{f}({\bf z}_{n})$

4: $\alpha_{\lambda}=\frac{2}{\lambda_{min}+\lambda_{max}}$ $\triangleright$
Scaling factor: $\lambda_{mim/max}$ are the min/max eigenvalues of
${J}_{f}({\bf z}_{n})$

5: $K_{n}=I-\alpha_{\lambda}{J}_{f}({\bf z}_{n})$ $\triangleright$ Kernel that
is fed to DCM machine

6: ${\bf d}_{n}={J}^{-1}_{f}({\bf z}_{n}){f}({\bf z}_{n})$ $\triangleright$
Compute matrix inverse with the DCM machine

7: ${\bf z}_{n+1}={\bf z}_{n}-\alpha{\bf d}_{n}$

8:end for

##  2 Details on the inverse design algorithm

In this section we present the details for the inverse design algorithm
implemented in text. The algorithm consist of a part of the DDA methodology
for the quantification of the problem and an its adaptation to a Lagrange
formalism for solving the require inverse scattering problem, the
determination of the permittivity of the scatterers. Note that both methods
are arguably the simplest methods to follow, since they offer an intuitive
understanding on the formulated problem and the coorresponding inverse-design
(constraints optimization) problem.

###  2.1 Notes on the DDA method

In this section we present a few details regarding the DDA method used in the
main text. The details can be found also in [2, 3, 4, 5]. A similar
methodological approach was also used in [6].

We start by assuming that each 2D scatterer (assuming a point in the x-y
plane) acquires its z-oriented dipole moment due to the local electric field,
i.e.,

$p=\alpha e_{\text{loc}}$ (11)

where the $e_{\text{loc}}$ is the vector of local z-polarized electric fields
at the center of each point and $\alpha$ is the polarizability that depends on
the shape and the material composition of each 2D scatterer. The local field
is the sum of the incident field and the secondary fields generated from all
the other dipoles such that:

$e_{\text{loc}}=e_{\text{inc}}+Gp$ (12)

where $e_{\text{inc}}$ is the incident field vector, ${p}$ is the induced
polarization vector and ${G}$ is the 2D Green’s function. In our case we
consider a two-dimensional (2D) problem with a transverse electric (TE)
excitation (the field is normal (z-direction) to the x-y plane). Therefore the
corresponding Green’s function reads

${G}={G}({r}_{i}-{r}_{k})=-j\frac{k_{0}^{2}}{4\pi\varepsilon_{0}}H^{(2)}_{0}(k_{0}|{r}_{i}-{r}_{k}|)$
(13)

where $H^{(2)}_{0}(k_{0}|{r}_{i}-{r}_{k}|)$ is the Hankel function of the
second type (with the time harmonic convention $e^{+j\omega t}$) and 0-th
order and $k_{0}=\omega_{0}\sqrt{\mu_{0}\varepsilon_{0}}$ is the free-space
wavenumber [7]. The $G$ is a $\mathbb{C}^{N\times N}$ Toeplitz matrix with
zero diagonal entries since the $|{r}_{i}-{r}_{k}|$ is treated as in (assuming
a uniformly spaced discrete grid) [2, 3, 4, 5].

By combining Eqs. (11) and (12) we obtain the following expression, arranged
using the matrix formulation as follows

$p=\left(A^{-1}-G\right)^{-1}e_{\text{inc}}$ (14)

where the lowercase quantities $p=[{p}_{1},{p}_{2},...,{p}_{N}]^{T}$,
$A=\text{diag}(\alpha)$,
$\alpha=A_{\text{cell}}\varepsilon_{0}[\varepsilon_{1}-1,\varepsilon_{2}-1,...,\varepsilon_{N}-1]$
($A_{\text{cell}}$ is the cross-sectional area of a cylinders) and
$e_{\text{inc}}=[e^{\text{inc}}_{1},e^{\text{inc}}_{2},...,e^{\text{inc}}_{N}]^{T}$
are $\mathbb{C}^{N\times 1}$ vectors, $\text{diag}(\cdot)$ is the diagonal
matrix operator.

Finally, the scattered field observed at $M$ specified discrete detection
points (in general $M\neq N$) is given by:

$e_{\text{sca}}=G_{\text{pr}}\leavevmode\nobreak\
p=G_{\text{pr}}\left(\text{diag}(\alpha^{-1})-G\right)^{-1}e_{\text{inc}}$
(15)

where $G_{\text{pr}}\in\mathbb{C}^{M\times N}$ is the “propagator" Green’s
function matrix. This propagator function connects the induced dipole
polarization vectors of the scatterers with the desired detection (or
objective) points. The above matrix representation of the scattering problem
allow us to have a clear inspection of the unknown quantities. These
quantities are the ones that will be formulated as a Lagrange multiplier
algorithm for the solution of the desired constrained optimization problem. We
note here that, as seen in Eq. (15), the forward scattering problem requires a
matrix inversion to evaluate the polarization density vectors induced in each
scattering cell, as we have discussed in our previous work [6] in which we
utilized the same DDA approach for the evaluation Eq. (14) and the matrix-
vector operation of Eq. (15) for different excitation and for different
scattering scenarios.

###  2.2 Notes on the Lagrange multiplier algorithm

For this, we utilize the DDA algorithm (where we closely follow the contrast
source inversion method![8]) and the Lagrange multiplier method for applying
the constraints and finding the optimal solution. First, in terms of the
defined problem, we have that the polarization is connected with the following
expressions

$p=A(e_{\text{inc}}+Gp)$ (16)

and

$e_{\text{sca}}=G_{\text{pr}}p$ (17)

A typical constrained minimization problem (primal) can be written as [9, 1,
10]

$\displaystyle\min_{\begin{subarray}{c}x,y\\\ y\in\mathbb{R}\\\ 0\leq y\leq
1\\\ \end{subarray}}$ $\displaystyle f(x,y)$ (18) s.t. $\displaystyle
g(x,y)\leq 0$

where $f(x,y)$ is the objective and $g(x,y)$ are the constraints also subject
to further requirements of the problem such as $y\in\mathbb{R}$ and $0\geq
y\geq 1$ For such problems the dual Lagrangian problem is expressed as

$\displaystyle\max_{\lambda}\min_{\begin{subarray}{c}x,y\\\ y\in\mathbb{R}\\\
0\leq y\leq 1\\\ \end{subarray}}$
$\displaystyle\mathcal{L}(x,y,\lambda)=f(x,y)+\lambda g(x,y)$ (19)

which is essentially a dual unconstrained problem (since all the constraints
are encapsulated to the $\lambda$ term). It is worth noting that the Lagrange
multiplier should be positive real, $\lambda\in R^{+}$. Finding an approximate
solution to the primal inverse scattering problem is therefore reduced to
finding a solution to the above dual problem. Notice that the Lagrange
multiplier can be applied to either $f(x,y)$ or $g(x,y)$ without affecting the
outcome of the overall process.

The algorithm for solving the above dual problem is the following:

  * •

Step 0: initial $x_{0}$ and $\lambda_{0}$

  * •

Step 1: minimize $y_{n}$, i.e., via
$\nabla_{y}\mathcal{L}(x_{n-1},y_{n},\lambda_{n-1})=0$

  * •

Step 2: project $y_{n}$ into $y\in\mathbb{R}$ and $0\leq y\leq 1$

  * •

Step 3: minimize $x_{n}$, i.e.,
$\nabla_{x}\mathcal{L}(x,y_{n},\lambda_{n-1})=0$

  * •

Step 4: maximize $\lambda_{n}$, i.e.,
$\nabla_{\lambda}\mathcal{L}(x_{n},y_{n},\lambda)=0$

  * •

Step 5: Repeat steps 1-4 until the error is minimized

For our particular example we have that $x=p$,
$y=A=\text{diag}(\varepsilon-1)$, and
$f(p,A)=1/2||(A^{-1}-G)p-e_{\text{inc}}||^{2}$ and
$g(p)=1/2||G_{\text{pr}}p-e_{\text{obj}}||^{2}$, and Lagrange function reads

$\mathcal{L}(p,A,\lambda)=||(A^{-1}-G)p-Ae_{\text{inc}}||^{2}+\lambda||G_{p}p-e_{obj}||^{2}$
(20)

The corresponding algorithmic steps are:

  * •

Step 0: initial $p_{0}$ and $\lambda_{0}$

  * •

Step 1: minimize $A_{n}$ via
$\nabla_{A}\mathcal{L}(p_{n-1},A,\lambda_{n-1})=0$

    * –

We have that $\nabla_{A}\mathcal{L}(p_{n-1},A,\lambda_{n-1})=\nabla
f(p,A)^{*}||(A^{-1}-G)p-e_{\text{inc}}||$ ($*$ is complex conjugate). This
expression lead to $A_{n}=p/(Gp-e_{\text{inc}})$. In practice this is a simple
calculation since $A$ is a diagonal matrix, i.e.,
$A=\text{diag}(\varepsilon-1)$.

  * •

Step 2: project $A_{n}$ into $A_{n}\in\mathbb{R}$ and $0\leq A\leq 4$ (for the
range $\varepsilon\in[1,5]$)

    * –

this is the point where essentially the required properties and bound of the
permittivity can be implemented. These bounds or constrains can be general

    * –

the above projection is rather a simple projection that does not guarantee
always the minimum within the projection domain. A more accurate projection
would be of the form
$A_{n}=\text{proj}[A_{n-1}-\eta\nabla_{A}||(A_{n-1}^{-1}-G)p_{n-1}-e_{\text{inc}}||^{2}]$.

  * •

Step 3: minimize $p_{n}$, i.e.,
$\nabla_{p}\mathcal{L}(p,A_{n},\lambda_{n-1})=0$ (DCM metadevice).

    * –

$p_{n}=K_{n}^{-1}e_{\text{L}}^{n}$

    * –

$K_{n}=(A^{-1}_{n}-G)^{*}(A^{-1}_{n}-G)+\lambda_{n-1}G^{*}_{\text{pr}}G_{\text{pr}}$

    * –

$e^{L}_{n}=\lambda_{n-1}G_{\text{pr}}^{*}e_{\text{obj}}+(A^{-1}_{n}-G)^{*}e_{\text{inc}}$

    * –

The matrix inversion $p_{n}$ is performed with our DCM metadevice

    * –

Due to noise error a simple weighted average filtering is applied, i.e.,
$p_{n}=(1-\alpha_{F})p_{n-1}+\alpha_{F}p_{n}$ with $\alpha_{F}=0.25$

  * •

Step 4: maximize $\lambda_{n}$, i.e.,
$\nabla_{\lambda}\mathcal{L}(A_{n},p_{n},\lambda)=0$

    * –

This maximization can be calculated by a simple gradient descent, i.e.,
$\lambda_{n}=\lambda_{n-1}+\eta\left(\nabla_{\lambda}\mathcal{L}(p_{n},A_{n},\lambda)-\delta\right)$
or
$\lambda_{n}=\lambda_{n-1}+\eta\left(||G_{p}p_{n}-e_{obj}||^{2}-\delta\right)$

    * –

Notice that this is an gradient _ascent_ since we assume $\eta>0$, therefore
we maximize the problem.

  * •

Step 5: Repeat steps 1-4 until the error is minimized. In our case we used the
following error

    * –

$||e_{\text{sca}}-e_{\text{obj}}||^{2}/||e_{\text{obj}}||^{2}$

Note that the quantities $\eta$ and $\delta$ are the step and minimal error
quantities that are user determined. The whole process stop either when
$\lambda$ reaches a plateau, or when the required error criterion is met. The
optimization goal was set as
$\frac{||e_{\text{sca}}-e_{\text{obj}}||^{2}}{||e_{\text{obj}}||^{2}}<\delta$,
where $e_{\text{sca}}=G_{p}p_{\text{m}}$ with
$p_{\text{m}}=(A_{m}^{-1}-G)^{-1}e_{\text{inc}}$ being the final $m$-th
evaluation of the iteration.

Notice that our approach has several similarities with the contrast source
inversion method and other similar inverse scattering methods [11, 8, 12, 13].

Undoubtedly this approach is only one of the available methods for
approximating the inverse design problem. This is rather an attempt to
showcase the ability of our device for performing inverse design with desired
objectives and constraints by exposing the crucial parts of the algorithm,
such as the matrix inversions. This part is usually implicit within
commercially available FDTD or FEM software. Hence here we developed our own
methodology so we can have deeper inspection to quantities. As a remark, the
field of inverse design and inverse scattering is a very rich field with a
plethora of methodologies that try to address similar problems [14].

##  3 RF Design, PCB, Device Implementation

A photograph of the experimental setup is shown in Fig S1, where all parts are
designated accordingly.

Figure S1: Photograph of the experimental setup with the corresponding
components.

###  3.1 Measurement

Measurements were performed using an ENA-5071C two port VNA. In order to avoid
the saturation of the amplifiers (multiplier module) the VNA power level was
set to be $-20$dBm for the open loop configuration and $-10$dBm for the closed
loop configuration. The VNA was set to have an IF bandwidth of 10 kHz. The
single frequency measurements (1601 point) at 45MHz with averaging applied
after obtaining the measured signal from VNA.

###  3.2 Multiplier

Figure S2: Schematics for the multiplier: The PCB layout design (top figure),
and the corresponding subcircuit (pictures from
Altium${}^{\text{\textregistered}}$). Bottom figure represent the AWR
Microwave Office${}^{\text{\textregistered}}$ schematic with the realistic
data

The schematic of the multiplier is depicted in Fig. S2. The multiplier was
designed to perform multiplication on the incoming complex amplitude such that
a new complex amplitude is rendered at the output. In other words, the output
is $V_{\mathrm{out}}=zV_{\mathrm{in}}$. This involves changing both the
amplitude and phase of the incoming signal. Phase change was performed using a
pair of serially connected Minicircuit JSPHS-51+ Phase Shifters (PS). Each
phase shifter provides slightly over 180 degrees of rotation. The amplitude
change was performed using the Analog Devices AD603ARZ Variable Gain Amplifier
(VGA). The Multiplier design contain the appropriate loads such that both the
input and output of the device externally appears as 50 Ohm.

Both of these devices are controlled using analog voltages with ranges of
$[-0.5\mathrm{V},+0.5\mathrm{V}]$ and $[0\mathrm{V},12\mathrm{V}]$ for the VGA
and PS, respectively. In order to create a common control mechanism, op-amp
level shifting circuits were used to put these on a common
$[0\mathrm{V},5\mathrm{V}]$ interface. The Multiplier board has a connection
that allows for a daughter board. The daughter board is supplied with 0V and
+5V and is responsible for returning two control voltages in the range of
$[0\mathrm{V},5\mathrm{V}]$.

This simple interface allows for a number of possible control schematics. At
its most simple scenario, the control board can consist of a pair of
potentiometers. However, we will present another control board which utilizes
a microcontroller to receive UART input and render the two analog voltages.

The VGA’s dynamic range could be shifted using an external resistor. This was
set so that the Multipliers’s range (including load elements, PS losses, etc)
was [-30dB – +17dB]. The multiplier effectively saturates if the input is
greater than -10dBm. Therefore for all measurements the reference input signal
that was used was -30dBm for avoiding any saturation effects.

It should be stated that the VGA imparts a varying phase change and the PS
pair imparts an amplitude change. This will be addressed later.

###  3.3 1-5 splitter (5-1 combiner)

Figure S3: Schematic and layout for 1-5 splitter based on the Minicircuits
AD5PS-1+ (pictures from Altium${}^{\text{\textregistered}}$)

The schematic of the 1-5 splitter is depicted in Fig. S3. An ideal passive
$n$-way splitter is comprised of a summation port and $n$ feed ports. The
scattering parameters are expected to be reciprocal such that for the
$i^{\mathrm{th}}$ feed port $|S_{\mathrm{S}i}|^{2}=|S_{i\mathrm{S}}|^{2}=1/n$
and all other elements within the matrix are zero. Due to losses, a real
splitter will fall short of this precise definition. Our splitter was based on
the Minicircuits AD5PS-1+, which yielded good performance at 45MHz with
approximately -7.2dB split ratio for all outputs.Note that $1/5\approx-7.0dB$.

###  3.4 Feedback coupler

Figure S4: Schematic and layout for the feedback coupler (pictures from
Altium${}^{\text{\textregistered}}$)).

The schematic of the feedback coupler is depicted in Fig. S4 The Feedback
coupler must perform several tasks.

  * •

Provide near unity feedback

  * •

Introduce the input signal

  * •

Sample the output signal

Therefore, the feedback coupler is a four-port device wherein the primary path
has near unity transmission such that the feedback is strong.

###  3.5 Switches

In order to replicate having a 10 port VNA, we utilized two demo boards
(EV1HMC253AQS24), which acted as RF SP8T RF switches, i.e., an analog
multiplexer. For one SP8T, we utilized five of these ports for illuminating
the bank of five couplers. The other SP8T was used to receive signals from the
couplers. The remaining three ports on each were used for system sanity
checks. Note that the stock high-pass 100pF capacitors on these boards were
switched to 470pF for better transmission at 45MHz.

While the off ports were nominally matched to 50 Ohm from DC-2.5GHz, there was
significant reflections. Deeper inspection of the datasheet indicated that the
"off" ports were only matched above 500MHz. Measurements indicated that the
off ports were approximately "open" at the design frequency and therefore
reflections from the off ports could be significantly reduced with parallel
50Ohm terminations. However, this was not done as this it would have reduced
power within the system on the "on" port. Rather, we note that any polluting
signal from these "open" off ports will have crossed through the coupler
twice. Due to the small coupling coefficient of the feedback coupler, these
values will have become very small.

The VNA was calibrated to the end of the switch ports. Measurements indicated
that transmission through each of the switch ports was similar enough as to
not warrant individual calibrations on each.

Each of the switches was actuated by three digital inputs to address the
$2^{3}=8$ ports on each switch. These digital signals were created by a micro-
controller which was programmed to respond to UART commands from an attached
computer. Code is available at
github.com/brianedw/RFMath/Arduino/mcu_control_V2/mcu_control_V2.ino.

###  3.6 Micro Controller Unit (MCU)

The two analog input control voltages for each Multiplier was created by an
MCU Control Board, which attached directly to the Multiplier. The heart of
this board is a Metro-Mini MCU.

Each control line was connected to both an 8-bit PWM DAC pin (labeled “fast”)
and a 10-bit PWM DAC pin (labeled “slow”). While both pins connected to the
control line through a high-pass filter, the fast DAC utilized a lower
capacitance and resistance than that of the slow DAC. During a set operation,
both pins would drive to their appropriate values, during this time, the
behavior of the collective output would be dominated by the fast DAC and
rapidly converge, but exhibit large ripples. After 20ms, the fast PWM DAC pin
would switch to a high-impedance state, leaving the voltage to settle in the
remaining difference utilizing the slow DAC alone. The high-pass filter was
designed to maintain accuracy of 10bit. Since the Metro-Mini is a 5V compliant
device, the generated voltages nicely matched to the expected inputs of the
Multiplier.

Each MCU board had two 3-pin UART input connectors. These were shorted such
that one could be used to receive a command from "upstream" while the other
would effectively passively repeat the signal. Additionally, each MCU board
had two 3-pin UART output connector which were similarly shorted together,
allowing it to transmit the same message to two devices. Each MCU was
programmed with a unique identification number. Upon receiving a UART command,
it would either act on that command or repeat the command on its output UART
pins for downstream devices. This input/output configuration created a lot of
possibilities for control topologies. However, in practice we found that we
could use a single MCU board (no multiplier attached), as a bridge between the
computer and the array of MCU Boards and that this array could all be
connected in parallel such that the output of the bridge was effectively
driving 25 inputs. Note that the required time complexity is of the order of
$\mathcal{O}(n^{2})$. Possibly this complexity can be further reduced by
implementing different connectivity schemes than the simple serial one that we
used. Code is available at
github.com/brianedw/RFMath/Arduino/mcu_control_V2/mcu_control_V2.ino.

##  4 Tuning/Calibration

As stated in 3.2, the VGA has a minor effect on the phase and the PS has a
minor effect on the amplitude. In other words, the phase and amplitude
responses are coupled. Additionally, other systematic errors are present such
as nonidealities in the level shifting circuits due to resistor tolerances.
When connected in a network that includes RF jumper cables of varying length,
there will also be phase shifts that naturally arise. In short, the
relationship between the control voltages and the response of the Multiplier
_in situ_ , are repeatable, but difficult to predict without developing a more
complex model.

We found that an effective strategy to capture, model, and invert the
relationship between control voltages and system response goes as follows.

  1. 1.

A collection of Multipliers are swept across their input values to map the
relationship between control voltage and complex multiplier response.

  2. 2.

These responses were analyzed using Principle Component Analysis (PCA) [15].

  3. 3.

The multipliers were assembled into the open-loop configuration and the
response of the entire open-loop network was measured under many sets of input
control voltages.

  4. 4.

These results were compared to a theoretical model of the network wherein the
weights of the components could be adjusted until the theoretical results
matched the experimental results.

  5. 5.

With accurate PCA weights in hand, the Multipliers can be immediately adjusted
to achieve a desired multiplication factor by inverting the model to achieve
any open-loop kernel.

  6. 6.

Additional refinement can be obtained by changing the device configuration
into the closed loop, which now includes the feedback couplers. Again, we
measure the response of the closed-loop network under many input conditions.

  7. 7.

We further refine the PCA weights of each multiplier to match this more
demanding data set. This becomes our final device model for both the open- and
closed-loop configurations.

We will go into detail on each one of these items in the following sections.

###  4.1 Multiplier PCA

A collection of 35 multipliers were each mapped using the MCU control boards,
capable of 10-bit resolution on both control voltages. The mapping occurred
with a grid of values based on [0, 11, …, 1012, 1023] on both controls.
Ideally, the mapping of two Multipliers would yield identical responses.
However, for all the reasons stated above, they do not. All of the mappings
were compared using a complex domain PCA analysis. Typically, in PCA, one
would examine deviations from the mean, but here we take another approach.
Rather, the collection of mappings were analyzed directly to yield a set of 4
PCA components. The response of any individual Multiplier could then be found
as the linear superposition of these components given by:

$m(d_{\mathrm{vga}},d_{\mathrm{PS}})=\sum_{i=0}^{3}w_{i}c_{i}(d_{\mathrm{vga}},d_{\mathrm{PS}})$

The term $c_{0}(d_{\mathrm{vga}},d_{\mathrm{PS}})$ is effectively the
“average” response scaled by a complex factor, while the next several
components represent likely deviations due to the systematic errors described
above. Within a PCA analysis the final PCA components (i.e.
$c_{34}(d_{\mathrm{vga}},d_{\mathrm{PS}})$, not shown) should be nearly pure
noise. We found that only the first four terms were needed to effectively
model any given Multiplier.

Given any randomly chosen multiplier, we can find the complex valued PCA
weights $w_{i}$ through a least-squares analysis. As opposed to the "deviation
from the mean" approach, the above formulation is particularly useful for RF
engineering. While the Multipliers were measured directly at their input and
output ports and analyzed as such, the model can easily account for the
addition of RF cables which would provide attenuation and phase rotation.
These will appear as a complex scaling of all of the components weights and
the Multiplier’s behavior (RF jumpers cables included) can still be captured
as the simple linear superposition of the PCA components. In fact, any losses
or phase rotations along the Multipliers flow path can be incorporated into
these weights. Therefore, we do _not_ characterize the individual multipliers,
but delay this until the architecture is fully assembled, as described in the
next section.

Regardless, we will use least-squares to find the set of $w_{i}$ which
characterizes the average multiplier response. We call these the “base
weights”.

Figure S5: PCA Components and Average Multiplier Response. The first four
panels represent $c_{0}(d_{\mathrm{vga}},d_{\mathrm{PS}})$,
$c_{1}(d_{\mathrm{vga}},d_{\mathrm{PS}})$,
$c_{2}(d_{\mathrm{vga}},d_{\mathrm{PS}})$, and
$c_{3}(d_{\mathrm{vga}},d_{\mathrm{PS}})$ and have a maximum saturation of
0.05. The final image shows the response of the “average” Multiplier with a
maximum saturation at 7.5

###  4.2 Open-Loop Device Fitting

The goal of the this section is to determine the PCA weights that characterize
each Multiplier _in situ_ , so that the system errors can be captured and
modeled. The open-loop DCM system was fully assembled including jumper cables,
splitters, and couplers. All multipliers within the array were set to the same
input value $(d_{\mathrm{vga}},d_{\mathrm{PS}})$ pair. The transmission matrix
of the system was then measured. This was repeated for all possible
combinations of 10 evenly spaced values in the range $[0,1023]$ to yield 100
measured transmission matrices, $\mathcal{T}_{\mathrm{meas}}$. Note that not
all of these 100 transmission matrices represent "passive" operators.

The same system was modeled using Scikit-RF, wherein the following assumptions
were made:

  * •

The 5-1 splitters were ideal such that power was evenly split with no phase

  * •

All jumpers were zero-length

  * •

The coupler feedback path was ideal with no power removed

  * •

The multipliers were all assumed to be “average” and the their responses were
assumed to be given by the “base weights”.

The system was simulated using SciKit-RF for each input pair
$(d_{\mathrm{vga}},d_{\mathrm{PS}})$ to yield 100 measured transmission
matrices $\mathcal{T}_{\mathrm{sim}}(\overline{w})$, which are naturally a
function of each Multipliers PCA weight. We can then define an error
$\mathrm{error}(\overline{w})=\left|\mathcal{T}_{\mathrm{sim}}(\overline{w})-\mathcal{T}_{\mathrm{meas}}\right|^{2}$
and optimize $\overline{w}$ until that error is minimized. It should be noted,
that with only four PCA weights per Multiplier, in theory, only 4 transmission
matrices are required to fully define the system. Using 100 helps guarantee
that normal measurement noise does not unduly influence the fitting.
Additionally, if a low error can be achieved across 100 measurements using
only 4 weights, then we can be confident that the model was sufficient to
capture the entire open loop system response, $\mathbb{K}$.

###  4.3 Setting the Open Loop System Response

Given a desired open-loop system response, $\mathbb{K}$, we need to calculate
the necessary multiplier values for the DCM architecture, $m_{i,j}$, gathered
to form $\mathbb{M}$. In this case, the simplicity of the DCM architecture
makes this trivial. If we assume an idealized passive five port splitters such
that given an input of 1W at the summation port, s, we will observe 1/5W on
each branch port, $i$. Put in terms of Scattering Parameters,
$S_{\mathrm{s},i}=1/\sqrt{5}$ and via reciprocity
$S_{i,\mathrm{s}}=1/\sqrt{5}$. Since we have such splitters at the input and
output of the Multiplier array,
$\mathbb{K}=(1/\sqrt{5})\mathbb{M}(1/\sqrt{5})$ and therefore
$\mathbb{M}=5\mathbb{K}$. Note that since we fitted the PCA weights of the
Multipliers under the assumption of ideal components, it is appropriate to
assume ideal components here.

With each of the desired $m_{i,j}$ in hand to achieve a given $\mathbb{K}$,
the next step is to determine the required
$(d_{\mathrm{vga}},d_{\mathrm{PS}})$. This can be done using a number of
function inversion schemes such a gradient descent. In practice, this could be
very fast as it is likely that in many applications, each new $\mathbb{M}$
will be a small step from the previous $\mathbb{M}$ and therefore each
multiplier will change only slightly.

###  4.4 Closed-Loop Device Fitting

Due to the recursive nature of the closed-loop configuration (Matrix
Inversion), the accuracy requirements are more stringent than for the open-
loop configuration. Moreover, additional degrees of freedom are introduced in
the form of coupler coefficients. These can be considered part of
$\overline{w}$. In short, the devices must be fitted again.

We will employ a similar strategy as was used in the _Open-Loop Device
Fitting_. Using the open-loop calibrated device models, a sequence of randomly
generated passive transmission matrices, $\mathbb{K}$, are shown to the
system. Note, unlike the open-loop matrices, in order to guarantee
convergence, these matrices must be passive. We model the closed-loop system
using Scikit-RF. Using the open-loop weights as a starting point, we optimize
the multiplier weights and coupling coefficients until the simulated
$\mathcal{T}_{\mathrm{sim}}(\overline{w})$ matches the measured
$\mathcal{T}_{\mathrm{meas}}$. This represents a small, but necessary,
refinement from the open-loop device model and can be used for both open- and
closed-loop applications.

###  4.5 Setting the Closed Loop System Response

Setting the closed-loop system response, $\mathbb{K}$ is identical to setting
the open-loop response. In both cases, each desired $m_{i,j}$ is used to find
the required $(d_{\mathrm{vga}},d_{\mathrm{PS}})$ using a function inversion
scheme.

##  5 System Accuracy

We performed an open loop measurement on 100 complex-valued random matrices
with (eigenvalues) values within the unit circle. For these, we configured the
open-loop with the target (or ideal kernel) $A_{e}$ and retrieved the measured
results $A_{m}$. We define as error the quantity

$\frac{||A_{m}-A_{e}||^{2}}{||A_{e}||^{2}}100\%$ (21)

In Fig. S6, we can see the difference between the two matrices for 100 random
cases. We observe that all the results are within a $0.05-0.3\%$ percent
error. Similarly, we performed the same error analysis for the same 100 random
matrices, only this time on a closed-loop setup (matrix-inversion). The
results (Fig. S7) reveal that the error can climb up to 20%, but for most of
the results, we get a matrix inversion with less than 2% error. Finally, we
assess the matrix inversion fidelity by evaluating the trace of the
$A_{m}^{-1}A_{e}$ product. Ideally the trace of the product
$\text{tr}(A_{m}^{-1}A_{e})/5=1$. In Fig. S8, we observe that this product
spans between $0.5-1.5$. However, for the particular examples we used in the
manuscript, this accuracy can be maintained at reasonably high levels once
error-correcting and filtering techniques are applied. Note that for the
closed-loop case, the level of the measured voltage is in the order of $\mu$V,
very close to the noise floor of the VNA device we used. For the open loop
operation, the measured voltage was hundreds of mV.

Figure S6: The error between the exact and the measured matrices, open loop
configuration, for 100 random complex matrices. Figure S7: The error between
the exact and the measured matrices, closed loop configuration (matrix
inversion), for 100 random complex matrices. Figure S8: The fidelity of the
matrix inversion expressed in terms of the normalized trace of the
$A_{m}^{-1}A_{e}$ product for 100 random complex matrices.

##  6 System Transient Analysis

###  6.1 Single Multiplier

In terms of the time response of the multiplier module the transient analysis
reveal (Fig. S9) that the module obtained the desired value approximately
within 3-4 signal periods, i.e., $T=22.2$ns. The measurements were performed
using the RIGOL DG4062 pulse generator (15 sinusoidal pulses at 45MHz), and
the measured response extracted with the RIGOL DS1104 oscilloscope.

Figure S9: The transient response of a single multiplier module. The blue
curves correspond to the input signal, while the red curves are the measured
(top) and simulated (bottom) using AWR Microwave
Office${}^{\text{\textregistered}}$ results. The agreement is excellent. It is
evident that it takes approximately 3 to 4 signal periods for the multiplier
to obtain the desired output signal. Here we assumed small signal
amplification (VGA voltage is +.05) and the phase shift voltage is 0V.

The open loop response is therefore assumed to be very close to the single
multipliers response since both splitters and connecting cables introduce a
small phase shift to the signal. The closed loop transient response is
affected by both the multiplier timing and the condition number of the input
matrix (kernel) as shown in [16].

##  7 De-embedding the solution

###  7.1 Open Loop

Let us define the open-loop response as

$V_{\mathrm{out}}=\mathbb{K}V_{\mathrm{in}}$

Note that this includes not only the DCM architecture (multipliers, splitters,
jumpers), but also the through channel of the input/output coupler. In other
words, the open-loop is defined using all of the components of the closed-
loop. However, the loop has been broken "open" just after the coupler array
and measured at this point. Since in the closed configuration, these
measurement planes were coincident, upon "closing" the loop, these measurement
will then represent the complete response of the loop. While a minor
perturbation to the results, this definition assumes that the weakly coupled
additional ports on the coupler are properly terminated.

Let us further define response of only the DCM architecture as
$\mathbb{K}^{\prime}$. When the system is in a closed loop configuration, this
relates the vector exiting the coupler array ($V_{4}$) to the vector incident
on the coupler array ($V_{2}$).

$V_{2}=\mathbb{K}^{\prime}V_{4}$

The coupler array introduces a small loss as the input is introduced and the
output is measured. The near unity transmission is named $\alpha_{1}$. It is
clear then that $\mathbb{K}=\alpha_{1}\mathbb{K}^{\prime}$.

###  7.2 Closed Loop

The closed loop response is fully defined by the open-loop response and the
definition of the scattering parameters of the coupler.

$\displaystyle V_{2}$ $\displaystyle=\mathbb{K}^{\prime}V_{4}$ (22)
$\displaystyle V_{3}$ $\displaystyle=\alpha_{2}V_{1}+\beta V_{2}$ (23)
$\displaystyle V_{4}$ $\displaystyle=\beta V_{1}+\alpha_{1}V_{2}$ (24)

Our goal is to solve the equations for $V_{4}$, which represents the vectorial
solution of the problem in question. For the _expected solution_ , this should
be done such that the solution depends only on the kernel $\mathbb{K}$ and the
input vector ($V_{1}$). For the _measured solution_ , this should be only in
terms of the measured results ($V_{3}$) and the known input ($V_{1}$).

###  7.3 Expected Solution

We begin by applying the definitions above

$\displaystyle V_{4}$ $\displaystyle=\beta V_{1}+\alpha_{1}V_{2}$
$\displaystyle V_{4}$ $\displaystyle=\beta
V_{1}+\alpha_{1}\mathbb{K}^{\prime}V_{4}$ $\displaystyle V_{4}$
$\displaystyle=\beta V_{1}+\mathbb{K}V_{4}$

and then solve the final equation for the $V_{4}$.

$V_{4}=(I-\mathbb{K})^{-1}\beta V_{1}$

###  7.4 Measured Solution

We begin with Eq 23:

$V_{3}=\alpha_{2}V_{1}+\beta V_{2}$

and then solve it for $V_{2}$

$V_{2}=\frac{1}{\beta}V_{3}-\frac{\alpha_{2}}{\beta}V_{1}$

and then substitute the above into 24

$V_{4}=\beta
V_{1}+\alpha_{1}(\frac{1}{\beta}V_{3}-\frac{\alpha_{2}}{\beta}V_{1})$

and then simplify

$V_{4}=(\beta-\frac{\alpha_{1}\alpha_{2}}{\beta})V_{1}+\frac{\alpha_{1}}{\beta}V_{3}$

Note that in many real world cases, the coupler will be defined such that we
can assume $\alpha_{2}\rightarrow 0$

$V_{4}=\beta V_{1}+\frac{\alpha_{1}}{\beta}V_{3}$

## References

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results was the lack of a good notion of singular support for varieties of
positive characteristic.

Beilinson, in [Bei16], generalized the notion of singular support for
constructible sheaves defined over any characteristic. Since then the above
singular support characterizations have found positive characteristic
variants. Namely, Psarmoligkos [Psa22] proved that character sheaves on a
reductive group have nilpotent singular support. Shortly after Zhou in
[Zho24a], proved Mirković’s singular support characterization of character
sheaves on Lie algebras using similar methods. The methods of Zhou should
presumably extend to the modular case as well. We will work through the
details for one direction of the implication.

###### Proposition 7.1

Let $\mathcal{A}$ be a simple perverse sheaf in
$\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{E})$. If $\mathcal{A}$ is a
character sheaf, then
$SS(\mathcal{A})\subset\mathfrak{g}\times\mathcal{N}_{\mathfrak{g}}$.

We will closely follow the proof of Theorem 5.1 in [Zho24a].

###### Lemma 7.2

Suppose $\mathfrak{g}$ is semisimple. Let
$\mathcal{C}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{E})$ be a simple
perverse sheaf. Then $\mathcal{C}$ is cuspidal if and only if
$SS(\mathcal{C})\subseteq\mathfrak{g}\times\mathcal{N}_{\mathfrak{g}}$.

###### Proof

By Lemma 8.2.7 (3) of [Ach21], $\mathcal{C}$ is $\mathbb{G}_{m}$-equivariant
for the weight 2 action. By [Zho24b],
$CC(\mathcal{C})=CC(\mathbb{T}\mathcal{C})$, and hence
$SS(\mathcal{C})=SS(\mathbb{T}\mathcal{C})$. In particular, $SS(\mathcal{C})$
is nilpotent if and only if
$\operatorname{supp}(\mathcal{C})\subseteq\mathcal{N}_{\mathfrak{g}}$ and
$\operatorname{supp}(\mathbb{T}\mathcal{C})\subseteq\mathcal{N}_{\mathfrak{g}}$.
By Lemma 5.8, we have that
$SS(\mathcal{C})\subseteq\mathfrak{g}\times\mathcal{N}_{\mathfrak{g}}$ if and
only if $\mathcal{C}$ is cuspidal. ■

###### Lemma 7.3

If $\mathcal{C}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{E})$ is
cuspidal, then
$SS(\mathcal{C})\subseteq\mathfrak{g}\times\mathcal{N}_{\mathfrak{g}}$.

###### Proof

We can find a $\boxtimes$-factorization of $\mathcal{C}$ as
$\mathcal{C}\cong\mathcal{L}\boxtimes\mathcal{B}$. Let
$\operatorname{pr}_{2}:Z(\mathfrak{g})\times[\mathfrak{g},\mathfrak{g}]\to[\mathfrak{g},\mathfrak{g}]$
be the projection. One has that

$SS(\mathcal{C})=SS(\mathcal{L}\boxtimes\mathcal{B})=\operatorname{pr}_{2}^{\circ}SS(\mathcal{B})$

As we saw in the proof of Theorem 5.3, $\mathcal{B}$ is cuspidal. So by Lemma
7.2, we have that $SS(\mathcal{B})$ is nilpotent. It is clear that $p^{\circ}$
takes $[\mathfrak{g},\mathfrak{g}]\times\mathcal{N}_{\mathfrak{g}}$ to a
subset contained in $\mathfrak{g}\times\mathcal{N}_{\mathfrak{g}}$ (since the
nilpotent cone for a reductive Lie algebra is a subset of its semisimple
part). Therefore, $SS(\mathcal{C})$ is nilpotent. ■

Proof of Proposition 7.1.

Let $\mathcal{A}$ be a character sheaf. By 6.9, $\mathcal{A}$ is quasi-
admissible. By Lemma 6.11, $\mathcal{A}$ is a subobject of
$\underline{\textnormal{Ind}}_{L\subset P}^{G}\mathcal{C}$ for some cuspidal
sheaf $\mathcal{C}$ on $\mathfrak{l}$. It then suffices to show that
$SS(\operatorname{Av}_{P*}^{G}\circ i_{*}\pi^{!}\mathcal{C})$ has nilpotent
singular support. By functoriality of singular support,

$SS(i_{*}\pi^{!}\mathcal{C})=i_{\circ}\pi^{\circ}SS(\mathcal{C})$

which preserves nilpotency.

By Lemma 3.3 of [Psa22], one has that

$SS(\operatorname{Av}_{P*}^{G}i_{*}\pi^{!}\mathcal{C})\subseteq\overline{G\cdot
SS(i_{*}\pi^{!}\mathcal{C})}.$

Since $\mathcal{N}_{\mathfrak{g}}$ is a union of $G$-orbits, we have that
$SS(\underline{\textnormal{Ind}}_{P}^{G}\mathcal{C})$ is nilpotent. $\square$

##  8 Induction Series

Cuspidal and admissible sheaves on the nilpotent cone $\mathcal{N}$ are well
studied in the context of the modular generalized Springer correspondence of
[AHJR16], [AHJR17a], and [AHJR17b]. In this section, we will describe some
analogous results for cuspidal and admissible sheaves on all of
$\mathfrak{g}$. We will also study the compatibility between admissible
sheaves on $\mathfrak{g}$ and on $\mathcal{N}$.

Let $a_{\mathfrak{g}}:\mathcal{N}_{\mathfrak{g}}\to\mathfrak{g}$ be the
inclusion of the nilpotent cone into $\mathfrak{g}$. Consider the diagram

${\mathfrak{g}}$${\mathfrak{p}}$${\mathfrak{l}}$${\mathcal{N}_{\mathfrak{g}}}$${\mathcal{N}_{\mathfrak{p}}}$${\mathcal{N}_{\mathfrak{l}}}$$\scriptstyle{i}$$\scriptstyle{\pi}$$\scriptstyle{a_{\mathfrak{g}}}$$\scriptstyle{a_{\mathfrak{p}}}$$\scriptstyle{i^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{a_{\mathfrak{l}}}$

The top row of this diagram was used in defining induction and restriction
functors. We will define analogous induction and restriction functors on the
nilpotent cone that are compatible with restriction. Namely, if
$A^{\prime}\in\operatorname{Perv}_{G}(\mathcal{N},\mathbb{E})$, we can define
the induction and restriction functors via

${\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\textnormal{Res}$\cr}}}}_{P}^{G}=\operatorname{For}_{L}^{P}\circ\pi^{\prime}_{!}\circ
i^{\prime*}\circ\operatorname{For}_{P}^{G}$
$\vec{\textnormal{Res}}_{P}^{G}=\operatorname{For}_{L}^{P}\circ\pi^{\prime}_{*}\circ
i^{\prime!}\circ\operatorname{For}_{P}^{G}$
$\operatorname{Ind}_{P}^{G}=\operatorname{Av}_{P!}^{G}\circ
i^{\prime}_{!}\circ\pi^{\prime*}\circ\operatorname{Av}_{L!}^{P}$

These induction and restriction functors behave largely in the same way as the
induction and restriction functors discussed before. Namely, they satisfy
variants of Lemmas 4.10, 4.11, 4.12. The following lemma explains how
induction and restriction on the nilpotent cone can be computed via induction
and restriction of all of $\mathfrak{g}$.

###### Lemma 8.1

Let $A\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{E})$ and
$B\in\operatorname{Perv}_{L}(\mathfrak{l},\mathbb{E})$. The following
compatibilities between the nilpotent restriction (resp. induction) and
restriction (resp. induction) functors hold

  1. 1.

${\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\textnormal{Res}$\cr}}}}_{P}^{G}\circ
a_{\mathfrak{g}}^{*}\cong
a_{\mathfrak{l}}^{*}{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\underline{\textnormal{Res}}$\cr}}}}_{P}^{G}$,

  2. 2.

${\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\underline{\textnormal{Res}}$\cr}}}}_{P}^{G}\circ
a_{\mathfrak{g}!}\cong
a_{\mathfrak{l}!}{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\textnormal{Res}$\cr}}}}_{P}^{G}$,

  3. 3.

$\operatorname{Ind}_{P}^{G}\circ a_{\mathfrak{l}}^{*}\cong
a_{\mathfrak{g}}^{*}\underline{\textnormal{Ind}}_{P}^{G}$,

  4. 4.

$\underline{\textnormal{Ind}}_{P}^{G}\circ a_{\mathfrak{l}!}\cong
a_{\mathfrak{g}!}\operatorname{Ind}_{P}^{G}$.

###### Proof

Facts (2) and (4) can be found in [AHJR16] 2.9. Facts (1) and (3) are an easy
consequence of commutativity of the earlier diagram and proper base change. ■

We recall here the notion of a cuspidal sheaf as used in the modular
generalized Springer correspondence.

###### Definition 8.2

Let $\mathcal{C}\in\operatorname{Perv}_{G}(\mathcal{N},\mathbb{E})$. We call
$\mathcal{C}$ cuspidal if
${\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\textnormal{Res}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\textnormal{Res}$\cr}}}}_{P}^{G}\mathcal{C}=0$
for all proper parabolic subgroups $P\subset G$.

###### Remark 2

Observe the absence of a central factor condition in the definition of
cuspidal sheaves on the nilpotent cone. This is not surprising as
$\mathcal{N}_{\mathfrak{g}}=\mathcal{N}_{[\mathfrak{g},\mathfrak{g}]}$. In
particular, if $\mathcal{C}\in\operatorname{Perv}_{G}(\mathcal{N},\mathbb{E})$
is cuspidal, then by Lemma 8.1 (2), $a_{\mathfrak{g}!}\mathcal{C}$ is res-
small in $\mathfrak{g}$. Clearly $a_{\mathfrak{g}!}\mathcal{C}$ is supported
on $[\mathfrak{g},\mathfrak{g}]$, and hence is cuspidal. On the other hand, if
$\mathcal{C}^{\prime}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{E})$ is
cuspidal, then by Lemma 8.1 (1), $\mathcal{C}|_{\mathcal{N}}[-\dim
Z(\mathfrak{g})]$ is cuspidal. □

The structure of the modular generalized Springer correspondence ultimately is
determined by understanding the induction series. Before reviewing the modular
generalized Springer correspondence, we will review the definition of
induction series on the nilpotent cone as well as the generalization for the
Lie algebra setting.

###### Definition 8.3

A cuspidal datum for $\mathcal{N}_{\mathfrak{g}}$ is a triple
$(L,\mathcal{O},\mathcal{E})$ where $L\subseteq G$ is a Levi subgroup,
$\mathcal{O}$ is a distinguished $L$-orbit in $\mathcal{N}_{\mathfrak{l}}$,
and $\mathcal{E}\in\operatorname{Loc}_{L}(\mathcal{O})$ such that
$\operatorname{IC}(\mathcal{O},\mathcal{E})$ is a cuspidal sheaf on
$\mathfrak{l}$. Let $\mathfrak{C}_{\mathcal{N}_{\mathfrak{g}},\mathbb{E}}$ be
the set of all cuspidal datums on $\mathcal{N}_{\mathfrak{g}}$. We can define
the induction series
$\operatorname{Adm}_{(L,\mathcal{O},\mathcal{E})}(\mathcal{N}_{\mathfrak{g}},\mathbb{E})$
for a cuspidal datum $(L,\mathcal{O},\mathcal{E})$ as the full Serre
subcategory of
$\operatorname{Perv}_{G}(\mathcal{N}_{\mathfrak{g}},\mathbb{E})$ generated by
subobjects (or quotients) of
$\operatorname{Ind}_{L}^{G}\operatorname{IC}(\mathcal{O},\mathcal{E})$.

There is a $G$-action on
$\mathfrak{C}_{\mathcal{N}_{\mathfrak{g}},\mathbb{E}}$ given by

$g\cdot(L,\mathcal{O},\mathcal{E})=({}^{g}L,{}^{g}\mathcal{O},\operatorname{Ad}(g^{-1})^{*}\mathcal{E})$

Note that the induction series is invariant under this $G$-action (Corollary
2.2, [AHJR17a]).

###### Definition 8.4

A cuspidal datum for $\mathfrak{g}$ is a triple
$(L,\alpha,\mathcal{O},\mathcal{E})$ where $L\subseteq G$ is a Levi subgroup,
$\mathcal{O}\subseteq[\mathfrak{l},\mathfrak{l}]$ is a nilpotent $L$-orbit of
$\mathfrak{l}$, $\alpha\in Z(\mathfrak{l})$, and
$\mathcal{E}\in\operatorname{Loc}_{L}(\mathcal{O})$ such that
$\mathcal{L}_{\alpha}\boxtimes\operatorname{IC}(\mathcal{O},\mathcal{E})$ is a
cuspidal sheaf on $\mathfrak{l}$ (where
$\mathcal{L}_{\alpha}=\mathbb{T}_{Z(\mathfrak{l})}i_{\alpha*}\underline{\mathbb{E}}$
). Let $\mathfrak{C}_{\mathfrak{g},\mathbb{E}}$ be the set of all cuspidal
datums on $\mathfrak{g}$. We can define the induction series
$\operatorname{Adm}_{(L,\alpha,\mathcal{O},\mathcal{E})}(\mathfrak{g},\mathbb{E})$
for a cuspidal datum $(L,\alpha,\mathcal{O},\mathcal{E})$ as the full Serre
subcategory of $\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{E})$ generated by
subobjects (or quotients) of
$\underline{\textnormal{Ind}}_{L}^{G}(\mathcal{L}_{\alpha}\boxtimes\operatorname{IC}(\mathcal{O},\mathcal{E}))$.

Note that by Lemma 5.8 it is clear that every cuspidal sheaf on $\mathfrak{l}$
occurs in some cuspidal datum corresponding to $L$.

There is a $G$-action on $\mathfrak{C}_{\mathfrak{g},\mathbb{E}}$ given by

$g\cdot(L,\alpha,\mathcal{O},\mathcal{E})=({}^{g}L,{}^{g}\alpha,{}^{g}\mathcal{O},\operatorname{Ad}(g^{-1})^{*}\mathcal{E})$

Note that the induction series is invariant under this $G$-action by the same
argument of Corollary 2.2 in [AHJR17a]. It is clear that every irreducible
sheaf in an induction series is admissible.

Note that in both of the previous two definitions, there is seemingly a choice
between using subobjects or quotients. By Lemma 2.3 of [AHJR17a] for the
nilpotent case and Proposition 6.12 for the Lie algebra case, we can freely
interchange these notions.

###### Remark 3

Let $(L,\alpha,\mathcal{O},\mathcal{E})$ be a cuspidal datum for
$\mathfrak{g}$, and consider the cuspidal sheaf
$\mathcal{C}=\mathcal{L}_{\alpha}\boxtimes\operatorname{IC}(\mathcal{O},\mathcal{E})$
on $\mathfrak{l}$. Note that by Remark 2,
$\operatorname{IC}(\mathcal{O},\mathcal{E})$ is cuspidal on
$N_{\mathfrak{l}}$. In particular, we can associate to this cuspidal datum for
$\mathfrak{g}$ a corresponding cuspidal datum for $\mathcal{N}_{\mathfrak{g}}$
given by $(L,\mathcal{O},\mathcal{E})$. □

The modular generalized Springer correspondence can be thought of as a 3-step
process of increasingly more difficult statements:

  1. (S1)

There is a decomposition of the simple perverse sheaves in
$\operatorname{Perv}_{G}(\mathcal{N},\mathbb{E})$ into induction series:

$\operatorname{Irr}(\operatorname{Perv}_{G}(\mathcal{N},\mathbb{E}))=\bigcup_{(L,\mathcal{O},\mathcal{E})\in\mathfrak{C}_{\mathcal{N},\mathbb{E}}/G}\operatorname{Irr}(\operatorname{Adm}_{(L,\mathcal{O},\mathcal{E})}(\mathcal{N},\mathbb{E})).$

  2. (S2)

The union in (S1) is, in fact, disjoint.

  3. (S3)

There is a bijection between the irreducible sheaves in the induction series
and irreducible representations of the Weyl group of $L$:

$\operatorname{Irr}(\operatorname{Adm}_{L,\mathcal{O},\mathcal{E}}(\mathcal{N},\mathbb{E}))\cong\operatorname{Irr}(\mathbb{E}[N_{G}(L)/L]).$

We can rephrase (S1) into the language of admissible complexes as follows:
every simple perverse sheaf in
$\operatorname{Perv}_{G}(\mathcal{N},\mathbb{E})$ is an admissible complex.
This statement fails horrifically when directly translated to the entire Lie
algebra. For example, consider the skyscraper sheaf $\delta_{0}$ at $0$ in
$\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{E})$. Its Fourier transform is
the constant sheaf $\underline{\mathbb{E}}_{\mathfrak{g}^{\prime}}$ which is
not orbital; therefore, $\delta_{0}$ is not an admissible complex.
Nonetheless, if one instead replaces all perverse sheaves with admissible
complexes, the result does hold (Lemma 6.10 and Lemma 6.11).

After the refining of (S1), we can show that (S2) will hold as well. Both of
these results are stated in Theorem 8.10. The proof of the Lie algebra version
of (S2) will follow from a similar argument as the nilpotent cone case covered
in [AHJR17b]. Namely, one must leverage a version of Mackey’s formula in
positive characteristic. We provide a version of Mackey’s formula in Theorem
8.7.

There likely should be a variant for (S3) as well, although we do not yet have
a sensible replacement. If one looks at all of
$\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{E})$ with
$\textnormal{char}(\mathbb{E})=0$, there is a replacement given in [Gun18],
but it is not clear how to adapt these methods in our case.

###  8.1 Mackey’s Formula

We will establish a variation of Mackey’s formula which holds for character
sheaves on $\mathfrak{g}$. The proof will follow very closely from Theorem 2.2
of [AHJR17b] and Proposition 10.1.2 of [MS89].

Let $G$ be an algebraic group with normal subgroup $H\trianglelefteq G$. Let
$X$ be a $G$-variety such that $X$ is a principal $H$-variety. The quotient
equivalence states that for $\pi:X\to X/H$,
$\pi^{*}\circ\textnormal{Infl}_{G/H}^{G}:D_{G/H}(X/H,\mathbb{E})\to
D_{G}(X,\mathbb{E})$ is an equivalence of categories. We will write
$\pi_{\flat}$ for the inverse of this equivalence. The following lemma gives a
reformulation of the parabolic induction functor. The new formulation closely
models the definition of parabolic induction given in Lusztig [Lus85a].

######  Lemma 8.5 ([AHJR16], Lemma 2.14)

Consider the diagram

${\mathfrak{l}}$${G\times\mathfrak{p}}$${G\times^{P}\mathfrak{p}}$${\mathfrak{g}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

where $\alpha(g,x)=\pi_{P}(x)$ is the projection of $x$ onto $\mathfrak{l}$,
$\beta$ is the obvious quotient, and $\gamma$ is the map such that
$\gamma\circ\beta=\sigma$ is the action map $\sigma(g,x)={}^{g}x$. Then there
is a natural isomorphism of functors,

$\underline{\textnormal{Ind}}_{P}^{G}\cong\gamma_{!}\beta_{\flat}\alpha^{*}[2\dim
U_{P}]$

where $U_{P}$ is the unipotent radical of $P$.

We will first state a result about double cosets. We will use the notation
${}^{g}L=gLg^{-1}$.

######  Lemma 8.6 ([DM20], Lemma 5.6(i))

Let $P$, $Q$ be parabolic subgroups of $G$ with Levi factors $L$, $M$
respectively. Define

$\Sigma_{M,L}=\\{g\in G\mid M\cap{}^{g}L\textnormal{ contains a maximal torus
of }G\\}.$

Then $\Sigma_{M,L}$ is a union of finitely many $M\text{-}L$ double cosets and
there is a bijection

$M\setminus\Sigma_{M,L}/L\leftrightarrow Q\setminus G/P.$

Given parabolics $P,Q$ with Levis $L,M$, let $g_{1},g_{2},\ldots,g_{s}$ be a
set of representatives for the $M\text{-}L$ double cosets in $\Sigma_{M,L}$
ordered such that $Qg_{i}P\subseteq\overline{Qg_{j}P}$ when $i\leq j$. Note
that for $g_{i}\in\Sigma_{M,L}$, the group $M\cap{}^{g_{i}}L$ is a Levi
subgroup of both $M$ and ${}^{g_{i}}L$. Precisely, it is a Levi factor of the
parabolic subgroup of $M\cap{}^{g_{i}}P$ of $M$ and a Levi factor of the
parabolic subgroup $Q\cap{}^{g_{i}}L$ of ${}^{g_{i}}L$.

For a parabolic subgroup $P$ with Levi factor $L$, denote the projections
$\pi_{L\subset P}:P\to L$ and
$\pi_{\mathfrak{l}\subset\mathfrak{p}}:\mathfrak{p}\to\mathfrak{l}$.

We will state Mackey’s formula in the modular setting. The statement is
similar to the characteristic 0 case, but we cannot obtain a direct sum due to
the failure of the theory of weights.

###### Theorem 8.7

Let $L\subset P$ be a parabolic subgroup with Levi subgroup and $M\subset Q$
be another parabolic subgroup with Levi subgroup. The corresponding Lie
algebras will be $\mathfrak{l},\mathfrak{p},\mathfrak{m},$ and $\mathfrak{q}$
respectively. Let $g_{1},g_{2},\ldots,g_{s}$ be as above. Let
$\mathcal{A}\in\operatorname{Perv}_{L}(\mathfrak{l},\mathbb{E})$. Then in
$\operatorname{Perv}_{M}(\mathfrak{m},\mathbb{E})$, we have a filtration

${\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\underline{\textnormal{Res}}$\cr}}}}_{M\subset
Q}^{G}\underline{\textnormal{Ind}}_{L\subset
P}^{G}\mathcal{A}=\mathcal{A}_{0}\supset\mathcal{A}_{1}\supset
A_{2}\supset\ldots\supset A_{s}=0,$

in which the successive quotients have the form

$\mathcal{A}_{i-1}/\mathcal{A}_{i}\cong\underline{\textnormal{Ind}}_{M\cap^{g_{i}}L\subset
M\cap^{g_{i}}P}^{M}{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\underline{\textnormal{Res}}$\cr}}}}_{M\cap^{g_{i}}L\subset
Q\cap^{g_{i}}L}^{{}^{g_{i}}L}\left(\operatorname{Ad}(g_{i}^{-1})^{*}\mathcal{A}\right),\>\>\>\>\text{for
}i=1,\ldots,s.$

###### Proof

The role of equivariance in this proof is often subtle. To have this proof
mimic the one of Mars and Springer, we will frequently regard
$\operatorname{Perv}_{L}(\mathfrak{l},\mathbb{E})$ as a subcategory of
$D(\mathfrak{l},\mathbb{E})$. Throughout the proof, we will use the
reformulation of parabolic induction, as in Lemma 8.5.

Define a variety

$Z=\\{(g,x)\in G\times\mathfrak{p}\mid{}^{g}x\in\mathfrak{q}\\}$

We will define maps analogous to $\alpha,\beta,\gamma$ on $Z$. First, define
$a:Z\to\mathfrak{l}$ by $a(g,x)=\pi_{\mathfrak{l}\subset\mathfrak{p}}(x)$ and
$\tau:Z\to\mathfrak{m}$ by
$\tau(g,x)=\pi_{\mathfrak{m}\subset\mathfrak{q}}({}^{g}x)$. Just as with
$\sigma$, we can factor $\tau$ into maps

${Z}$${\mathfrak{m}}$${Z^{\prime}}$${\mathfrak{q}}$$\scriptstyle{b}$$\scriptstyle{\tau}$$\scriptstyle{\phi}$$\scriptstyle{d}$$\scriptstyle{\pi_{\mathfrak{q}\subset\mathfrak{m}}}$

where $Z^{\prime}=\\{(g,x)\in
G\times^{P}\mathfrak{p}\mid{}^{g}x\in\mathfrak{q}\\}$, and $d$ is the obvious
restriction of $\gamma$ onto $Z^{\prime}$. This gives the following
commutative diagram

${\mathfrak{m}}$${Z}$${Z^{\prime}}$${\mathfrak{q}}$${\mathfrak{l}}$${G\times\mathfrak{p}}$${G\times^{P}\mathfrak{p}}$${\mathfrak{g}}$$\scriptstyle{b}$$\scriptstyle{\tau}$$\scriptstyle{a}$$\scriptstyle{i_{Z}}$$\scriptstyle{\phi}$$\scriptstyle{d}$$\scriptstyle{i_{Z^{\prime}}}$$\scriptstyle{\pi_{\mathfrak{q}\subset\mathfrak{m}}}$$\scriptstyle{i_{\mathfrak{q}\subset\mathfrak{g}}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

We can now reduce computing the restriction of induction to computing sheaf
functors on $Z$ and its related varieties. In particular, we can compute

$\displaystyle{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\underline{\textnormal{Res}}$\cr}}}}_{\mathfrak{q}}^{\mathfrak{g}}\underline{\textnormal{Ind}}_{P}^{G}\mathcal{A}$
$\displaystyle=(\pi_{\mathfrak{q}\subset\mathfrak{m}})_{!}i_{\mathfrak{q}\subset\mathfrak{g}}^{*}\delta_{!}\beta_{\flat}\alpha^{*}\mathcal{A}[2\dim
U_{P}]$ (82)
$\displaystyle=(\pi_{\mathfrak{q}\subset\mathfrak{m}})_{!}d_{!}i_{Z^{\prime}}^{*}\beta_{\flat}\alpha^{*}\mathcal{A}[2\dim
U_{P}]$
$\displaystyle=\phi_{!}i_{Z^{\prime}}^{*}\beta_{\flat}\alpha^{*}\mathcal{A}[2\dim
U_{P}]$ $\displaystyle=\phi_{!}b_{\flat}i_{Z}^{*}\alpha^{*}\mathcal{A}[2\dim
U_{P}]$ $\displaystyle=\phi_{!}b_{\flat}a^{*}\mathcal{A}[2\dim U_{P}]$

As it stands, the variety $Z$ needs to be broken up so we can inductively
prove the theorem on the length of the filtration. We will first define closed
subvarieties of $G$,

$G_{i}=\bigcup_{j=1}^{i}Qg_{j}P$

for $i=0,\ldots,s$. Note that $G_{0}=\emptyset$ and $G_{s}=G$. We will now use
these to restrict $Z$. Define

$Z_{i}=\\{(g,x)\in Z\mid g\in G_{i}\\}=\\{(g,x)\in
G_{i}\times\mathfrak{p}\mid{}^{g}x\in\mathfrak{p}\\}$

Similarly, $Z_{0}=\emptyset$ and $Z_{s}=Z$. To obtain the piece of $Z$ only
consisting of the $g_{i}$ coset, we can define

$V_{i}=Z_{i}\setminus Z_{i-1}.$

We can create subsets $Z_{i}^{\prime}$ and $V_{i}^{\prime}$ in the same way
that $Z^{\prime}$ was defined. We similarly get factorization maps. We can
encode all this information in the following commutative diagrams

${\mathfrak{l}}$${Z_{1}}$${Z_{1}^{\prime}}$${\mathfrak{m}}$${\mathfrak{l}}$${Z_{2}}$${Z_{2}^{\prime}}$${\mathfrak{m}}$${\mathfrak{l}}$${\vdots}$${\vdots}$${\mathfrak{m}}$${\mathfrak{l}}$${Z_{s-1}}$${Z_{s-1}^{\prime}}$${\mathfrak{m}}$${\mathfrak{l}}$${Z}$${Z^{\prime}}$${\mathfrak{m}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{a_{1}}$$\scriptstyle{b_{1}}$$\scriptstyle{\tau_{1}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\iota_{1}^{\prime}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{a_{2}}$$\scriptstyle{b_{2}}$$\scriptstyle{\tau_{2}}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\phi_{2}}$$\scriptstyle{\iota_{2}^{\prime}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{a_{i}}$$\scriptstyle{b_{i}}$$\scriptstyle{\tau_{i}}$$\scriptstyle{\iota_{i}}$$\scriptstyle{\phi_{i}}$$\scriptstyle{\iota_{i}^{\prime}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{a_{s-1}}$$\scriptstyle{b_{s-1}}$$\scriptstyle{\tau_{s_{1}}}$$\scriptstyle{\iota_{s-1}}$$\scriptstyle{\phi_{s-1}}$$\scriptstyle{\iota_{s-1}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{\tau}$$\scriptstyle{\phi}$

and

${\mathfrak{l}}$${V_{i}}$${V_{i}^{\prime}}$${\mathfrak{m}}$${Z_{i}}$${Z_{i}^{\prime}}$$\scriptstyle{j}$$\scriptstyle{\operatorname{b}_{i}}$$\scriptstyle{\operatorname{a}_{i}}$$\scriptstyle{j^{\prime}}$$\scriptstyle{\varphi_{i}}$$\scriptstyle{a_{i}}$$\scriptstyle{b_{i}}$$\scriptstyle{\phi_{i}}$

To make the notation a little cleaner for the next part, let $\iota:Z_{i-1}\to
Z_{i}$ be the closed embedding normally denoted $\iota_{i-1}$ and $j:V_{i}\to
Z_{i}$ be the open embedding. We also denote
$\iota^{\prime}=\iota_{i-1}^{\prime}$. We then have an open-closed
distinguished triangle associated with these embeddings

${{j_{!}j^{*}a_{i}^{*}\mathcal{A}[2\dim U_{P}]}}$${{a_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{\iota_{!}\iota^{*}a_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{j_{!}\operatorname{a}_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{a_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{\iota_{!}a_{i-1}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(b_{i})_{\flat}j_{!}\operatorname{a}_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(b_{i})_{\flat}a_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(b_{i})_{\flat}\iota_{!}a_{i-1}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{j_{!}^{\prime}(\operatorname{b}_{i})_{\flat}\operatorname{a}_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(b_{i})_{\flat}a_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{\iota^{\prime}_{!}(b_{i-1})_{\flat}a_{i-1}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(\phi_{i})_{!}j_{!}^{\prime}(\operatorname{b}_{i})_{\flat}\operatorname{a}_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(\phi_{i})_{!}(b_{i})_{\flat}a_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(\phi_{i})_{!}\iota_{!}^{\prime}(b_{i-1})_{\flat}a_{i-1}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(\varphi_{i})_{!}(\operatorname{b}_{i})_{\flat}\operatorname{a}_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(\phi_{i})_{!}(b_{i})_{\flat}a_{i}^{*}\mathcal{A}[2\dim
U_{P}]}}$${{(\phi_{i-1})_{!}(b_{i-1})_{\flat}a_{i-1}^{*}\mathcal{A}[2\dim
U_{P}]}}$

The following lemma will be crucial to the proof and can be thought of as the
inductive step of our proof. We will delay the proof until later.

###### Lemma 8.8

In the same setup as the proof of the theorem,

$(\varphi_{i})_{!}(\operatorname{b}_{i})_{\flat}\operatorname{a}_{i}^{*}\mathcal{A}[2\dim
U_{P}]\cong\underline{\textnormal{Ind}}_{M\cap^{g_{i}}L\subset
M\cap^{g_{i}}P}^{M}{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\underline{\textnormal{Res}}$\cr}}}}_{M\cap^{g_{i}}L\subset
Q\cap^{g_{i}}L}^{{}^{g_{i}}L}(\operatorname{Ad}(g_{i}^{-1})^{*}\mathcal{A})$

We will first perform an induction to show that
$\mathcal{C}_{i}=(\phi_{i})_{!}(b_{i})_{\flat}a_{i}^{*}\mathcal{A}[2\dim
U_{P}]$ is perverse for all $i$. We will also denote
$\mathcal{B}_{i}=(\varphi_{i})_{!}(\operatorname{b}_{i})_{\flat}\operatorname{a}^{*}\mathcal{A}[2\dim
U_{P}]$. By Lemma 8.8, $\mathcal{B}_{i}$ is perverse for all $i$. The base
case is easy as the distinguished triangle becomes the exact sequence,

$0\to\mathcal{B}_{1}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{C}_{1}\to 0$

and hence $\mathcal{C}_{1}$ is perverse. Now assume that $\mathcal{C}_{i-1}$
is perverse. Then the distinguished triangle is

$\mathcal{B}_{i}\to\mathcal{C}_{i}\to\mathcal{C}_{i-1}\to$

Since perverse sheaves are closed under extensions, we have that
$\mathcal{C}_{i}$ is perverse. In particular, the distinguished triangle
becomes a short exact sequence of perverse sheaves

$0\to\mathcal{B}_{i}\to\mathcal{C}_{i}\to\mathcal{C}_{i-1}\to 0.$

It remains to prove the actual substance of the theorem. We will do this by
inducting on the length of the filtration. For the base case of $i=1$, the
theorem obviously holds since $\mathcal{C}_{1}\cong\mathcal{B}_{1}$ by the
above short exact sequence. Now assume that $\mathcal{C}_{i-1}$ has a
descending filtration of length $i-1$ with the same successive quotients as
the theorem. By (Proof), we can extend this to $\mathcal{C}$. Hence by
induction, the result holds for all $i$. We can then take $i=s$ and then we
are done by Lemma 8.8. ■

Proof of Lemma 8.8. Fix $i=1,\ldots,s$. We can perform a short computation for
$V_{i}$,

$\displaystyle V_{i}$ $\displaystyle=\\{(g,x)\in G_{i}\setminus
G_{i-1}\times\mathfrak{p}\mid{}^{g}x\in\mathfrak{q}\\}$
$\displaystyle=\left\\{(g,x)\in\left(\bigcup_{j=1}^{i}Qg_{j}P\setminus\bigcup_{j=1}^{i-1}Qg_{j}P\right)\times\mathfrak{p}\>\bigg{|}\>{}^{g}x\in\mathfrak{q}\right\\}$
$\displaystyle=\\{(g,x)\in Qg_{i}P\mid{}^{g}x\in\mathfrak{q}\\}.$

Define $Y_{i}=Q\times P\times(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})$ which
is a $Q\cap{}^{g_{i}}P$-variety via the action,

$g\cdot(q,p,y)=(qg^{-1},pg_{i}^{-1}g^{-1}g_{i},{}^{g}y)$

for $g\in Q\cap{}^{g_{i}}P$, $(q,p,y)\in Y_{i}$. In fact, $Y_{i}$ is a
principal $Q\cap{}^{g_{i}}P$-bundle over $V_{i}$ via the map

$\sigma:Y_{i}\to V_{i}\hskip
28.45274pt(q,p,y)\mapsto(qg_{i}p^{-1},{}^{pg_{i}^{-1}}y).$

We can define variations of $\varphi_{i}$, $\operatorname{b}_{i}$ and
$\operatorname{a}_{i}$ for $Y_{i}$. Namely, define maps

$\tilde{a}:Y_{i}\to\mathfrak{l}\hskip
28.45274pt(q,p,y)\mapsto\pi_{\mathfrak{l}\subset\mathfrak{p}}({}^{pg_{i}^{-1}}y),$
$\tilde{\tau}:Y_{i}\to\mathfrak{m}\hskip
28.45274pt(q,p,y)\mapsto\pi_{\mathfrak{m}\subset\mathfrak{q}}({}^{q}y).$

We can factor $\tilde{\tau}=\tilde{\varphi}\circ\tilde{b}$ where
$\tilde{b}:Y_{i}\to
Q\times^{Q\cap{}^{g_{i}}P}(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})$ is the
projection map. We summarize all these maps in the following commutative
diagram,

${\mathfrak{l}}$${Y_{i}}$${Q\times^{Q\cap{}^{g_{i}}P}(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})}$${\mathfrak{m}}$${\mathfrak{l}}$${V_{i}}$${V_{i}^{\prime}}$${\mathfrak{m}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\tilde{a}}$$\scriptstyle{\tilde{\tau}}$$\scriptstyle{\tilde{b}}$$\scriptstyle{\sigma}$$\scriptstyle{\tilde{\varphi}}$$\scriptstyle{\sigma^{\prime}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{a}_{i}}$$\scriptstyle{\operatorname{b}_{i}}$$\scriptstyle{\varphi_{i}}$

As we have done before, we can find an isomorphism of sheaves,

$(\varphi_{i})_{!}(\operatorname{b}_{i})_{\flat}\operatorname{a}_{i}^{*}\mathcal{A}[2\dim
U_{P}]\cong\tilde{\varphi}_{!}\tilde{b}_{\flat}\tilde{a}^{*}\mathcal{A}[2\dim
U_{P}]$ (83)

Define a morphism

$q_{L}:Y_{i}\to L\times\mathfrak{l}\hskip
28.45274pt(q,p,y)\mapsto(\pi_{L\subset
P}(p),\pi_{\mathfrak{l}\subset\mathfrak{p}}({}^{g_{i}^{-1}}y))$

It is easy to see that $\tilde{\alpha}$ factors as $a_{\mathfrak{l}}\circ
q_{L}$ where $a_{\mathfrak{l}}$ is the action map. We can then consider the
diagram,

${Y_{i}}$${L\times\mathfrak{l}}$${\mathfrak{l}}$$\scriptstyle{q_{L}}$$\scriptstyle{\tilde{a}}$$\scriptstyle{\tilde{a}^{\prime}}$$\scriptstyle{a_{\mathfrak{l}}}$$\scriptstyle{\operatorname{pr}_{2}}$

where $\tilde{a}^{\prime}$ is the composition $\operatorname{pr}_{2}\circ
q_{L}$. It is clear that both the top and bottom triangles of the diagram
commute. Since $\mathcal{A}$ is $L$-equivariant, we have an isomorphism
$a_{\mathfrak{l}}^{*}\mathcal{A}\cong\operatorname{pr}_{2}^{*}\mathcal{A}$.
Therefore, we get an isomorphism
$\tilde{a}^{*}\mathcal{A}\cong(\tilde{a}^{\prime})^{*}\mathcal{A}$. The main
benefit now is that $\tilde{a}^{\prime}$ is independent of the $P$-factor.
Therefore, we can factor $\tilde{a}^{\prime}$ through the quotient $Y_{i}\to
Q\times(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})$ to get a map
$\hat{a}:Q\times(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})\to\mathfrak{l}$.
Likewise, we can factor $\tilde{\tau}$ through the quotient $Y_{i}\to
Q\times(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})$ to get a map
$\hat{\tau}:Q\times(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})\to\mathfrak{m}$.
We can further factor $\hat{\tau}$ as $\tilde{\varphi}\hat{b}$ where
$\hat{b}:Q\times(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})\to
Q\times^{Q\cap{}^{g_{i}}P}(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})$ is the
usual quotient map. All of these maps can be encoded in the commutative
diagram:

${\mathfrak{l}}$${Y_{i}}$${Q\times^{Q\cap{}^{g_{i}}P}(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})}$${\mathfrak{m}}$${\mathfrak{l}}$${Q\times(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})}$${Q\times^{Q\cap{}^{g_{i}}P}(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})}$${\mathfrak{m}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\tilde{b}}$$\scriptstyle{\tilde{a}}$$\scriptstyle{\tilde{\tau}}$$\scriptstyle{\tilde{\varphi}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\hat{b}}$$\scriptstyle{\hat{a}}$$\scriptstyle{\hat{\tau}}$$\scriptstyle{\tilde{\varphi}}$

By basic sheaf functor relations, we get an isomorphism of sheaves,

$\tilde{\varphi}_{!}\tilde{b}_{\flat}\tilde{a}^{*}\mathcal{A}\cong\tilde{\varphi}_{!}\hat{b}_{\flat}\hat{a}^{*}\mathcal{A}$
(84)

Define

$Y_{i}^{\prime}=(\mathfrak{m}\cap{}^{g^{i}}\mathfrak{l})\oplus(\mathfrak{m}\cap{}^{g^{i}}\mathfrak{u}_{P})\oplus(\mathfrak{u}_{Q}\cap{}^{g^{i}}\mathfrak{l})$

which comes equipped with an obvious projection
$\rho:\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p}\to Y_{i}^{\prime}$. This
projection map is $(Q\cap{}^{g_{i}}P)$-equivariant. We also get an induced
vector bundle projection
$\operatorname{id}\times\rho:Q\times(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})\to
Q\times Y_{i}^{\prime}$. Factor $\hat{a}$ and $\hat{\tau}$ through
$(\operatorname{id}\times\rho)$ to get maps
$\hat{a}=\check{a}(\operatorname{id}\times\rho)$ and
$\hat{\tau}=\check{\tau}(\operatorname{id}\times\rho)$. We can further factor
$\check{\tau}$ as $\check{\varphi}\check{b}$ where $\check{b}:Q\times
Y_{i}^{\prime}\to Q\times^{Q\cap{}^{g_{i}}P}Y_{i}^{\prime}$ is the obvious
projection. All of these maps can be encoded in the commutative diagram:

${\mathfrak{l}}$${Q\times(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})}$${Q\times^{Q\cap{}^{g_{i}}P}(\mathfrak{q}\cap{}^{g_{i}}\mathfrak{p})}$${\mathfrak{m}}$${\mathfrak{l}}$${Q\times
Y_{i}^{\prime}}$${Q\times^{Q\cap{}^{g_{i}}P}Y_{i}^{\prime}}$${\mathfrak{m}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\hat{b}}$$\scriptstyle{\hat{a}}$$\scriptstyle{\hat{\tau}}$$\scriptstyle{\operatorname{id}\times\rho}$$\scriptstyle{\tilde{\varphi}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\check{b}}$$\scriptstyle{\check{\tau}}$$\scriptstyle{\check{a}}$$\scriptstyle{\check{\varphi}}$

We then obtain an isomorphism of sheaves,

$\tilde{\varphi}_{!}\hat{b}_{\flat}\hat{a}^{*}\mathcal{A}\cong\check{\varphi}_{!}\check{b}_{\flat}\check{a}^{*}\mathcal{A}[-2\dim(U_{Q}\cap{}^{g_{i}}U_{P})].$
(85)

We could have instead factored $\check{\tau}$ as
$\check{\varphi}^{\prime}\check{b}^{\prime}$ where $\check{b}^{\prime}:Q\times
Y_{i}^{\prime}\to Q\times^{M\cap{}^{g_{i}}P}Y_{i}^{\prime}$. Note that
$U_{Q}\cap{}^{g_{i}}P$ is affine, so we get an isomorphism

$\check{\varphi}_{!}\check{b}_{\flat}\check{a}^{*}\mathcal{A}\cong\check{\varphi}_{!}^{\prime}\check{b}_{\flat}^{\prime}\check{a}^{*}\mathcal{A}[2\dim(U_{Q}\cap{}^{g_{i}}U_{P})].$
(86)

Consider the affine bundle $p_{M\subset Q}\times\operatorname{id}:Q\times
Y_{i}^{\prime}\to M\times Y_{i}^{\prime}$. It is not hard to check that both
$\check{a}$ and $\check{\tau}$ both factor through $p_{M\subset
Q}\times\operatorname{id}$. We then get factorizations
$\check{a}=\overline{a}(p_{M\subset Q}\times\operatorname{id})$ and
$\check{\tau}=\overline{\tau}(p_{M\subset Q}\times\operatorname{id})$. Again,
we can further factor $\overline{\tau}$ as
$\overline{\tau}=\overline{\varphi}\overline{b}$ where $\overline{b}:M\times
Y_{i}^{\prime}\to M\times^{M\cap{}^{g_{i}}P}Y_{i}^{\prime}$ is the quotient
map. These maps fit into the commutative diagram,

${\mathfrak{l}}$${Q\times
Y_{i}^{\prime}}$${Q\times^{M\cap{}^{g_{i}}P}Y_{i}^{\prime}}$${\mathfrak{m}}$${\mathfrak{l}}$${M\times
Y_{i}^{\prime}}$${M\times^{M\cap{}^{g_{i}}P}Y_{i}^{\prime}}$${\mathfrak{m}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\check{b}^{\prime}}$$\scriptstyle{\check{\tau}}$$\scriptstyle{\check{a}}$$\scriptstyle{p_{M\subset
Q}\times\operatorname{id}}$$\scriptstyle{\check{\varphi}^{\prime}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\overline{a}}$$\scriptstyle{\overline{b}}$$\scriptstyle{\overline{\tau}}$$\scriptstyle{\overline{\varphi}}$

By another series of basic sheaf functor relations, we obtain an isomorphism,

$\check{\varphi}_{!}^{\prime}\check{b}_{\flat}^{\prime}\check{a}^{*}\mathcal{A}\cong\overline{\varphi}_{!}\overline{b}_{\flat}\overline{a}^{*}\mathcal{A}[-2\dim
U_{Q}].$ (87)

Finally, we will compare the right hand side of (87) with the definition of
$\underline{\textnormal{Ind}}_{M\cap^{g_{i}}L\subset
M\cap^{g_{i}}P}^{M}{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\underline{\textnormal{Res}}$\cr}}}}_{M\cap^{g_{i}}L\subset
Q\cap^{g_{i}}L}^{{}^{g_{i}}L}(\operatorname{Ad}(g_{i}^{-1})^{*}\mathcal{A})$.
Namely, we can fit all of the relevant information into the following
commutative diagram,

${\mathfrak{l}}$${{}^{g_{i}}\mathfrak{l}}$${\mathfrak{q}\cap{}^{g_{i}}\mathfrak{l}}$${\mathfrak{m}\cap^{g_{i}}\mathfrak{l}}$${M\times(\mathfrak{m}\cap{}^{g_{i}}\mathfrak{p})}$${M\times^{M\cap{}^{g_{i}}P}(\mathfrak{m}\cap{}^{g_{i}}\mathfrak{p})}$${\mathfrak{m}}$${Y_{i}^{\prime}}$${\mathfrak{m}\cap{}^{g_{i}}\mathfrak{p}}$${\mathfrak{l}}$${M\times
Y_{i}^{\prime}}$${M\times^{M\cap{}^{g_{i}}P}Y_{i}^{\prime}}$${\mathfrak{m}}$$\scriptstyle{\operatorname{Ad}(g_{i}^{-1})}$$\scriptstyle{i}$$\scriptstyle{\pi}$$\scriptstyle{\overline{\alpha}}$$\scriptstyle{\overline{\beta}}$$\scriptstyle{\overline{\delta}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\overline{p}^{\prime}}$$\scriptstyle{\overline{p}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\overline{b}}$$\scriptstyle{\overline{a}}$$\scriptstyle{p_{1}}$$\scriptstyle{\overline{\varphi}}$$\scriptstyle{p_{1}^{\prime}}$$\scriptstyle{\operatorname{id}}$

First, note that by definition,

$\underline{\textnormal{Ind}}_{M\cap^{g_{i}}L\subset
M\cap^{g_{i}}P}^{M}{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\underline{\textnormal{Res}}$\cr}}}}_{M\cap^{g_{i}}L\subset
Q\cap^{g_{i}}L}^{{}^{g_{i}}L}(\operatorname{Ad}(g_{i}^{-1})^{*}\mathcal{A})=\overline{\delta}_{!}\overline{\beta}_{\flat}\overline{\alpha}^{*}\pi_{!}i^{*}\operatorname{Ad}(g_{i}^{-1})^{*}\mathcal{A}$

By the same arguments we have performed throughout this proof, we can reduce
to showing that

$\overline{a}^{*}=p_{1}^{*}\overline{\alpha}^{*}\pi_{!}i^{*}\operatorname{Ad}(g_{i}^{-1})^{*}$

which follows from a combination of proper base change of the middle square
and the fact that $\pi^{\prime}$ is an affine bundle. Therefore, we get an
isomorphism of sheaves,

$\underline{\textnormal{Ind}}_{M\cap^{g_{i}}L\subset
M\cap^{g_{i}}P}^{M}{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\displaystyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-4.30554pt\cr$\textstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\scriptstyle\underline{\textnormal{Res}}$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.15277pt\cr$\scriptscriptstyle\underline{\textnormal{Res}}$\cr}}}}_{M\cap^{g_{i}}L\subset
Q\cap^{g_{i}}L}^{{}^{g_{i}}L}(\operatorname{Ad}(g_{i}^{-1})^{*}\mathcal{A})\cong\overline{\varphi}_{!}\overline{b}_{\flat}\overline{a}^{*}\mathcal{A}[2\dim(M\cap{}^{g_{i}}P)].$
(88)

We can then combine the myriad of isomorphisms from (83), (84), (85), (86),
(87), (88). It only remains to check that the shifts accumulated throughout
cancel out. I.e., we must check that

$\dim
U_{P}-\dim(U_{Q}\cap{}^{g_{i}}U_{P})+\dim(U_{Q}\cap{}^{g_{i}}P)-\dim(U_{Q})=\dim(M\cap{}^{g_{i}}U_{P}).$
(89)

Note that (89) follows from the proof of Proposition 10.1.2 of [MS89].
$\square$

###  8.2 Decomposition of Admissible Sheaves

To simplify notation, for a cuspidal datum
$(L,\alpha,\mathcal{O},\mathcal{E})$ for $\mathfrak{g}$, we will write
$\mathcal{I}_{L,\mathcal{O}}^{\alpha,\mathcal{E}}=\mathcal{L}_{\alpha}\boxtimes\operatorname{IC}(\mathcal{O},\mathcal{E})$.

###### Lemma 8.9

Let $P_{1},P_{2}$ be Levi subgroups of $G$ with Levi radicals $L_{1},L_{2}$.
Let $(L_{1},\alpha_{1},\mathcal{O}_{1},\mathcal{E}_{1})$ and
$(L_{2},\alpha_{2},\mathcal{O}_{2},\mathcal{E}_{2})$ be two cuspidal datums
for $\mathfrak{g}$. If
$\mathcal{A}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{E})$ belongs to
the induction series of both
$(L_{1},\alpha_{1},\mathcal{O}_{1},\mathcal{E}_{1})$ and
$(L_{2},\alpha_{2},\mathcal{O}_{2},\mathcal{E}_{2})$, then

$\operatorname{Hom}\left(\underline{\textnormal{Ind}}_{P_{1}}^{G}\mathcal{I}_{L_{1},\mathcal{O}_{1}}^{\alpha_{1},\mathcal{E}_{1}},\underline{\textnormal{Ind}}_{P_{2}}^{G}\mathcal{I}_{L_{2},\mathcal{O}_{2}}^{\alpha_{2},\mathcal{E}_{2}}\right)\neq
0$

###### Proof

By definition, $\mathcal{A}$ is a subobject of
$\underline{\textnormal{Ind}}_{P_{2}}^{G}\mathcal{I}_{L_{2},\mathcal{O}_{2}}^{\alpha_{2},\mathcal{E}_{2}}$
and a quotient of
$\underline{\textnormal{Ind}}_{P_{1}}^{G}\mathcal{I}_{L_{1},\mathcal{O}_{1}}^{\alpha_{1},\mathcal{E}_{1}}$.
We then have the composition of maps

$\underline{\textnormal{Ind}}_{P_{1}}^{G}\mathcal{I}_{L_{1},\mathcal{O}_{1}}^{\alpha_{1},\mathcal{E}_{1}}\twoheadrightarrow\mathcal{A}\hookrightarrow\underline{\textnormal{Ind}}_{P_{2}}^{G}\mathcal{I}_{L_{2},\mathcal{O}_{2}}^{\alpha_{2},\mathcal{E}_{2}}$

In particular, the composition is nonzero, so the hom-space is nonzero. ■

###### Theorem 8.10

There is a decomposition of the admissible sheaves into induction series,

$\operatorname{Irr}(\operatorname{Adm}_{G}(\mathfrak{g},\mathbb{E}))=\bigsqcup_{(L,\alpha,\mathcal{O},\mathcal{E})\in\mathfrak{C}_{\mathfrak{g},\mathbb{E}}/G}\operatorname{Irr}(\operatorname{Adm}_{(L,\alpha,\mathcal{O},\mathcal{E})}(\mathfrak{g},\mathbb{E})).$

###### Proof

By Lemma 6.10 and Lemma 6.11, we have that every admissible sheaf occurs in
some induction series. As a result, it only remains to show that this union is
disjoint.

Suppose $(L_{1},\alpha_{1},\mathcal{O}_{1},\mathcal{E}_{1})$ and
$(L_{2},\alpha_{2},\mathcal{O}_{2},\mathcal{E}_{2})$ are cuspidal datums for
$\mathfrak{g}$ which are not in the same $G$-orbit. By adjunction,

$\operatorname{Hom}\left(\underline{\textnormal{Ind}}_{P_{1}}^{G}\mathcal{I}_{L_{1},\mathcal{O}_{1}}^{\alpha_{1},\mathcal{E}_{1}},\underline{\textnormal{Ind}}_{P_{2}}^{G}\mathcal{I}_{L_{2},\mathcal{O}_{2}}^{\alpha_{2},\mathcal{E}_{2}}\right)=\operatorname{Hom}\left(\mathcal{I}_{L_{1},\mathcal{O}_{1}}^{\alpha_{1},\mathcal{E}_{1}},\vec{\underline{\textnormal{Res}}}_{P_{1}}^{G}\underline{\textnormal{Ind}}_{P_{2}}^{G}\mathcal{I}_{L_{2},\mathcal{O}_{2}}^{\alpha_{2},\mathcal{E}_{2}}\right)$

By Mackey’s theorem, we have a filtration

$\vec{\underline{\textnormal{Res}}}_{L_{1}\subset
P_{1}}^{G}\underline{\textnormal{Ind}}_{L_{2}\subset
P_{2}}^{G}\mathcal{I}_{L_{2},\mathcal{O}_{2}}^{\alpha_{2},\mathcal{E}_{2}}=\mathcal{A}_{0}\supset\mathcal{A}_{1}\supset
A_{2}\supset\ldots\supset A_{s}=0,$

in which the successive quotients have the form

$\mathcal{A}_{i-1}/\mathcal{A}_{i}\cong\underline{\textnormal{Ind}}_{L_{1}\cap^{g}L_{2}\subset
L_{1}\cap^{g}P_{2}g^{-1}}^{M}\vec{\underline{\textnormal{Res}}}_{L_{1}\cap^{g}L_{2}\subset
P_{1}\cap^{g}L_{2}}^{{}^{g}L_{2}}\left(\operatorname{Ad}(g^{-1})^{*}\mathcal{I}_{L_{2},\mathcal{O}_{2}}^{\alpha_{2},\mathcal{E}_{2}}\right)$

Since $\operatorname{Ad}(g^{-1})$ is a smooth morphism,
$\operatorname{Ad}(g^{-1})^{*}\mathcal{I}_{L_{2},\mathcal{O}_{2}}^{\alpha_{2},\mathcal{E}_{2}}$
is also cuspidal. Hence, $\mathcal{A}_{i-1}/\mathcal{A}_{i}$ must be zero.
Since all of the composition factors are zero, we must have that
$\vec{\underline{\textnormal{Res}}}_{L_{1}\subset
P_{1}}^{G}\underline{\textnormal{Ind}}_{L_{2}\subset
P_{2}}^{G}\mathcal{I}_{L_{2},\mathcal{O}_{2}}^{\alpha_{2},\mathcal{E}_{2}}=0$.
But by Lemma 8.9, this can only happen if the induction series are disjoint. ■

##  9 Modular Reduction

For an algebraic variety $X$, consider the extension by scalars functor
$\mathbb{K}(-)=\mathbb{K}\otimes_{\mathbb{O}}(-):D(X,\mathbb{O})\to
D(X,\mathbb{K})$. We can use $\mathbb{K}(-)$ to identify the objects of
$D(X,\mathbb{K})$ with the objects of $D(X,\mathbb{O})$. We can compute the
hom-spaces of $D(X,\mathbb{K})$ from $D(X,\mathbb{O})$ using the formula

$\operatorname{Hom}_{D(X,\mathbb{K})}(A,B)=\mathbb{K}\otimes_{\mathbb{O}}\operatorname{Hom}_{D(X,\mathbb{O})}(A,B)$

under this identification of objects. Moreover, a similar relationship holds
when restricting to the heart of the perverse $t$-structure since
$\mathbb{K}(-)$ is $t$-exact.

On the other hand, we have a restriction of scalars functor
$\mathbb{F}(-)=\mathbb{F}\otimes_{\mathbb{O}}^{L}(-)$ induced by the quotient
$\mathbb{O}\twoheadrightarrow\mathbb{F}$.

We have an induced map on Grothendieck groups, called the modular reduction
map

$d:K_{0}(\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{K}))\to
K_{0}(\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{F}))$

For $\mathcal{A}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{K})$, let
$\mathcal{A}_{\mathbb{O}}$ be the torsion-free sheaf in
$\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{O})$ such that
$\mathcal{A}=\mathbb{K}(\mathcal{A}_{\mathbb{O}})$. Then
$d([\mathcal{A}])=[\mathbb{F}(\mathcal{A}_{\mathbb{O}})]$. We say that
$\mathcal{A}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{F})$ belongs to
the modular reduction of
$\mathcal{B}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{K})$ if
$[\mathcal{A}]$ appears with non-zero multiplicity in $d([\mathcal{B}])$.

We end with a short proposition giving a tool for finding cuspidal, character,
and orbital sheaves in the modular setting from the characteristic 0 setting.

###### Proposition 9.1

[AHJR16] Let $\mathcal{G}$ be a simple object in
$\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{F})$.

  1. 1.

If $\mathcal{G}$ occurs in the modular reduction of a cuspidal sheaf
$\mathcal{F}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{K})$. Then
$\mathcal{G}$ is cuspidal.

  2. 2.

If $\mathcal{G}$ occurs in the modular reduction of a character sheaf
$\mathcal{F}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{K})$. Then
$\mathcal{G}$ is a character sheaf.

  3. 3.

If $\mathcal{G}$ occurs in the modular reduction of an orbital sheaf
$\mathcal{F}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{K})$. Then
$\mathcal{G}$ is an orbital sheaf.

###### Proof

(1):  Let $\mathcal{G}$ be a simple perverse sheaf in
$\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{F})$ occurring in the modular
reduction of a cuspidal sheaf
$\mathcal{F}\in\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{K})$. In
particular, let $\mathcal{F}_{\mathbb{O}}$ be a torsion-free sheaf in
$\operatorname{Perv}_{G}(\mathfrak{g},\mathbb{O})$ such that
$\mathcal{F}\cong\mathbb{K}\otimes_{\mathbb{O}}\mathcal{F}_{\mathbb{O}}$. We
have that $\mathcal{G}$ is a composition factor of
$\mathbb{F}\otimes_{\mathbb{O}}^{L}\mathcal{F}_{\mathbb{O}}$. By performing
the obvious modifications to Proposition 2.22 of [AHJR16], one obtains that if
$\mathcal{G}$ is res-small. Let
$\mathcal{F}\cong\mathcal{L}\boxtimes\mathcal{B}$ where $\mathcal{L}$ is a
character sheaf on $Z(\mathfrak{g})$. Since, $\mathbb{K}(-)$ gives a bijection
at the level of objects, we must have that
$\mathcal{F}_{\mathbb{O}}\cong\mathcal{L}_{\mathbb{O}}\boxtimes\mathcal{B}_{\mathbb{O}}$
for torsion-free sheaves
$\mathcal{L}_{\mathbb{O}}\in\operatorname{Perv}_{G}(Z(\mathfrak{g}),\mathbb{O})$
and
$\mathcal{B}_{\mathbb{O}}\in\operatorname{Perv}_{G}([\mathfrak{g},\mathfrak{g}],\mathbb{O})$.
The Fourier transform commutes with extension of scalars, hence
$(\mathbb{T}(\mathcal{L}_{\mathbb{O}})\cong\mathbb{T}(\mathcal{L}))_{\mathbb{O}}$.
To check if $\mathcal{L}_{\mathbb{O}}$ is a character sheaf, one only needs to
compute that $\mathbb{T}(\mathcal{L}_{\mathbb{O}})$ is supported at a point.
Since
$\operatorname{supp}((\mathbb{T}(\mathcal{L}))_{\mathbb{O}})=\operatorname{supp}(\mathbb{T}(\mathcal{L}))$,
we have shown that $\mathcal{L}_{\mathbb{O}}$ is a character sheaf on
$Z(\mathfrak{g})$. Since $\boxtimes$ commutes with
$\mathbb{F}\otimes_{\mathbb{O}}^{L}(-)$, we get a $\boxtimes$-decomposition of
$\mathbb{F}\otimes_{\mathbb{O}}^{L}\mathcal{F}_{\mathbb{O}}$,

$\mathbb{F}\otimes_{\mathbb{O}}^{L}\mathcal{F}_{\mathbb{O}}\cong(\mathbb{F}\otimes_{\mathbb{O}}^{L}\mathcal{L}_{\mathbb{O}})\boxtimes(\mathbb{F}\otimes_{\mathbb{O}}\mathcal{B}_{\mathbb{O}})$

Let
$\mathcal{L}_{\mathbb{F}}=\mathbb{F}\otimes_{\mathbb{O}}^{L}\mathcal{L}_{\mathbb{O}}$.
It remains to check that is a character sheaf, but this follows from the fact
that extension of scalars commutes with pullbacks:

$+^{*}\mathcal{L}_{\mathbb{F}}\cong\mathbb{F}\otimes_{\mathbb{O}}^{L}+^{*}\mathcal{L}_{\mathbb{O}}\cong\mathbb{F}\otimes_{\mathbb{O}}^{L}\mathcal{L}_{\mathbb{O}}\cong\mathcal{L}_{F}$

(2):  By Remark 2.23 of [AHJR16], one has that induction commutes with modular
reduction. The claim then follows from Theorem 6.2 and (1).

(3):  The Fourier transform is an equivalence of categories and commutes with
scalars, and hence, commutes with modular reduction. The claim then follows by
taking the Fourier transform of (2). ■

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