# A modular architecture for creating multimodal agents

Thomas Baier
Faculty of Humanities
Vrije Universiteit Amsterdam
Amsterdam, The Netherlands
<EMAIL_ADDRESS>
& Selene Baez Santamaria
Faculty of Humanities
Vrije Universiteit Amsterdam
Amsterdam, The Netherlands
<EMAIL_ADDRESS>
& Piek Vossen
Faculty of Humanities
Vrije Universiteit Amsterdam
Amsterdam, The Netherlands
<EMAIL_ADDRESS>

###### Abstract

The paper describes a flexible and modular platform to create multimodal
interactive agents. The platform operates through an event-bus on which
signals and interpretations are posted in a sequence in time. Different
sensors and interpretation components can be integrated by defining their
input and output as topics, which results in a logical workflow for further
interpretations. We explain a broad range of components that have been
developed so far and integrated into a range of interactive agents. We also
explain how the actual interaction is recorded as multimodal data as well as
in a so-called episodic Knowledge Graph. By analysing the recorded
interaction, we can analyse and compare different agents and agent components.

_Keywords_ Multimodal agents $\cdot$ Multimodal interaction $\cdot$ Data
sharing

##  1 Introduction

Interaction amongst agents, both human and AI, is especially complex when it
occurs in real-world contexts. It involves a complex psycho-social process
between the agents with their intentions, capabilities, and roles, and it
relates to the shared surroundings and space within which the interaction
occurs. To study such interactions, test different agents, and share data and
systems, two things are essential: 1) we need to be able to record the
interaction as multimodal data, and 2) we need to be able to freely experiment
with different components to test their impact on the interaction. This is
specifically challenging because multimodal signals, such as images, sound,
speech, faces and their expressions and gestures are not aligned, noisy and
can be interpreted in many different ways partially dependent on each other.
Building and testing systems that deal with such complex contexts requires a
multitude of different technologies and software components that need to work
together in a single framework to make sense.

Previously, Báez Santamaría et al. (2021) introduced a) EMISSOR for storing
sequences of multimodal signals with their annotations and b) episodic
Knowledge Graphs (eKGs) for storing the accumulation of interpretations as
symbolic triples. EMISSOR registers interactions as scenarios in which
multimodal signals align using a temporal ruler to mark signals’ start and
endpoints. Segments within each signal are annotated with interpretations,
such as detected objects or faces through cameras or entities and properties
mentioned in speech. For example, recorded images in time can be annoted as a
human face that is next identified as "Carl", who is the speaker of an audio
signal (human voice), annotated as a text signal "I am from Amsterdam" which
is further processed for its content. A visualization of an interaction
represented in EMISSOR is included in appendix A.

Episodic Knowledge Graphs (eKGs) represent the accumulation of interpretations
of signals over time as RDF111Resource Description Framework,
https://www.w3.org/RDF/ triples consisting of subject, predicate and object
identifiers (so-called Internationalized Resource Identifiers or IRIs) e.g.,
the speaker Carl claiming: [leolaniWorld:Carl, n2mu:live-in,
leolaniWorld:Amsterdam]. Such triples combine into more complex and rich
knowledge structures and can easily be connected with other external knowledge
such as DBpedia222https://www.dbpedia.org specifying what is Amsterdam through
another triple. By interpreting conversations and perceptions as triples, the
agent not only records signals but also learns about people and the world.

These components capture multimodal interactions both as the interpretation of
discrete signals in a specific modality and as an integrated symbolic
representation of situations in time. As explained in (Báez Santamaría et al.,
2021), EMISSOR aligns dispatched multimodal signals through a temporal ruler
to which each signal is grounded. What is seen, the person Carl or any object,
can be related to what is said by whom even at different points in time.
Whereas EMISSOR saves temporarily grounded signals as a stream of data on disk
with meta data, the "awareness of situations" and the "content of
communication" are additionally pushed to the knowledge graph as perceptions,
mentions, and properties, where we use JSON-LD to connect data across EMISSOR
and the eKG. The knowledge in the eKG is not only represented as a collection
of claims and perceptions, but also as a process, that tracks when what was
claimed or perceived by whom and with which certainty, belief, or emotion.
Thus, the graph represents an episodic memory333Note that the agent’s memory
does not forget anything and is limitless whose time frame is aligned with the
temporal ruler of EMISSOR.

This paper describes a modular platform for creating different interactive
agents whose interactions can be registered in both EMISSOR and eKGS. The
platform’s core is an event-bus, which can be considered a multi-topic
input/output queue that connects time-grounded signals with interpretation
components. Component APIs define the expected input and and output event
types, and the topics a component is connected to, such as "audio-in" or
"text-out", can be defined freely and specified in the configuration of an
agent. Likewise, different agents can be created easily by assembling
components and configuring their connection to the event-bus.

The primary signals themselves originate from backend components that read
data from sensors at time points and add them to a queue in the event-bus as
certain topics. Different interpretation components consume these signal
events (as defined by their input topic) and publish their output topic to the
event-bus as new events. By defining a simple API to and from the event-bus,
adding new signals and processing components is straightforward. These
components can annotate segments from signals with interpretations, such as
human speech or text, objects, and people perceived, ultimately pushing these
interpretations to the eKG.

This paper explains the overall architecture and implementation of the event-
bus and demonstrates how it renders interaction data to the EMISSOR format and
eKGs. We also explain how different interactive agents can be created within
the same platform using various components. Currently, we implemented a simple
Eliza agent444https://github.com/leolani/eliza-parent as well as a more
complex Leolani agent (Vossen et al.,
2019)555https://github.com/leolani/leolani-mmai-parent. Different agents still
render compatible multimodal data from their interactions within our
framework. Therefore, our platform can be used to compare agents by analysing
the rendered data, both in EMISSOR and as an eKG.

This paper is further structured as follows. In Section 2, we explain the
overall architecture of the platform and its connection to EMISSOR and eKGs.
Section 3 describes the components that are currently available. Section 4
explains how you can create your own agent either by combining existing
components or by defining your own component. All our code is available on
Github under the Apache2.0 open source license from:
https://github.com/leolani.

##  2 Event-Bus architecture

Figure 1 shows a schematic representation of the event-bus architecture. The
central component is the event-bus itself, to which signals and
interpretations are pushed either by backend components (at the left side) or
interpretation components (at the right side). The event-bus is represented as
a vertical (gray) bus to emphasize that it has a temporal dimension where
events are processed in a sequence. The labels on the arrows between the
event-bus and the components denote the type of event transmitted. Vertical
lines in the event-bus symbolize different topics used by the individual
components to define the event flow through the application. The topic labels
components are connected to are denoted in italic next to the arrows.

Figure 1: Schematic representation of the event-bus architecture showing
backend components to read from sensors and fill the queue in the event-bus
and interpretation components for annotating published signals. In this
schema, only audio signals are processed with an Eliza module for generating a
response. Interaction data is recorded from events by the EMISSOR component.

Examples of backend components are microphones, audio controllers, and
cameras. The sensors connected to backend components can be directly linked to
the system or obtained from remote servers. This schema shows how a microphone
produces an audio signal with a start and stop point. Some examples of
interpretation components are shown on the right side of the event-bus. The
VAD component at the top applies voice-activity detection to an audio signal
published by the microphone. It reports back to the event-bus whether the
audio is human speech, annotating the corresponding segment in the audio. The
speech-to-text (STT) component applies speech recognition to any audio signal
annotated as human speech and publishes the transcript as a text signal to the
event-bus. The text signal is picked up by an implementation of the Eliza
chatbot (Weizenbaum, 1966), which checks it for triggers and returns a
response as another text signal. Finally, the text-to-speech (TTS) component
picks up the text response and converts it to an audio signal. The backend
speaker listens to the event-bus for any audio signals to output. An audio
controller between the microphone and speaker mutes the mic when speaking and
vice-versa. Any data produced by the sensors or the components as signals is
stored in a backend container.

###  2.1 Enabling reasoning

The main application can function with just the event-bus and some simple
components, taking in signals and generating signals as a response. This
behavior, however, is primarily reactive and not reflective of the information
conveyed during an interaction. If we want an agent capable of reasoning over
the interpretations, we first need to capture the interaction, including
metadata and signals, together with their annotations and the knowledge
conveyed such that we can reason over it. For this, we included specific
components to connect the event-bus data to EMISSOR and the eKG.

Figure 2 shows the interconnections across the three components: event-bus,
EMISSOR and the eKG. The three components are aligned such that each
interaction in the event-bus generates a corresponding multimodal signal
representation in EMISSOR and, if the interpretation yields knowledge, also an
RDF triple in the eKG.

EMISSOR stores interactions as scenarios with metadata in JSON files for
signals in each modality: images, audio, text, and RDF triples. Further
details are described in Báez Santamaría et al. (2021). The metadata grounds
the signals to a temporal ruler and defines any annotation of segments within
a signal as interpretations. The signals themselves are stored as separate
files on disk. We consider the RDF triples extracted from text as a modality
stored on disk in EMISSOR, but they are also pushed to a separate triple store
functioning as the eKG.

In the eKG model, interactions are initialized as instances of situational
eps:contexts grounded in time and space. Contexts correspond to scenarios in
EMISSOR. Within a context, a series of sem:events are represented, which are
grounded in time like the signals in EMISSOR. We distinguish conversation and
perception events.

#### Conversation:

Conversation events in the eKG are structured as grasp:Chat instances
containing grasp:Utterances in which participants of the interaction make
grasp:Statements or formulate questions. The information conveyed in
grasp:Statements is stored in gaf:Claim named graphs attributed to speakers
(grasp:wasAttributedTo) that hold certain perspectives (grasp:Attribution).
These perspectives are properties such as grasp:CertaintyValue,
grasp:PolarityValue, grasp:SentimentValue, grasp:EmotionValue.666Currently, we
use NLP techniques to extract such values from the text generated by speech-
recognition. In future work, modules will be added that extract these from the
audio signal or face expressions as well As such the model can express that
people believe or deny certain facts with some certainty at moments in time.
Claims correspond to text signals in EMISSOR, which are derived from audio
signals.

#### Perceptions:

In contrast, perception events in the eKG are structured as grasp:Visual
instances containing grasp:Detectionss which could be objects, or people.
These detections are considered to be instances of grasp:Experiences. Similar
to conversation events, the information obtained from these grasp:Experiences
is stored in gaf:Claim named graphs with triples for the output of
object/face-recognition components. Therefore, the information is
grasp:wasAttributedTo sensors, and the grasp:Attributions are restricted to
grasp:CertaintyValues only. Perception events therefore reflect the awareness
of an agent of objects and people in the context. Perceptions correspond with
image signals in EMISSOR.

Figure 2: Schematic overview of relations across data structures in EMISSOR,
the episodic Knowledge Graph and the event-bus.

As is shown in Figure 2, the event-bus supports specific components that
render the data for EMISSOR and for the eKG. These components can also push
data, such as annotations or triples, to the event-bus itself so that other
components can take these as input for further processing. The event-bus is
thus not only populated by the sensors but also by interpretation components.

In the next section, we give an overview of all the components that are
currently available in our platform and we explain in more detail how they
interact. After that, we explain how new components can be added and how you
can create your own agent.

##  3 Components available

The task of interaction is broken down into a large number of smaller tasks
operating on specific input events taken from the event-bus and pushing their
output back onto the event-bus. So far, we developed the following components,
grouped by their underlying input signal:

  1. 1.

Audio signal

voice activity detection

     777https://github.com/leolani/cltl-vad

classifying an audio signal as speech, output is an annotated audio signal
pushed to the event-bus and saved to EMISSOR as an audio annotation.

automatic speech recognition

     888https://github.com/leolani/cltl-asr

transforming speech to text, three different models can be chosen which
generate a text output signal to the event-bus and EMISSOR.

speaker identification (under development)

identifying the speaker based on a sample of their voice, output is a
speaker’s identity (as an IRI and their name), and an audio signal annotated
with the source pushed to the event-bus and EMISSOR but also registered in the
eKG as an encounter.

  2. 2.

Text signal

mention detection

     999https://github.com/leolani/cltl-knowledgeextraction

detecting names and potential objects in a text signal, output is an annotated
signal with named entities and object mentions, pushed to the event-bus and
EMISSOR.

mention identification

     101010https://github.com/leolani/cltl-knowledgelinking

identifying instances in the knowledge graph for mention annotations in a text
signal, output is an identifier (IRI), pushed to the event-bus and EMISSOR.
Corresponding RDF triples (IRI gaf:denotedBy gaf:utterance#offset) are pushed
to the eKG.

triple extraction

     111111https://github.com/leolani/cltl-knowledgeextraction

detecting RDF triples in a text signal; three different modules can be
selected; output are RDF triples pushed to the event-bus, EMISSOR and the eKG.

query extraction

     121212https://github.com/leolani/cltl-questionprocessor

detecting a question in a text signal and converting this into a SPARQL query;
output is a SPARQL query send to the eKG and possibly other knowledge graphs.

emotion detection

     131313https://github.com/leolani/cltl-emotionrecognition

detecting basic emotions in a text signal; output is an emotion label pushed
to the event-bus and EMISSOR as an annotation and possibly as a perspective to
the eKG.

gesture generation

     141414https://github.com/leolani/cltl-gesturegeneration

returns a gesture that is an appropriate response to the emotion annotations
in a text signal; output is an emotion label pushed to the event-bus and
EMISSOR as an annotation and possibly as a perspective to the eKG.

  3. 3.

RDF signal

knowledge representation for claims

     151515https://github.com/leolani/cltl-knowledgerepresentation

posting RDF claims to the eKG; output is JSON-LD as responses to the changes
in the eKG by digesting new claims pushed to the event-bus.

knowledge representation for queries

     161616https://github.com/leolani/cltl-knowledgerepresentation

posting SPARQL queries to the eKG or other graphs; output are RDF triples as
the result of the query pushed to the event-bus.

response generation

     171717https://github.com/leolani/cltl-languagegeneration

verbalising qualitative reflections on changes in the eKG (responses); output
is a natural language response as a text signal, pushed to the event-bus and
stored in EMISSOR.

language generation

     181818https://github.com/leolani/cltl-languagegeneration

verbalising SPARQL query results; output is natural language as a text signal,
pushed to the event-bus and stored in EMISSOR.

  4. 4.

Image signal

object recognition

     191919https://github.com/leolani/cltl-object-recognition

detecting multiple objects in images; output is bounding boxes and object
labels as an annotated image signal, pushed to the event-bus and stored in
EMISSOR.

face detection

     202020https://github.com/tae898/age-gender

detecting human faces in an image, the output are bounding boxes with the
estimated age and gender as an annotated image, pushed to the event-bus and
stored in EMISSOR.

face identification

     212121https://github.com/leolani/cltl-face-recognition

identifying people from their face, the output is an identifier (IRI and a
label as name), pushed to the event-bus and stored in EMISSOR. Encounters are
also registered in the eKG as perception events.

###  3.1 Processing audio signals

The components that process the audio signal were discussed in detail in
Section 2 when explaining Figure 1. In a nutshell, the voice-activity-
detection component detect portions of human speech in audio, which the
automatic-speech-recognition transcribes. Eventually, these components push a
text signal to the event-bus, which is rendered from an audio signal for text
processing components.

###  3.2 Processing text and rdf signals

The components that operate on text signals are more complex. They generate
different interpretations and types of output signals and combine with
components that operate on RDF triple signals. Various Natural Language
Processing (NLP) modules are called within different components, among which
spaCy(Vasiliev, 2020), NLTK, StanfordNLP (Angeli et al., 2015) and various
transformer models. In addition to their basic processing, triple-extraction
components output various annotations, for example, mentions of people and
objects for which referring expressions need to be resolved to people and
objects known. Referring expressions can be names (Carl), common noun phrases
(the waiter), or ambiguous pronouns. By reasoning over the context and
previous encounters, mention-identification components aim to establish the
referent of these expressions, possibly involving the human interlocutor in
case of doubt (e.g., by posting a question as a text signal "Which waiter?").
Resolved expressions are converted to IRIs incorporated in triples. These can
be registrations of mentions, e.g., Carl talking about Carla, or as part of
triples expressing a property from an individual: leolaniWorld:Carla
leolaniWorld:live-in leolaniWorld:Amsterdam.

Whenever a gaf:Claim is posted to the eKG, the knowledge-representation-for-
claims component generates a reflective response to the claim. This involves
running various pre-coded SPARQL queries to detect knowledge gaps, conflicts,
uncertainties, novelties, analogies, and possible generalizations. The
response-generation component selects a result to formulate a response
attempting to improve the eKG, e.g., fill gaps or resolve uncertainty. The
JSON triples are verbalized as natural language text in a text signal.

The previous route through the event-bus applies to signals classified as
statements. If a text signal in the event-bus is classified as a question, the
query-extraction component extracts a SPARQL query from the text signal that
the knowledge-representation-for-queries component posts this to the eKG (or
any other triple store). The language-generation component takes the query’s
result (one, many, or none triples) to verbalize it as a text signal.
Different answers are generated according to the type and number of query
results. Furthermore, answers consider the status of the knowledge, e.g. how
certain, who is the source, and when was it mentioned.

In addition, some components only annotate mentions or emotions expressed in
the text (without necessarily being involved in a triple). The mention-
detection component detects things that (can) exist in the physical world,
which are entities that the object recognition can classify. Mentions are
saved as annotations of the text signal in EMISSOR and pushed to the event-bus
for further processing. Next, the mention-identification component picks up
mentions and tries to resolve its identity given the entities registered in
the eKG, e.g. by checking names, the contexts, or resolving pronouns’ co-
reference. Identities are pushed to the event-bus and registered as
gaf:Mentions in the eKG. Possibly, a new identity can be established and a new
IRI is stored in the eKG with the properties that can be inferred.

The emotion-detection component interprets a text signal and pushes an emotion
to the event-bus and as annotation in EMISSOR. Emotions expressed by the
interlocutor represent potential perspectives of the source on the situation
for the eKG. The same component can be applied to text signals of the agent.
Especially when creating a response to new information, the corresponding
emotion can be expressed by the gesture-generation component or audio
properties such as volume and pitch.

###  3.3 Processing image signals

Finally, the object recognition and face-detection components take image
signals as input to detect objects and faces with bounding boxes. These are
saved as annotations of image signals in EMISSOR and pushed to the event-bus.
From the event-bus, the face-identification component establishes the identity
of people by comparing it with the faces of known people. Unknown faces are
saved as new identities with inferred properties, such as gender and age. A
new identity pushed to the event-bus is picked up by a component that asks for
a person’s name. The image-signal-processing components either produce
annotations pushed to the event-bus and/or encoded in EMISSOR or produce an
IRI with properties: name, age, gender, and registered perception event in the
eKG.

###  3.4 Sample input-output signal sequences

We show some schematic representations of the sequence of input and output
events with their payload type and input/output topics denoted above the
arrows222222Topic names are exemplary and can be configured in the
application.

  1. 1.

audio (signal) $\xrightarrow{\text{mic}}$ speech (annotation)
$\xrightarrow{\text{vad}}$ text (annotation)
$\xrightarrow{\text{text-in}}$ mention (annotation)
$\xrightarrow{\text{entities}}$ IRI (annotation)
$\xrightarrow{\text{linking}}$ claim triple (eKG)
$\xrightarrow{\text{triples}}$ response triples (eKG)
$\xrightarrow{\text{response}}$ text (signal) $\xrightarrow{\text{text-out}}$
audio (signal)

  2. 2.

audio (signal) $\xrightarrow{\text{mic}}$ speech (annotation)
$\xrightarrow{\text{vad}}$ text (annotation)
$\xrightarrow{\text{text-in}}$ mention (annotation)
$\xrightarrow{\text{entities}}$ IRI (annotation)
$\xrightarrow{\text{linking}}$ query triple (eKG)
$\xrightarrow{\text{triples}}$ response triples (eKG)
$\xrightarrow{\text{response}}$ text (signal) $\xrightarrow{\text{text-out}}$
audio (signal)

  3. 3.

audio (signal) $\xrightarrow{\text{mic}}$ speaker (annotation)
$\xrightarrow{\text{speaker}}$ IRI (annotation)
$\xrightarrow{\text{identity}}$ perception (eKG)

  4. 4.

image (signal) $\xrightarrow{\text{cam}}$ face (annotation)
$\xrightarrow{\text{face}}$ IRI (annotation) $\xrightarrow{\text{identity}}$
perception (eKG)

  5. 5.

image (signal) $\xrightarrow{\text{cam}}$ object (annotation)
$\xrightarrow{\text{object}}$ IRI (annotation) $\xrightarrow{\text{identity}}$
perception (eKG)

The above components become active when the input-topic requirements are met.
As such they do not represent higher-level goals or targets but are data-
driven, i.e. always respond to the presence of input topics in the event-bus.
The event-bus architecture however also allows to define higher-level
intentions that represent behaviour of the agent to fulfill a given task or
reach a goal. These intentions can be prioritised and remain active until
resolved. For each intention, we can specify which components are actively
listening to the event-bus and which follow-up intentions are published once a
task is completed or a certain goal is reached. The above components thus are
only activated within these intentions.

We implemented the following high-level intentions in the platform:

Greeting

After detecting a new human face in an image signal, the agent tries to
identify the person to great him/her by name or to get to know a new person by
asking for a name.

Giving consent

Ask people’s permission to keep the data and share it for research. If no
consent is given, the agent will remove the episodic data (EMISSOR and eKG)
after the interaction.

Eliza

Takes text signals as input and generate a text response using the Eliza
trigger patterns and hard-coded responses.

About agent

Takes text signals as input that contain a question about the agent, and
generates a text signal that informs the participant about the agent.

What do you see

Takes text signals as input that contain a question about the visual context
and generates a text signal that describes what has been seen in the recent
context (where recency is defined in the configuration).

Leolani

Interaction with an eKG as described in the beginning of this section,
relating interpretations to identities (IRIs) and triples (Vossen et al.,
2019).

Blenderbot

Process text signals with Blenderbot (Roller et al., 2020). Blenderbot is a
generative model trained with different conversational data to generate a
human-like response.

Goodbye

When a face is no longer detected or the text signal contains a goodbye clue
it ends the scenario and says goodbye to the participant.

Both components and higher-level intentions can be created and combined
freely. Partly, this provides control over certain goals or tasks to be
completed but it also allows for flexible and spontaneous behavior. In the
next section, we explain how you can design your own agent by adding
components and high-level intentions. All the code for the platform with the
currently developed components are available on Github:
https://github.com/leolani under Apache2.0 license. A good starting point for
building an agent is the repository: https://github.com/leolani/cltl-combot.

##  4 Developing your own agent

As components are self-contained and loosely coupled through the event-bus,
developing an agent is reduced to composing the necessary components and
defining a flow of events, which is done by configuring the input and output
topics of the components appropriately.

###  4.1 Adding, replacing and mixing components

To enable compatibility between components, we use EMISSOR also as the
preferred data format of the payload carried by the events. Signal components
publish EMISSOR Signals, eventually split up into a start and stop event.
Interpretation components publish EMISSOR Mentions, defining segments and
their annotations. Like this, any component that can process e.g. EMISSOR text
signals or interpretations of a certain type, can be integrated into an agent
by simply configuring it to listen to a topic that provides events with a text
signal or the desired type of interpretation as payload. Furthermore, data
from components that provide EMISSOR-based event payloads can be directly
recorded in EMISSOR by a central storage component without additional
adaption.

As long as component implementations process the same type of events, one
implementation can simply be swapped for another. Even multiple
implementations of the same component can be included in the same application,
either for comparison or to increase performance. In the current version of
the platform, we have for example various components for extracting triples
and for speech recognition that can be applied simultaneously to increase
recall or precision through voting. The source of signals and annotations is
included in all EMISSOR data structures, which allows downstream components to
determine the origin of an event.

Our modular architecture aims for easy integration of components that were not
designed for our platform. The overhead to turn a software application into a
component compatible with our platform consists of converting input and output
of the application from and to EMISSOR format and connecting it to the event-
bus. This can usually be achieved by adding these steps on top of existing
code, without the need for modification.

###  4.2 Application types

By choosing different implementations of the event-bus (local vs. remote),
agent implementations can reach from e.g. a monolithic Python application to a
containerized setup, running each component as a separate process on a single
machine, for instance using docker-
compose232323https://docs.docker.com/compose, or even running in a cluster
consisting of many physical machines using e.g.
Kubernetes242424https://kubernetes.io. These different setups do not require
any code adaption of the components itself, as those only communicate through
the event-bus interface, which remains identical. In a containerized setup,
our architecture also allows mixing components from different platforms, e.g.
using Python next to Java components, or components running different Python
environments in the same agent to resolve dependency conflicts, which is a
common problem in software development.

###  4.3 Out of the box

For the components listed in Section 3 we provide Python implementations, each
in its own repository published in the Leolani Github organization
252525https://github.com/leolani. Each component can be packaged as Python
package and needs to provide a service module containing the EMISSOR and
event-bus integration, which can be run from a Python application. Packages
also provide the possibility to reuse component code across components. In the
future we plan to add Docker configurations to enable running each component
in a Docker container.

Our current agent implementations are realized as monolithic Python
applications, which initialize and run the service modules from the each of
the included component packages, and use a local implementation of the event-
bus. We provide a Python implementation of the backend server which can be run
on Linux/OS X/Windows platforms262626https://github.com/leolani/cltl-backend,
as well as a backend server that can be run on Pepper
robots272727https://github.com/leolani/cltl-backend-naoqi. Our agents can be
used as a skeleton to build further agents, and in the future we will provide
docker–compose and Kubernetes setups based on component
Docker282828https://www.docker.com images to run the agents as fully
containerized applications. In the containerized setting we will be able to
run components either locally or remotely in the cloud.

To collect all code needed for an agent we created parent repositories for
each agent which contain all of the agent’s components as git submodules. We
provide build tooling to package each of our Python components, share the
packages between components and setup Python environments with all
dependencies needed to run the agent application and individual components.
This process can be executed centrally from the parent repository using our
build setup. Following this pattern, new agents can be created also outside of
our organisation, mixing components under different governance.

To build components, the https://github.com/leolani/cltl-combot provides
infrastructure libraries for event-bus integration, configuration management
and resource management (locking). The cltl.combot.infra.event module provides
our event-bus interface with a local implementation as well as an
implementation based on the kombu292929https://github.com/celery/kombu library
supporting the AMQP303030https://www.amqp.org protocol. This allows to connect
our framework to various available messaging servers like for instance
RabbitMQ313131https://www.rabbitmq.com. Furthermore,
cltl.combot.infra.topic_worker.TopicWorker provides a utility class to listen
to a configurable set of topics and process incoming events sequentially in a
single thread, such that only the actual processing function, accepting a
single event as input, needs to be implemented by the user of the library. Our
own components are structured to separate functionality, using the cltl
namespace, from the event-bus integration, using the cltl_service namespace.

A simple example is presented in Appendix B, C and D. For a full example
including configuration and resource management we provide a template
component at https://github.com/leolani/cltl-template with more detailed code
templates, makefiles for our build setup and in the future a Dockerfile
template.

##  5 Conclusions

We described a platform for creating interactive agents which also captures
the interaction at the signal level, the interpretation level and the
integrated knowledge level. Central to our platform is an event-bus on which
signals and interpretations can be published in temporal sequence as so-called
topics. By defining components as services that take certain topics as input
and produce other topics as output, we have the flexibility to engineer any
combination of components to create agent behaviour. Our platform also
supports the specification of higher-level intentions that can be carried out
to fulfill a task or goal. Such intentions define which low-level components
should be active to achieve this.

Multimodal interaction is extremely complex and rich. The agents that are
currently built through our platform are far from what is needed to deal with
all facets of human interaction. A lot of research and development is needed
not only on individual components but also on their interaction and
integration. Our platform will specifically help developing such complex
integrated agents.

In the near future, we will use the platform to create different agents that
can be used in interaction experiments. The rendered multimodal data and eKGs
can be analysed and compared to assess and evaluate the agents and their
components. As such, we hope it will function as a laboratory for agent
development and testing.

## References

  * Báez Santamaría et al. [2021] Selene Báez Santamaría, Thomas Baier, Taewoon Kim, Lea Krause, Jaap Kruijt, and Piek Vossen.  Emissor: A platform for capturing multimodal interactions as episodic memories and interpretations with situated scenario-based ontological references.  _Proceedings of the 1st Workshop on Multimodal Semantic Representations (MMSR)_ , pages 56–77, 2021.
  * Vossen et al. [2019] Piek Vossen, Lenka Bajčetić, Selene Báez Santamaria, Suzana Basić, and Bram Kraaijeveld.  Modelling context awareness for a situated semantic agent.  In _Proceedings of 11th International and Interdisciplinary Conference on Modeling and Using Context, CONTEXT 2019_ , 2019.
  * Weizenbaum [1966] Joseph Weizenbaum.  Eliza—a computer program for the study of natural language communication between man and machine.  _Communications of the ACM_ , 9(1):36–45, 1966\.
  * Vasiliev [2020] Yuli Vasiliev.  _Natural Language Processing with Python and SpaCy: A Practical Introduction_.  No Starch Press, 2020.
  * Angeli et al. [2015] Gabor Angeli, Melvin Jose Johnson Premkumar, and Christopher D Manning.  Leveraging linguistic structure for open domain information extraction.  In _Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 344–354, 2015.
  * Roller et al. [2020] Stephen Roller, Emily Dinan, Naman Goyal, Da Ju, Mary Williamson, Yinhan Liu, Jing Xu, Myle Ott, Kurt Shuster, Eric M Smith, et al.  Recipes for building an open-domain chatbot.  _arXiv preprint arXiv:2004.13637_ , 2020.

##  Appendix A EMISSOR multimodal data representation

Figure 3: Visualization of an interaction in the EMISSOR data format.

Figure 3, taken from [Báez Santamaría et al., 2021], shows a visualization of
four modalities (text, audio, visual, and knowledge). Signals are grounded in
a temporal container on the horizontal axis, with bars marking alignments
through the temporal ruler. Red boxes mark segments annotated as mentions of
objects (pills and the table). Text segments highlighted in red are annotated
as mentions of triples. The upper graphs represent corresponding triples from
eKG generated from the annotated source modalities along the temporal
sequence. The visual modality shows two different camera viewpoints (left is
what Carl sees and right is what Leolani sees) concatenated side by side.

##  Appendix B Example component structure

.1 cltl-mycomponent. .2 config/. .2 requirements.txt. .2 setup.py. .2 src/. .3
app.py. .4 cltl/. .5 mycomponent/. .6 api.py. .6 implementation.py. .4
cltl_service/. .5 mycomponent/. .6 service.py.

##  Appendix C Example service.py module

⬇

from cltl.combot.infra.event import Event, EventBus

from cltl.combot.infra.time_util import timestamp_now

from cltl.combot.infra.topic_worker import TopicWorker

from cltl.combot.event.emissor import TextSignalEvent

from emissor.representation.scenario import TextSignal

class SimpleService:

def __init__(self, input_topic: str, output_topic: str, event_bus: EventBus):

self._input_topic = input_topic

self._output_topic = output_topic

self._event_bus = event_bus

self._topic_worker = None

def start(self):

self._topic_worker = TopicWorker([self._input_topic], self._event_bus,

provides=[self._output_topic],

processor=self._process)

self._topic_worker.start().wait()

def stop(self):

if not self._topic_worker:

pass

self._topic_worker.stop()

self._topic_worker.await_stop()

self._topic_worker = None

def _process(self, event: Event[TextSignalEvent]):

input = event.payload.signal.text)

response = "" ### PROCESS THE INPUT

if response:

output_event = self._create_payload(response)

self._event_bus.publish(self._output_topic, Event.for_payload(output_event))

def _create_payload(self, response):

signal = TextSignal.for_scenario(None, timestamp_now(), timestamp_now(), None,
response)

return TextSignalEvent.create(signal)

##  Appendix D Example app.py application

⬇

from cltl.combot.infra.event.memory import SynchronousEventBus

from cltl_service.mycomponent import SimpleService

event_bus = SynchronousEventBus()

service = SimpleService("text_in", "text_out", event_bus)

try:

service.start()

while True:

time.sleep(1)

except KeyboardInterrupt:

self.service.stop()


# Proper $p$-harmonic functions and harmonic morphisms on the classical non-
compact semi-Riemannian Lie groups

Elsa Ghandour  Mathematics, Faculty of Science
University of Lund
Box 118, Lund 221 00
Sweden<EMAIL_ADDRESS>and  Sigmundur Gudmundsson  Mathematics,
Faculty of Science
University of Lund
Box 118, Lund 221 00
Sweden<EMAIL_ADDRESS>
###### Abstract.

We apply the method of eigenfamilies to construct new explicit complex-valued
$p$-harmonic functions on the non-compact classical Lie groups, equipped with
their natural semi-Riemannian metrics. We then employ this same approach to
manufacture explicit complex-valued harmonic morphisms on these groups.

###### Key words and phrases:

$p$-harmonic functions, harmonic morphisms, semi-Riemannian classical groups

###### 2020 Mathematics Subject Classification:

31B30, 53C43, 58E20

##  1\. Introduction

For a positive integer $p$, the complex-valued $p$-harmonic functions are
solutions to a partial differential equation of order $2\,p$. This equation
arises in various contexts, see for example the extensive analysis in [5] and
a historic account in [15]. The best known applications are in physics e.g.
for $p=2$ in the areas of continuum mechanics, including elasticity theory and
the solution of Stokes flows. The literature on $2$-harmonic functions is
vast, but until quiet recently, the domains were either surfaces or open
subsets of flat Euclidean space, with only very few exceptions. For this see
the regularly updated online bibliography [8] maintained by the second author.

In their recent article [11], the authors produce $p$-harmonic functions on
the classical Lie groups equipped with their standard Riemannian metrics. The
primary goal of this work is to extend the study to the semi-Riemannian
situation. By Theorem 3.2 we show how the problem can be reduced to finding an
eigenfamily, i.e. a collection of complex-valued functions which are eigen
both with respect to the Laplace-Beltrami operator $\tau$ and the conformality
operator $\kappa$, on the semi-Riemannian manifolds involved. The main part of
this paper is devoted to the construction of such families on the following
classical Lie groups equipped with their natural semi-Riemannian metrics

$\text{\bf GL}_{n}({\mathbb{C}}),\,\text{\bf GL}_{n}({\mathbb{R}}),\,\text{\bf
GL}_{n}({\mathbb{H}}),\,\text{\bf SL}_{n}({\mathbb{C}}),\,\text{\bf
SL}_{n}({\mathbb{R}}),\,\text{\bf SL}_{n}({\mathbb{H}}),$ $\text{\bf
SO}(n,{\mathbb{C}}),\,\text{\bf Sp}(n,{\mathbb{C}}),\,\text{\bf
Sp}(n,{\mathbb{R}}),\,\text{\bf SO}^{*}(2n),\,\text{\bf SU}(p,q),\,\text{\bf
SO}(p,q),\,\text{\bf Sp}(p,q).$

Our eigenfamilies can also be used to manufacture complex-valued harmonic
morphisms on these manifolds as explained in Theorem 4.3. They can therefore
be seen as an interesting byproduct of the process presented here.

For semi-Riemannian geometry we recommend O’Neill’s standard text [16].
Readers not familiar with harmonic morphisms are advised to consult the
standard text [2] by Baird and Wood, [3], [4], [13] and the regularly updated
online bibliography [7]. For the details from Lie theory, used in this paper,
we refer the reader to [12] and [14].

##  2\. Eigenfunctions and Eigenfamilies

In this paper we manufacture explicit complex-valued proper $p$-harmonic
functions and harmonic morphisms on semi-Riemannian manifolds. For this we
apply two different construction techniques which are presented in Theorem 3.2
and in Theorem 4.3, respectively. The main ingredients for both these recipes
are the common eigenfunctions for the tension field $\tau$ and the
conformality operator $\kappa$ which we now describe.

Let $(M,g)$ be an $m$-dimensional semi-Riemannian manifold and
$T^{{\mathbb{C}}}M$ be the complexification of the tangent bundle $TM$ of $M$.
We extend the metric $g$ to a complex-bilinear form on $T^{{\mathbb{C}}}M$.
Then the gradient $\nabla\phi$ of a complex-valued function
$\phi:(M,g)\to{\mathbb{C}}$ is a section of $T^{{\mathbb{C}}}M$. In this
situation, the well-known complex linear Laplace-Beltrami operator (alt.
tension field) $\tau$ on $(M,g)$ acts locally on $\phi$ as follows

$\tau(\phi)=\operatorname{div}(\nabla\phi)=\sum_{i,j=1}^{m}\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial
x_{j}}\Big{(}g^{ij}\,\sqrt{|g|}\,\frac{\partial\phi}{\partial x_{i}}\Big{)}.$

For two complex-valued functions $\phi,\psi:(M,g)\to{\mathbb{C}}$ we have the
following well-known fundamental relation

$\tau(\phi\cdot\psi)=\tau(\phi)\cdot\psi+2\cdot\kappa(\phi,\psi)+\phi\cdot\tau(\psi),$

where the complex bilinear conformality operator $\kappa$ is given by

$\kappa(\phi,\psi)=g(\nabla\phi,\nabla\psi).$

Locally this satisfies

$\kappa(\phi,\psi)=\sum_{i,j=1}^{m}g^{ij}\cdot\frac{\partial\phi}{\partial
x_{i}}\frac{\partial\psi}{\partial x_{j}}.$

######  Definition 2.1.

[10] Let $(M,g)$ be a semi-Riemannian manifold. Then a complex-valued function
$\phi:M\to{\mathbb{C}}$ is said to be an eigenfunction if it is eigen both
with respect to the Laplace-Beltrami operator $\tau$ and the conformality
operator $\kappa$ i.e. there exist complex numbers
$\lambda,\mu\in{\mathbb{C}}$ such that

$\tau(\phi)=\lambda\cdot\phi\ \ \text{and}\ \
\kappa(\phi,\phi)=\mu\cdot\phi^{2}.$

A set $\mathcal{E}=\\{\phi_{i}:M\to{\mathbb{C}}\,|\,i\in I\\}$ of complex-
valued functions is said to be an eigenfamily on $M$ if there exist complex
numbers $\lambda,\mu\in{\mathbb{C}}$ such that for all
$\phi,\psi\in\mathcal{E}$ we have

$\tau(\phi)=\lambda\cdot\phi\ \ \text{and}\ \
\kappa(\phi,\psi)=\mu\cdot\phi\cdot\psi.$

The following Theorem 2.2 shows that, given an eigenfamily $\mathcal{E}$, one
can employ this to produce an extensive collection
$\mathcal{H}^{d}_{\mathcal{E}}$ of further such objects. The result is a semi-
Riemannian version of Theorem 2.2 proven in [6] for the Riemannian case.

######  Theorem 2.2.

Let $(M,g)$ be a semi-Riemannian manifold and the set of complex-valued
functions

$\mathcal{E}=\\{\phi_{k}:M\to{\mathbb{C}}\,|\,k=1,2,\dots,n\\}$

be an eigenfamily on $M$ i.e. there exist complex numbers
$\lambda,\mu\in{\mathbb{C}}$ such that for all $\phi,\psi\in\mathcal{E}$

$\tau(\phi)=\lambda\cdot\phi\ \ \text{and}\ \
\kappa(\phi,\psi)=\mu\cdot\phi\cdot\psi.$

Then the set of complex homogeneous polynomials of degree $d$

$\mathcal{H}^{d}_{\mathcal{E}}=\\{P:M\to{\mathbb{C}}\,|\,P\in{\mathbb{C}}[\phi_{1},\phi_{2},\dots,\phi_{n}],\
P(\alpha\cdot\phi)=\alpha^{d}\cdot P(\phi),\ \alpha\in{\mathbb{C}}\\}$

is an eigenfamily on $M$ such that for all
$P,Q\in\mathcal{H}^{d}_{\mathcal{E}}$ we have

$\tau(P)=(d\cdot\lambda+d(d-1)\cdot\mu)\cdot P\ \ \text{and}\ \
\kappa(P,Q)=d^{2}\cdot\mu\cdot P\cdot Q.$

###### Proof.

The statement can be proven with exactly the same arguments as its Riemannian
counterpart, see Theorem 2.2 in [6]. ∎

##  3\. Proper $p$-Harmonic Functions

In this section we describe a method for manufacturing complex-valued proper
$p$-harmonic functions on semi-Riemannian manifolds. This method was recently
introduced for the Riemannian case in [11].

######  Definition 3.1.

Let $(M,g)$ be a semi-Riemannian manifold. For a positive integer $p$, the
iterated Laplace-Beltrami operator $\tau^{p}$ is given by

$\tau^{0}(\phi)=\phi\ \ \text{and}\ \
\tau^{p}(\phi)=\tau(\tau^{(p-1)}(\phi)).$

We say that a complex-valued function $\phi:(M,g)\to{\mathbb{C}}$ is

  1. (i)

$p$-harmonic if $\tau^{p}(\phi)=0$ and

  2. (ii)

proper $p$-harmonic if $\tau^{p}(\phi)=0$ and $\tau^{(p-1)}(\phi)$ does not
vanish identically.

The following Theorem 3.2 can be proven in exactly the same way as its
Riemannian counterpart found as Theorem 3.1 in [11].

######  Theorem 3.2.

Let $\phi:(M,g)\to{\mathbb{C}}$ be a complex-valued function on a semi-
Riemannian manifold and $(\lambda,\mu)\in{\mathbb{C}}^{2}\setminus\\{0\\}$ be
such that the tension field $\tau$ and the conformality operator $\kappa$
satisfy

$\tau(\phi)=\lambda\cdot\phi\ \ \text{and}\ \
\kappa(\phi,\phi)=\mu\cdot\phi^{2}.$

Then for any positive integer $p\in{\mathbb{Z}}^{+}$ the non-vanishing
function

$\Phi_{p}:W=\\{x\in M\,|\,\phi(x)\not\in(-\infty,0]\\}\to{\mathbb{C}}$

with

$\Phi_{p}(x)=\begin{cases}c_{1}\cdot\log(\phi(x))^{p-1},&\text{if
}\;\mu=0,\;\lambda\not=0\\\\[5.69046pt]
c_{1}\cdot\log(\phi(x))^{2p-1}+c_{2}\cdot\log(\phi(x))^{2p-2},&\text{if
}\;\mu\not=0,\;\lambda=\mu\\\\[5.69046pt]
c_{1}\cdot\phi(x)^{1-\frac{\lambda}{\mu}}\log(\phi(x))^{p-1}+c_{2}\cdot\log(\phi(x))^{p-1},&\text{if
}\;\mu\not=0,\;\lambda\not=\mu\end{cases}$

is a proper $p$-harmonic function. Here $c_{1},c_{2}$ are complex coefficients
not both zero.

##  4\. Complex-Valued Harmonic Morphisms

In this section we describe a method for constructing complex-valued harmonic
morphisms $\phi:(M,g)\to{\mathbb{C}}$ from semi-Riemannian manifolds. This is
a special case of the much studied harmonic morphisms $\phi:(M,g)\to(N,h)$
between semi-Riemannian manifolds. They are maps which pull back local real-
valued harmonic functions on $(N,h)$ to harmonic functions on $(M,g)$. The
standard reference for the extensive theory of harmonic morphisms is the book
[2], but we also recommend the updated online bibliography [7].

The following result is a direct consequence of Theorem 3 of the paper [4] by
B. Fuglede.

######  Proposition 4.1.

A function $\phi:(M,g)\to{\mathbb{C}}$ from a semi-Riemannian manifold to the
standard Euclidean complex plane, is a harmonic morphism if and only if it is
harmonic and horizontally conformal i.e.

$\tau(\phi)=0\ \ \text{and}\ \ \kappa(\phi,\phi)=0.$

The following Theorem 4.2 is a semi-Riemannian version of Theorem 5.2 of [1],
see also [2]. It gives the theory of complex-valued harmonic morphisms a
strong geometric flavour and provides a useful tool for the construction of
minimal submanifolds of codimension two. This is our main motivation for
studying these maps.

######  Theorem 4.2.

Let $\phi:(M,g)\to{\mathbb{C}}$ be a horizontally conformal map from a semi-
Riemannian manifold to the standard Euclidean complex plane. Then $\phi$ is
harmonic if and only if its fibres are minimal at regular points of $\phi$.

The next result shows that eigenfamilies can be utilised to manufacture a
variety of harmonic morphisms.

######  Theorem 4.3.

[10] Let $(M,g)$ be a semi-Riemannian manifold and

$\mathcal{E}=\\{\phi_{k}:M\to{\mathbb{C}}\,|\,k=1,2,\dots,n\\}$

be an eigenfamily of complex-valued functions on $M$. If
$P,Q:{\mathbb{C}}^{n}\to{\mathbb{C}}$ are linearily independent homogeneous
polynomials of the same positive degree then the quotient

$\frac{P(\phi_{1},\dots,\phi_{n})}{Q(\phi_{1},\dots,\phi_{n})}$

is a non-constant harmonic morphism on the open and dense subset

$\\{p\in M\,|\,Q(\phi_{1}(p),\dots,\phi_{n}(p))\neq 0\\}.$

##  5\. The Semi-Riemannian Lie Group $\text{\bf GL}_{n}({\mathbb{C}})$

The complex general linear group $\text{\bf GL}_{n}({\mathbb{C}})$ of
invertible $n\times n$ matrices is given by

$\text{\bf GL}_{n}({\mathbb{C}})=\\{z\in{\mathbb{C}}^{n\times n}\,|\,\det
z\neq 0\\}.$

Its Lie algebra $\mathfrak{gl}_{n}({\mathbb{C}})$ of left-invariant vector
fields on $\text{\bf GL}_{n}({\mathbb{C}})$ can be identified with
${\mathbb{C}}^{n\times n}$ i.e. the linear space of complex $n\times n$
matrices. We equip $\text{\bf GL}_{n}({\mathbb{C}})$ with its natural semi-
Riemannian metric $g$ induced by the semi-Euclidean inner product
$\mathfrak{gl}_{n}({\mathbb{C}})\times\mathfrak{gl}_{n}({\mathbb{C}})\to{\mathbb{R}}$
on $\mathfrak{gl}_{n}({\mathbb{C}})$ satisfying

$g(Z,W)\mapsto-\,\mathfrak{R}\mathfrak{e}\operatorname{trace}(Z\cdot W).$

For $\mathfrak{gl}_{n}({\mathbb{C}})$ we then have the orthogonal
decomposition

$\mathfrak{gl}_{n}({\mathbb{C}})=\mathfrak{gl}^{+}_{n}({\mathbb{C}})\oplus\mathfrak{gl}^{-}_{n}({\mathbb{C}}),$

where

$\mathfrak{gl}^{+}_{n}({\mathbb{C}})=\mathfrak{u}(n)=\\{Z\in\mathfrak{gl}_{n}({\mathbb{C}})\,|\,Z+\bar{Z}^{t}=0\\}$

is the set of skew-Hermitian matrices and

$\mathfrak{gl}^{-}_{n}({\mathbb{C}})=i\cdot\mathfrak{u}(n)=\\{Z\in\mathfrak{gl}_{n}({\mathbb{C}})\,|\,Z-\bar{Z}^{t}=0\\}$

the set of the Hermitian ones. Here $\mathfrak{u}(n)$ is the Lie algebra of
the unitary group $\text{\bf U}(n)$ which is the maximal compact subgroup of
$\text{\bf GL}_{n}({\mathbb{C}})$ satisfying

$\text{\bf U}(n)=\\{z\in\text{\bf
GL}_{n}({\mathbb{C}})\,|\,z\,\bar{z}^{t}=I_{n}\\}.$

For $1\leq r,s\leq n$, we shall by $E_{rs}\in{\mathbb{R}}^{n\times n}$ denote
the matrix given by

$(E_{rs})_{\alpha\beta}=\delta_{r\alpha}\delta_{s\beta}$

and for $r<s$ let $X_{rs},Y_{rs}$ be the symmetric and skew-symmetric matrices

$X_{rs}=\frac{1}{\sqrt{2}}(E_{rs}+E_{sr}),\ \
Y_{rs}=\frac{1}{\sqrt{2}}(E_{rs}-E_{sr}),$

respectively. Further, let $D_{r}$ be the diagonal elements with
$D_{t}=E_{tt}$. By $\mathcal{B}^{+}$ we denote the orthonormal basis for the
Lie algebra $\mathfrak{u}(n)$ satisfying

$\mathcal{B}^{+}=\\{Y_{rs},i\,X_{rs}\,|\,1\leq r<s\leq
n\\}\cup\\{i\,D_{t}\,|\,t=1,2,\dots,n\\}$

and $\mathcal{B}^{-}=i\,\mathcal{B}^{+}$. Then
$\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ is an orthonormal basis for
$\mathfrak{gl}_{n}({\mathbb{C}})$ such that $g(Z,Z)=1$ if
$Z\in\mathcal{B}^{+}$ and $g(Z,Z)=-1$ for $Z\in\mathcal{B}^{-}$. For later use
we define the two standard matrices $J_{n}$ and $I_{pq}$ by

$J_{n}=\begin{bmatrix}0&I_{n}\\\ -I_{n}&0\end{bmatrix}\ \ \text{and}\ \
I_{pq}=\begin{bmatrix}-I_{p}&0\\\ 0&I_{q}\end{bmatrix}.$

Let $G$ be a subgroup of $\text{\bf GL}_{n}({\mathbb{C}})$ with Lie algebra
$\mathfrak{g}$ inheriting the induced left-invariant semi-Riemannian metric
from $g$. Then employing the Koszul formula for the Levi-Civita connection
$\nabla$ on $(G,g)$, we see that for all $Z,W\in\mathfrak{g}$ we have

$\displaystyle g(\hbox{$\nabla$\kern-3.00003pt\lower
4.30554pt\hbox{$Z$}\kern-1.00006pt{$Z$}},W)$ $\displaystyle=$ $\displaystyle
g([W,Z],Z)$ $\displaystyle=$
$\displaystyle-\mathfrak{R}\mathfrak{e}\operatorname{trace}(WZ-ZW)Z$
$\displaystyle=$
$\displaystyle-\mathfrak{R}\mathfrak{e}\operatorname{trace}W(ZZ-ZZ)$
$\displaystyle=$ $\displaystyle 0.$

If $Z\in\mathfrak{g}$ is a left-invariant vector field on $G$ and
$\phi:U\to{\mathbb{C}}$ is a local complex-valued function on $G$ then the
first and second order derivatives satisfy

$Z(\phi)(p)=\frac{d}{ds}\bigl{(}\phi(p\cdot\exp(sZ))\bigr{)}\Big{|}_{s=0},$
$Z^{2}(\phi)(p)=\frac{d^{2}}{ds^{2}}\bigr{(}\phi(p\cdot\exp(sZ))\bigr{)}\Big{|}_{s=0}.$

This implies that the tension field $\tau$ and the conformality operator
$\kappa$ on $G$ fulfill

$\displaystyle\tau(\phi)$ $\displaystyle=$
$\displaystyle\sum_{Z\in\mathcal{B}_{\mathfrak{g}}}g(Z,Z)\cdot\bigl{(}Z^{2}(\phi)-\hbox{$\nabla$\kern-3.00003pt\lower
4.30554pt\hbox{$Z$}\kern-1.00006pt{$Z$}}(\phi)\bigr{)}$ $\displaystyle=$
$\displaystyle\sum_{Z\in\mathcal{B}_{\mathfrak{g}}}g(Z,Z)\cdot Z^{2}(\phi)$
$\displaystyle\kappa(\phi,\psi)$ $\displaystyle=$
$\displaystyle\sum_{Z\in\mathcal{B}_{\mathfrak{g}}}g(Z,Z)\cdot Z(\phi)\cdot
Z(\psi),$

where $\mathcal{B}_{\mathfrak{g}}$ is any orthonormal basis for the Lie
algebra $\mathfrak{g}$.

The restriction of the semi-Riemannian metric $g$ on $\text{\bf
GL}_{n}({\mathbb{C}})$ to its maximal compact subgroup $\text{\bf U}(n)$ is
its standard Riemannian metric. For this we have the following result, see
Lemma 5.1 of [10].

######  Lemma 5.1.

Let $z_{j\alpha}:\text{\bf U}(n)\to\mathbb{C}$ be the complex-valued matrix
elements of the standard representation of the unitary group $\text{\bf
U}(n)$. Then the tension field $\tau$ and the conformality operator $\kappa$
on $\text{\bf U}(n)$ satisfy the following relations

$\displaystyle\tau(z_{j\alpha})$ $\displaystyle=$ $\displaystyle-\,n\cdot
z_{j\alpha},$ $\displaystyle\kappa(z_{j\alpha},z_{k\beta})$ $\displaystyle=$
$\displaystyle-\,z_{k\alpha}\cdot z_{j\beta}.$

With this at hand we yield the following statement.

######  Proposition 5.2.

Let $z_{j\alpha}:\text{\bf GL}_{n}({\mathbb{C}})\to\mathbb{C}$ be the complex-
valued matrix elements of the standard representation of the general linear
group $\text{\bf GL}_{n}({\mathbb{C}})$. Then the tension field $\tau$ and the
conformality operator $\kappa$ on $\text{\bf GL}_{n}({\mathbb{C}})$ fulfill
the following relations

$\displaystyle\tau(z_{j\alpha})$ $\displaystyle=$ $\displaystyle-2\,n\cdot
z_{j\alpha},$ $\displaystyle\kappa(z_{j\alpha},z_{k\beta})$ $\displaystyle=$
$\displaystyle-\,2\cdot z_{k\alpha}\cdot z_{j\beta}.$

###### Proof.

This is an immediate consequence of Lemma 5.1 and how the semi-Riemannian
metric is defined on the complex Lie algebra

$\mathfrak{gl}_{n}({\mathbb{C}})=\mathfrak{u}(n)\oplus i\cdot\mathfrak{u}(n).$

∎

######  Theorem 5.3.

Let $v$ be a non-zero element of $\mathbb{C}^{n}$. Then the complex
$n$-dimensional vector space

$\mathcal{E}_{v}=\\{\phi_{a}:\text{\bf
GL}_{n}({\mathbb{C}})\to\mathbb{C}\,|\,\phi_{a}(z)=\operatorname{trace}(v^{t}az^{t}),\
a\in\mathbb{C}^{n}\\}$

is an eigenfamily on $\text{\bf GL}_{n}({\mathbb{C}})$ such that for all
$\phi,\psi\in\mathcal{E}_{v}$ we have

$\tau(\phi)=-\,2\,n\cdot\phi\ \ \text{and}\ \
\kappa(\phi,\psi)=-\,2\cdot\phi\cdot\psi.$

###### Proof.

This result can be proven in exactly the same way as Theorem 5.2 presented in
[10]. ∎

##  6\. The Semi-Riemannian Lie Group $\text{\bf GL}_{n}({\mathbb{R}})$

The real general linear group $\text{\bf GL}_{n}({\mathbb{R}})$ of invertible
$n\times n$ matrices is given by

$\text{\bf GL}_{n}({\mathbb{R}})=\\{x\in{\mathbb{R}}^{n\times n}\,|\,\det
x\neq 0\\}.$

The Lie algebra $\mathfrak{gl}_{n}({\mathbb{C}})$ of the complex general
linear group $\text{\bf GL}_{n}({\mathbb{C}})$ has a natural orthogonal
decomposition

$\mathfrak{gl}_{n}({\mathbb{C}})=\mathfrak{gl}_{n}({\mathbb{R}})\oplus
i\cdot\mathfrak{gl}_{n}({\mathbb{R}}),$

where $\mathfrak{gl}_{n}({\mathbb{R}})$ is the Lie algebra of $\text{\bf
GL}_{n}({\mathbb{R}})$ consisting of the real $n\times n$ matrices.

######  Proposition 6.1.

Let $x_{j\alpha}:\text{\bf GL}_{n}({\mathbb{R}})\to{\mathbb{R}}$ be the real-
valued matrix elements of the standard representation of the general linear
group $\text{\bf GL}_{n}({\mathbb{R}})$. Then the tension field $\tau$ and the
conformality operator $\kappa$ on $\text{\bf GL}_{n}({\mathbb{R}})$ satisfy
the following relations

$\displaystyle\tau(x_{j\alpha})$ $\displaystyle=$ $\displaystyle-\,n\cdot
x_{j\alpha},$ $\displaystyle\kappa(x_{j\alpha},x_{k\beta})$ $\displaystyle=$
$\displaystyle-\,x_{k\alpha}\cdot x_{j\beta}.$

###### Proof.

This is an immediate consequence of Proposition 5.2 and how the semi-
Riemannian metric is defined on the complex Lie algebra

$\mathfrak{gl}_{n}({\mathbb{C}})=\mathfrak{gl}_{n}({\mathbb{R}})\oplus
i\cdot\mathfrak{gl}_{n}({\mathbb{R}}).$

∎

As an immediate consequence of Proposition 6.1 we have the following result.

######  Theorem 6.2.

Let $v$ be a non-zero element of ${\mathbb{C}}^{n}$. Then the complex
$n$-dimensional vector space

$\mathcal{E}_{v}=\\{\phi_{a}:\text{\bf
GL}_{n}({\mathbb{R}})\to{\mathbb{C}}\,|\,\phi_{a}(x)=\operatorname{trace}(v^{t}ax^{t}),\
a\in\mathbb{C}^{n}\\}$

is an eigenfamily on $\text{\bf GL}_{n}({\mathbb{R}})$ such that for all
$\phi,\psi\in\mathcal{E}_{v}$ we have

$\tau(\phi)=-\,n\cdot\phi\ \ \text{and}\ \
\kappa(\phi,\psi)=-\,\phi\cdot\psi.$

###### Proof.

This is a direct consequence of Proposition 6.1 and Theorem 5.3. ∎

##  7\. The Semi-Riemannian Lie Group $\text{\bf GL}_{n}({\mathbb{H}})$

In this section we consider the quaternionic general linear group $\text{\bf
GL}_{n}({\mathbb{H}})$. Its standard complex representation $\pi:\text{\bf
GL}_{n}({\mathbb{H}})\to\text{\bf GL}_{2n}({\mathbb{C}})$ is given by

$\pi:(z+jw)\mapsto g=\begin{bmatrix}z_{11}&\dots&z_{1n}&w_{11}&\dots&w_{1n}\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
z_{n1}&\dots&z_{nn}&w_{n1}&\dots&w_{nn}\\\
-\bar{w}_{11}&\dots&-\bar{w}_{1n}&\bar{z}_{11}&\dots&\bar{z}_{1n}\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
-\bar{w}_{n1}&\dots&-\bar{w}_{nn}&\bar{z}_{n1}&\dots&\bar{z}_{nn}\\\
\end{bmatrix}.$

The Lie algebra $\mathfrak{gl}_{n}({\mathbb{H}})$ of $\text{\bf
GL}_{n}({\mathbb{H}})$ clearly satisfies

$\mathfrak{gl}_{n}({\mathbb{H}})=\Big{\\{}\begin{bmatrix}Z&W\\\
-\bar{W}&\bar{Z}\end{bmatrix}\Big{|}\
Z,W\in\mathfrak{gl}_{n}({\mathbb{C}})\Big{\\}}.$

As a subgroup of $\text{\bf GL}_{2n}({\mathbb{C}})$ the quaternionic general
linear group $\text{\bf GL}_{n}({\mathbb{H}})$ inherits its natural semi-
Riemannian metric $g$ induced by the semi-Euclidean inner product
$\mathfrak{gl}_{n}({\mathbb{H}})\times\mathfrak{gl}_{n}({\mathbb{H}})\to\mathfrak{gl}_{n}({\mathbb{H}})$
on $\mathfrak{gl}_{n}({\mathbb{H}})$ given by

$g(Z,W)=-\mathfrak{R}\mathfrak{e}\operatorname{trace}(Z\cdot W).$

For the Lie algebra $\mathfrak{gl}_{n}({\mathbb{H}})$ we have the orthogonal
splitting

$\mathfrak{gl}_{n}({\mathbb{H}})=\mathfrak{gl}^{+}_{n}({\mathbb{H}})\oplus\mathfrak{gl}^{-}_{n}({\mathbb{H}}),$

where

$\mathfrak{gl}^{+}_{n}({\mathbb{H}})=\mathfrak{sp}(n)=\Big{\\{}\begin{bmatrix}Z&W\\\
-\bar{W}&\bar{Z}\end{bmatrix}\in{\mathbb{C}}^{2n\times
2n}\,\Big{|}\,Z+\bar{Z}^{t}=0,W-W^{t}=0\Big{\\}}.$

By $\mathcal{B}^{+}$ we denote the following orthonormal basis for the Lie
algebra $\mathfrak{sp}(n)$ of the quaternionic unitary group $\text{\bf
Sp}(n)$ which is the maximal compact subgroup of $\text{\bf
GL}_{n}({\mathbb{H}})$. This satisfies

$\displaystyle\mathcal{B}^{+}$ $\displaystyle=$
$\displaystyle\Big{\\{}\frac{1}{\sqrt{2}}\begin{bmatrix}0&iX_{rs}\\\
iX_{rs}&0\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}0&X_{rs}\\\
-X_{rs}&0\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}iX_{rs}&0\\\
0&-iX_{rs}\end{bmatrix},$
$\displaystyle\qquad\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{rs}&0\\\
0&Y_{rs}\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}0&D_{t}\\\
-D_{t}&0\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}iD_{t}&0\\\
0&-iD_{t}\end{bmatrix},$
$\displaystyle\qquad\qquad\frac{1}{\sqrt{2}}\begin{bmatrix}0&iD_{t}\\\
iD_{t}&0\end{bmatrix}\Big{|}\,1\leq r<s\leq n,\ 1\leq t\leq n\Big{\\}}\,.$

For the orthogonal complement $\mathfrak{gl}^{-}_{n}({\mathbb{H}})$ of
$\mathfrak{sp}(n)$ in $\mathfrak{gl}_{n}({\mathbb{H}})$ we have the
orthonormal basis

$\displaystyle\mathcal{B}^{-}$ $\displaystyle=$
$\displaystyle\Big{\\{}\frac{1}{\sqrt{2}}\begin{bmatrix}0&Y_{rs}\\\
-Y_{rs}&0\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}0&iY_{rs}\\\
iY_{rs}&0\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}iY_{rs}&0\\\
0&-iY_{rs}\end{bmatrix},$
$\displaystyle\qquad\frac{1}{\sqrt{2}}\begin{bmatrix}X_{rs}&0\\\
0&X_{rs}\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}D_{t}&0\\\
0&D_{t}\end{bmatrix}\Big{|}\,1\leq r,s\leq n,\ 1\leq t\leq n\Big{\\}}\,.$

Then $\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ is an orthonormal basis
for $\mathfrak{gl}_{n}({\mathbb{H}})$ such that $g(Z,Z)=1$ if
$Z\in\mathcal{B}^{+}$ and $g(Z,Z)=-1$ if $Z\in\mathcal{B}^{-}.$

With this at our disposal we can now prove the following statement.

######  Proposition 7.1.

Let $z_{j\alpha},w_{j\alpha}:\text{\bf GL}_{n}({\mathbb{H}})\to{\mathbb{C}}$
be the complex-valued matrix elements of the standard representation of the
quaternionic general linear group $\text{\bf GL}_{n}({\mathbb{H}})$. Then the
tension field $\tau$ and the conformality operator $\kappa$ on $\text{\bf
GL}_{n}({\mathbb{H}})$ satisfy the following relations

$\tau(z_{j\alpha})=-\,2n\cdot z_{j\alpha},\ \ \tau(w_{j\alpha})=-\,2n\cdot
w_{j\alpha},$ $\kappa(z_{j\alpha},z_{k\beta})=-\,z_{k\alpha}\cdot z_{j\beta},\
\ \kappa(w_{j\alpha},w_{k\beta})=-\,w_{k\alpha}\cdot w_{j\beta},$
$\kappa(z_{j\alpha},w_{k\beta})=-\,z_{k\alpha}\cdot w_{j\beta}.$

###### Proof.

For the tension field $\tau$ on $\text{\bf GL}_{n}({\mathbb{H}})$ we have

$\displaystyle\tau(z_{j\alpha})$ $\displaystyle=$
$\displaystyle\sum_{Z\in\mathcal{B}}g(Z,Z)\cdot Z^{2}(z_{j\alpha})$
$\displaystyle=$
$\displaystyle\sum_{Z\in\mathcal{B}^{+}}Z^{2}(z_{j\alpha})-\sum_{Z\in\mathcal{B}^{-}}Z^{2}(z_{j\alpha})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,e_{j}z\Big{\\{}3\sum_{r<s}\begin{bmatrix}-X_{rs}^{2}&0\\\
0&-X_{rs}^{2}\end{bmatrix}+\sum_{r<s}\begin{bmatrix}Y_{rs}^{2}&0\\\
0&Y_{rs}^{2}\end{bmatrix}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+3\sum_{t=1}^{n}\begin{bmatrix}-D_{t}^{2}&0\\\
0&-D_{t}^{2}\end{bmatrix}\Big{\\}}e_{\alpha}^{t}$
$\displaystyle-\frac{1}{2}\,e_{j}z\Big{\\{}\sum_{r<s}\begin{bmatrix}X_{rs}^{2}&0\\\
0&X_{rs}^{2}\end{bmatrix}+3\sum_{r<s}\begin{bmatrix}-Y_{rs}^{2}&0\\\
0&-Y_{rs}^{2}\end{bmatrix}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad+\sum_{t=1}^{n}\begin{bmatrix}D_{t}^{2}&0\\\
0&D_{t}^{2}\end{bmatrix}\Big{\\}}e_{\alpha}^{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,e_{j}z\Big{\\{}-4\sum_{r<s}\begin{bmatrix}X_{rs}^{2}&0\\\
0&X_{rs}^{2}\end{bmatrix}+4\sum_{r<s}\begin{bmatrix}Y_{rs}^{2}&0\\\
0&Y_{rs}^{2}\end{bmatrix}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad-4\sum_{t=1}^{n}\begin{bmatrix}D_{t}^{2}&0\\\
0&D_{t}^{2}\end{bmatrix}\Big{\\}}e_{\alpha}^{t}$ $\displaystyle=$
$\displaystyle-2\,e_{j}z\begin{bmatrix}n\cdot I_{n}&0\\\ 0&n\cdot
I_{n}\end{bmatrix}e_{\alpha}^{t}$ $\displaystyle=$ $\displaystyle-2n\cdot
z_{j\alpha}.$

For the conformal operation $\kappa$ on $\text{\bf GL}_{n}({\mathbb{H}})$ we
similarly we yield

$\displaystyle\kappa(z_{j\alpha},z_{k\beta})$ $\displaystyle=$
$\displaystyle\sum_{Z\in\mathcal{B}}g(Z,Z)\cdot Z(z_{j\alpha})\cdot
Z(z_{k\beta})$ $\displaystyle=$
$\displaystyle\sum_{Z\in\mathcal{B}^{+}}Z(z_{j\alpha})\cdot
Z(z_{k\beta})-\sum_{Z\in\mathcal{B}^{-}}Z(z_{j\alpha})\cdot Z(z_{k\beta})$
$\displaystyle=$ $\displaystyle
e_{j}z\Big{\\{}\sum_{Z\in\mathcal{B}^{+}}Z\begin{bmatrix}E_{\alpha\beta}&0\\\
0&0\end{bmatrix}Z^{t}-\sum_{Z\in\mathcal{B}^{+}}Z\begin{bmatrix}E_{\alpha\beta}&0\\\
0&0\end{bmatrix}Z^{t}\Big{\\}}z^{t}e_{k}^{t}$ $\displaystyle=$ $\displaystyle
e_{j}z\Big{\\{}-\sum_{r<s}\begin{bmatrix}X_{rs}E_{\alpha\beta}X_{rs}&0\\\
0&0\end{bmatrix}+\sum_{r<s}\begin{bmatrix}Y_{rs}E_{\alpha\beta}Y_{rs}^{t}&0\\\
0&0\end{bmatrix}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\sum_{t=1}^{n}\begin{bmatrix}D_{t}E_{\alpha\beta}D_{t}&0\\\
0&0\end{bmatrix}\Big{\\}}z^{t}e_{k}^{t}$ $\displaystyle=$ $\displaystyle-
e_{j}z(E_{\beta\alpha})z^{t}e_{k}^{t}$ $\displaystyle=$ $\displaystyle-
z_{k\alpha}\cdot z_{j\beta}.$

The other identities can be proven in exactly the same way. ∎

######  Theorem 7.2.

Let $u,v$ be a non-zero elements of ${\mathbb{C}}^{n}$. Then the complex
$2n$-dimensional vector space

$\mathcal{E}_{uv}=\\{\phi_{ab}:\text{\bf GL}_{n}({\mathbb{H}})\to{\mathbb{C}}\
|\ \phi_{ab}(g)=\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t}),\
a,b\in{\mathbb{C}}^{n}\\}$

is an eigenfamily on $\text{\bf GL}_{n}({\mathbb{H}})$ such that for all
$\phi,\psi\in\mathcal{E}_{uv}$ we have

$\tau(\phi)=-\,2n\cdot\phi\ \ \text{and}\ \
\kappa(\phi,\psi)=-\,\phi\cdot\psi.$

##  8\. The Semi-Riemannian Lie group $\text{\bf SL}_{n}({\mathbb{C}})$

In this section we construct eigenfamilies on the semisimple non-compact
complex special linear group $\text{\bf SL}_{n}({\mathbb{C}})=\\{z\in\text{\bf
GL}_{n}({\mathbb{C}})\,|\,\det z=1\\}$ equipped with its semi-Riemannian
metric inherited from $\text{\bf GL}_{n}({\mathbb{C}})$. For the Lie algebra

$\mathfrak{sl}_{n}({\mathbb{C}})=\\{Z\in\mathfrak{gl}_{n}({\mathbb{C}})\,|\,\operatorname{trace}Z=0\\}$

we have the orthogonal decomposition
$\mathfrak{sl}_{n}({\mathbb{C}})=\mathfrak{su}(n)\oplus
i\cdot\mathfrak{su}(n)$. Here

$\mathfrak{su}(n)=\\{Z\in\mathfrak{u}(n)\,|\,\operatorname{trace}Z=0\\}$

is the Lie algebra of the special unitary group

$\text{\bf SU}(n)=\\{z\in\text{\bf U}(n)\,|\,\det z=1\\}$

which is the maximal compact subgroup of $\text{\bf SL}_{n}({\mathbb{C}})$.

######  Lemma 8.1.

Let $z_{j\alpha}:\text{\bf SU}(n)\to\mathbb{C}$ be the complex-valued matrix
elements of the standard representation of the special unitary group
$\text{\bf SU}(n)$. Then the tension field $\tau$ and the conformality
operator $\kappa$ on $\text{\bf SU}(n)$ satisfy the following relations

$\displaystyle\tau(z_{j\alpha})$ $\displaystyle=$
$\displaystyle-\,\frac{(n^{2}-1)}{n}\cdot z_{j\alpha},$
$\displaystyle\kappa(z_{j\alpha},z_{k\beta})$ $\displaystyle=$
$\displaystyle-\,(z_{k\alpha}\cdot z_{j\beta}-\frac{1}{n}\cdot
z_{j\alpha}\cdot z_{k\beta}).$

###### Proof.

For the Lie algebra $\mathfrak{u}(n)$ of the unitary group $\text{\bf U}(n)$
we have the orthogonal splitting

$\mathfrak{u}(n)=\mathfrak{su}(n)\oplus\mathfrak{l},$

where $\mathfrak{l}$ is the real line generated by the unit vector
$E_{n}=i\,I_{n}/\sqrt{n}$. Hence the tension field $\hat{\tau}$ on the unitary
group $\text{\bf U}(n)$ satisfies

$\displaystyle\hat{\tau}(\phi)$ $\displaystyle=$
$\displaystyle\tau(\phi)+E_{n}^{2}(\phi),$

so we have

$\tau(z_{j\alpha})=\hat{\tau}(z_{j\alpha})-E_{n}^{2}(z_{j\alpha})=-(n\cdot
z_{j\alpha}-\frac{1}{n}\cdot z_{j\alpha})=-\,\frac{(n^{2}-1)}{n}\cdot
z_{j\alpha}.$

For the conformality operator $\hat{\kappa}$ on $\text{\bf U}(n)$ we
similarily yield

$\displaystyle\hat{\kappa}(\phi,\psi)$ $\displaystyle=$
$\displaystyle\kappa(\phi,\psi)+E_{n}(\phi)\cdot E_{n}(\psi).$

Hence

$\displaystyle\kappa(z_{j\alpha},z_{k\beta})$ $\displaystyle=$
$\displaystyle\hat{\kappa}(z_{j\alpha},z_{k\beta})-E_{n}(z_{j\alpha})\cdot
E_{n}(z_{k\beta})$ $\displaystyle=$ $\displaystyle-\,(z_{k\alpha}\cdot
z_{j\beta}-\frac{1}{n}\cdot z_{j\alpha}\cdot z_{k\beta}).$

∎

For the special linear group $\text{\bf SL}_{n}({\mathbb{C}})$ we have the
following statement.

######  Proposition 8.2.

Let $z_{j\alpha}:\text{\bf SL}_{n}({\mathbb{C}})\to\mathbb{C}$ be the complex-
valued matrix elements of the standard representation of the special linear
group $\text{\bf SL}_{n}({\mathbb{C}})$. Then the tension field $\tau$ and the
conformality operator $\kappa$ on $\text{\bf SL}_{n}({\mathbb{C}})$ satisfy
the following relations

$\displaystyle\tau(z_{j\alpha})$ $\displaystyle=$
$\displaystyle-\,\frac{2\,(n^{2}-1)}{n}\cdot z_{j\alpha},$
$\displaystyle\kappa(z_{j\alpha},z_{k\beta})$ $\displaystyle=$
$\displaystyle-\,2\,(z_{k\alpha}\cdot z_{j\beta}-\frac{1}{n}\cdot
z_{j\alpha}\cdot z_{k\beta}).$

###### Proof.

This is an immediate consequence of Lemma 8.1 and how the semi-Riemannian
metric is defined on the complex Lie algebra

$\mathfrak{sl}_{n}({\mathbb{C}})=\mathfrak{su}(n)\oplus
i\cdot\mathfrak{su}(n).$

∎

Let $P,Q:\text{\bf SL}_{n}({\mathbb{C}})\to{\mathbb{C}}$ be homogeneous
polynomials of the matrix elemens $z_{j\alpha}:\text{\bf
SL}_{n}({\mathbb{C}})\to{\mathbb{C}}$ of degree one i.e. of the form

$P(z)=\operatorname{trace}(A\cdot
z^{t})=\sum_{j,\alpha=1}^{n}a_{j\alpha}z_{j\alpha},\ \
Q(z)=\operatorname{trace}(B\cdot
z^{t})=\sum_{k,\beta}^{n}b_{k\beta}z_{k\beta}$

for some $A,B\in{\mathbb{C}}^{n\times n}$. As a direct consequence of
Proposition 8.2 we see that

$\tau(P)=-\,\frac{2\,(n^{2}-1)}{n}\cdot P,\ \
\tau(Q)=-\,\frac{2\,(n^{2}-1)}{n}\cdot Q$

and

$\displaystyle\kappa(P,Q)+\frac{2\,(n-1)}{n}\cdot P\,Q$ $\displaystyle=$
$\displaystyle\sum_{j,\alpha,k,\beta=1}^{n}a_{j\alpha}b_{k\beta}\,\kappa(z_{j\alpha},z_{k\beta})+\frac{2\,(n-1)}{n}\sum_{j,\alpha,k,\beta=1}^{n}a_{j\alpha}b_{k\beta}\,z_{j\alpha}z_{k\beta}$
$\displaystyle=$
$\displaystyle-\,2\sum_{j,\alpha,k,\beta=1}^{n}a_{j\alpha}b_{k\beta}\,z_{j\beta}z_{k\alpha}+\frac{2}{n}\sum_{j,\alpha,k,\beta=1}^{n}a_{j\alpha}b_{k\beta}\,z_{j\alpha}z_{k\beta}$
$\displaystyle+\frac{2\,(n-1)}{n}\sum_{j,\alpha,k,\beta=1}^{n}a_{j\alpha}b_{k\beta}\,z_{j\alpha}z_{k\beta}$
$\displaystyle=$ $\displaystyle
2\sum_{j,\alpha,k,\beta=1}^{n}\left(a_{j\alpha}b_{k\beta}\,z_{j\alpha}z_{k\beta}-a_{j\alpha}b_{k\beta}\,z_{j\beta}z_{k\alpha}\right)$
$\displaystyle=$ $\displaystyle
2\sum_{j,\alpha,k,\beta=1}^{n}(a_{j\alpha}b_{k\beta}-a_{k\alpha}b_{j\beta})\,z_{j\alpha}z_{k\beta}.$

######  Theorem 8.3.

Let $v$ be a non-zero element of $\mathbb{C}^{n}$. Then the complex
$n$-dimensional vector space

$\mathcal{E}_{v}=\\{\phi_{a}:\text{\bf
SL}_{n}({\mathbb{C}})\to\mathbb{C}\,|\,\phi_{a}(z)=\operatorname{trace}(v^{t}az^{t}),\
a\in\mathbb{C}^{n}\\}$

is an eigenfamily on $\text{\bf SL}_{n}({\mathbb{C}})$ such that for all
$\phi,\psi\in\mathcal{E}_{v}$ we have

$\tau(\phi)=-\,\frac{2\,(n^{2}-1)}{n}\cdot\phi,\ \
\kappa(\phi,\psi)=-\frac{2\,(n-1)}{n}\cdot\phi\cdot\psi.$

###### Proof.

Assume that $a,b\in\mathbb{C}^{n}$ and define $A=v^{t}a$ and $B=v^{t}b\,.$ By
construction any two columns of the matrices $A$ and $B$ are linearly
dependent. This means that for all $1\leq j,\alpha,k,\beta\leq n$

$\det\begin{bmatrix}a_{j\alpha}&b_{j\beta}\\\
a_{k\alpha}&b_{k\beta}\end{bmatrix}=a_{j\alpha}b_{k\beta}-a_{k\alpha}b_{j\beta}=0.$

The statement now follows from the calculation above. ∎

##  9\. The Semi-Riemannian Lie Group $\text{\bf SL}_{n}({\mathbb{R}})$

In this section we construct eigenfamilies on the semisimple non-compact
special linear group $\text{\bf SL}_{n}({\mathbb{R}})$ equipped with its semi-
Riemannian metric inherited from $\text{\bf GL}_{n}({\mathbb{C}})$. The
special linear group $\text{\bf SL}_{n}({\mathbb{R}})$ is the subgroup of
$\text{\bf GL}_{n}({\mathbb{R}})$ satisfying

$\text{\bf SL}_{n}({\mathbb{R}})=\\{x\in\text{\bf
GL}_{n}({\mathbb{R}})\,|\,\det x=1\\}$

with Lie algebra

$\mathfrak{sl}_{n}({\mathbb{R}})=\\{X\in\mathfrak{gl}_{n}({\mathbb{R}})\,|\,\operatorname{trace}X=0\\}.$

######  Proposition 9.1.

Let $x_{j\alpha}:\text{\bf SL}_{n}({\mathbb{R}})\to\mathbb{R}$ be the real-
valued matrix elements of the standard representation of the special linear
group $\text{\bf SL}_{n}({\mathbb{R}})$. Then the tension field $\tau$ and the
conformality operator $\kappa$ on $\text{\bf SL}_{n}({\mathbb{R}})$ satisfy
the following relations

$\displaystyle\tau(x_{j\alpha})$ $\displaystyle=$
$\displaystyle-\,\frac{(n^{2}-1)}{n}\cdot x_{j\alpha},$
$\displaystyle\kappa(x_{j\alpha},x_{k\beta})$ $\displaystyle=$
$\displaystyle-\,(x_{j\beta}\cdot x_{k\alpha}-\frac{1}{n}\cdot
x_{j\alpha}\cdot x_{k\beta}).$

###### Proof.

The Lie algebra $\mathfrak{sl}_{n}({\mathbb{C}})$ of the complex special
linear group $\text{\bf SL}_{n}({\mathbb{C}})$ is the complexification of
$\mathfrak{sl}_{n}({\mathbb{R}})$ and we have the orthogonal decomposition

$\mathfrak{sl}_{n}({\mathbb{C}})=\mathfrak{sl}_{n}({\mathbb{R}})\oplus
i\cdot\mathfrak{sl}_{n}({\mathbb{R}}).$

Hence the statement is an immediate consequence of Lemma 8.2. ∎

The next result is a direct consequence of Theorem 8.3, Propositions 8.2 and
9.1.

######  Theorem 9.2.

Let $v$ be a non-zero element of ${\mathbb{C}}^{n}$. Then the complex
$n$-dimensional vector space

$\mathcal{E}_{v}=\\{\phi_{a}:\text{\bf
SL}_{n}({\mathbb{R}})\to{\mathbb{C}}\,|\,\phi_{a}(x)=\operatorname{trace}(v^{t}ax^{t}),\
a\in{\mathbb{C}}^{n}\\}$

is an eigenfamily on $\text{\bf SL}_{n}({\mathbb{R}})$ such that for all
$\phi,\psi\in\mathcal{E}_{v}$ we have

$\tau(\phi)=-\frac{(n^{2}-1)}{n}\cdot\phi,\quad\kappa(\phi,\psi)=-\frac{(n-1)}{n}\cdot\phi\cdot\psi\,.$

Lie group Eigenfunctions $\phi$ $\lambda$ $\mu$ Conditions $\text{\bf
GL}_{n}({\mathbb{C}})$ $\operatorname{trace}(v^{t}az^{t})$ ${-\,2\,n}$ $-\,2$
$a\in{\mathbb{C}}^{n}$ $\text{\bf GL}_{n}({\mathbb{R}})$
$\operatorname{trace}(v^{t}ax^{t})$ ${-\,n}$ $-\,1$ $a\in{\mathbb{C}}^{n}$
$\text{\bf GL}_{n}({\mathbb{H}})$
$\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t})$ $-\,2\,n$ $-\,1$
$a,b\in{\mathbb{C}}^{n}$ $\text{\bf SL}_{n}({\mathbb{C}})$
$\operatorname{trace}(v^{t}az^{t})$ $-\frac{2(n^{2}-1)}{n}$
$-\frac{2(n-1)}{n}$ $a\in{\mathbb{C}}^{n}$ $\text{\bf SL}_{n}({\mathbb{R}})$
$\operatorname{trace}(v^{t}ax^{t})$ $-\frac{(n^{2}-1)}{n}$ $-\frac{(n-1)}{n}$
$a\in{\mathbb{C}}^{n}$ $\text{\bf SL}_{n}({\mathbb{H}})$
$\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t})$ $-\,\frac{(4n^{2}-1)}{2n}$
$-\,\frac{(2n-1)}{2n}$ $a,b\in{\mathbb{C}}^{n}$

Table 1. Eigenfunctions on classical non-compact Lie groups.

##  10\. The Semi-Riemannian Lie Group $\text{\bf
SL}_{n}({\mathbb{H}})\cong\text{\bf SU}^{*}(2n)$

In this section we construct eigenfamilies on the semisimple non-compact
quaternionic special linear group $\text{\bf SL}_{n}({\mathbb{H}})$. This can
be realised as

$\text{\bf SU}^{*}(2n)=\Big{\\{}\begin{bmatrix}z&w\\\
-\bar{w}&\bar{z}\end{bmatrix}\in\text{\bf
GL}_{2n}({\mathbb{C}})\,\Big{|}\,(z+j\,w)\in\text{\bf
SL}_{n}({\mathbb{H}})\Big{\\}},$

with Lie algebra

$\mathfrak{su}^{*}(2n)=\Big{\\{}\begin{bmatrix}Z&W\\\
-\bar{W}&\bar{Z}\end{bmatrix}\in\mathfrak{gl}_{n}({\mathbb{H}})\,\Big{|}\,\mathfrak{R}\mathfrak{e}\operatorname{trace}Z=0\Big{\\}}.$

For the Lie algebra $\mathfrak{gl}_{n}({\mathbb{H}})$ of $\text{\bf
GL}_{n}({\mathbb{H}})$ we have the orthogonal decomposition

$\mathfrak{gl}_{n}({\mathbb{H}})=\mathfrak{su}^{*}(2n)\oplus\mathfrak{l}$

where $\mathfrak{l}$ is the real line in $\mathfrak{gl}_{n}({\mathbb{H}})$
generated by the unit vector $E_{2n}=I_{2n}/{\sqrt{2n}}$.

######  Proposition 10.1.

Let $z_{j\alpha},w_{k\beta}:\text{\bf SU}^{*}(2n)\to{\mathbb{C}}$ be the
complex-valued matrix elements of the standard representation of the Lie group
$\text{\bf SU}^{*}(2n)$. Then the tension field $\tau$ and the conformality
operator $\kappa$ on $\text{\bf SU}^{*}(2n)$ satisfy the following relations

$\tau(z_{j\alpha})=-\,\frac{(4n^{2}-1)}{2n}\cdot z_{j\alpha},\ \
\tau(w_{j\alpha})=-\,\frac{(4n^{2}-1)}{2n}\cdot w_{j\alpha},$
$\kappa(z_{j\alpha},z_{k\beta})=-\,(z_{k\alpha}\cdot
z_{j\beta}-\frac{1}{2n}\cdot z_{j\alpha}\cdot z_{k\beta}),$
$\kappa(z_{j\alpha},w_{k\beta})=-\,(z_{k\alpha}\cdot
w_{j\beta}-\frac{1}{2n}\cdot z_{j\alpha}\cdot w_{k\beta}),$
$\kappa(w_{j\alpha},w_{k\beta})=-\,(w_{k\alpha}\cdot
w_{j\beta}-\frac{1}{2n}\cdot w_{j\alpha}\cdot w_{k\beta}).$

###### Proof.

Let $\hat{\tau}$ and $\hat{\kappa}$ denote the tension field and the
conformality operator on $\text{\bf GL}_{n}({\mathbb{H}})$, respectively. Then
it follows from Proposition 7.1 and the orthogonal decomposition
$\mathfrak{gl}_{n}({\mathbb{H}})=\mathfrak{su}^{*}(2n)\oplus\mathfrak{l}$ that

$\displaystyle\tau(z_{j\alpha})$ $\displaystyle=$
$\displaystyle\hat{\tau}(z_{j\alpha})+E_{n}^{2}(z_{j\alpha})$ $\displaystyle=$
$\displaystyle-2n\cdot z_{j\alpha}+\frac{1}{2n}\cdot z_{j\alpha}$
$\displaystyle=$ $\displaystyle-\,\frac{(4n^{2}-1)}{2n}\cdot z_{j\alpha}.$

Similarly, we have

$\displaystyle\kappa(z_{j\alpha},z_{k\beta})$ $\displaystyle=$
$\displaystyle\hat{\kappa}(z_{j\alpha},z_{k\beta})+E_{n}(z_{j\alpha})\cdot
E_{n}(z_{k\beta})$ $\displaystyle=$ $\displaystyle-\,(z_{k\alpha}\cdot
z_{j\beta}-\frac{1}{2n}\cdot z_{j\alpha}\cdot z_{k\beta}).$

∎

######  Theorem 10.2.

Let $u,v$ be a non-zero elements of ${\mathbb{C}}^{n}$. Then the complex
$2n$-dimensional vector space

$\mathcal{E}_{uv}=\\{\phi_{ab}:\text{\bf SU}^{*}(2n)\to{\mathbb{C}}\ |\
\phi_{ab}(g)=\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t}),\
a,b\in{\mathbb{C}}^{n}\\}$

is an eigenfamily on $\text{\bf SU}^{*}(2n)$ such that for all
$\phi,\psi\in\mathcal{E}_{uv}$ we have

$\tau(\phi)=-\,\frac{(4n^{2}-1)}{2n}\cdot\phi\ \ \text{and}\ \
\kappa(\phi,\psi)=-\,\frac{(2n-1)}{2n}\cdot\phi\cdot\psi.$

###### Proof.

Here the statement can be proven in the exactly the same way as that of
Theorem 8.3. ∎

##  11\. The Semi-Riemannian Lie Group $\text{\bf SO}(n,{\mathbb{C}})$

The semisimple complex special orthogonal group $\text{\bf
SO}(n,{\mathbb{C}})$ is the subgroup of $\text{\bf GL}_{n}({\mathbb{C}})$
defined by

$\text{\bf SO}(n,{\mathbb{C}})=\\{z\in\text{\bf
SL}_{n}({\mathbb{C}})\,|\,z\cdot z^{t}=I_{n}\\}.$

Its Lie algebra

$\mathfrak{so}(n,{\mathbb{C}})=\\{Z\in\mathfrak{gl}_{n}({\mathbb{C}})\,|\,Z+Z^{t}=0\\}$

has the orthogonal decomposition

$\mathfrak{so}(n,{\mathbb{C}})=\mathfrak{so}(n)\oplus i\cdot\mathfrak{so}(n),$

where $\mathfrak{so}(n)$ is the Lie algebra of the special orthogonal group
$\text{\bf SO}(n)$ consisting of the real skew-symmetric $n\times n$ matrices.
The restriction of the semi-Riemannian metric $g$ on $\text{\bf
SO}(n,{\mathbb{C}})$ to its maximal compact subgroup $\text{\bf SO}(n)$ is its
standard Riemannian metric. For this we have the following result, see Lemma
4.1 of [10].

######  Lemma 11.1.

Let $x_{j\alpha}:\text{\bf SO}(n)\to\mathbb{R}$ be the real-valued matrix
elements of the standard representation of the special orthogonal group
$\text{\bf SO}(n)$. Then the tension field $\tau$ and the conformality
operator $\kappa$ on $\text{\bf SO}(n)$ satisfy the following relations

$\displaystyle\tau(x_{j\alpha})$ $\displaystyle=$
$\displaystyle-\,\frac{(n-1)}{2}\cdot x_{j\alpha},$
$\displaystyle\kappa(x_{j\alpha},x_{k\beta})$ $\displaystyle=$
$\displaystyle-\,\frac{1}{2}\cdot(x_{j\beta}\cdot
x_{k\alpha}-\delta_{jk}\cdot\delta_{\alpha\beta}).$

######  Proposition 11.2.

Let $z_{j\alpha}:\text{\bf SO}(n,{\mathbb{C}})\to\mathbb{C}$ be the complex-
valued matrix elements of the standard representation of the complex special
orthogonal group $\text{\bf SO}(n,{\mathbb{C}})$. Then the tension field
$\tau$ and the conformality operator $\kappa$ on $\text{\bf
SO}(n,{\mathbb{C}})$ satisfy the following relations

$\displaystyle\tau(z_{j\alpha})$ $\displaystyle=$ $\displaystyle-\,(n-1)\cdot
z_{j\alpha},$ $\displaystyle\kappa(z_{j\alpha},z_{k\beta})$ $\displaystyle=$
$\displaystyle-\,(z_{j\beta}\cdot
z_{k\alpha}-\delta_{jk}\cdot\delta_{\alpha\beta}).$

###### Proof.

This is an immediate consequence of Lemma 11.1 and the fact how the semi-
Riemannian metric is defined on the complex Lie algebra

$\mathfrak{so}(n,{\mathbb{C}})=\mathfrak{so}(n)\oplus i\cdot\mathfrak{so}(n).$

∎

######  Theorem 11.3.

Let $v\in{\mathbb{C}}^{n}$ be a non-zero isotropic element i.e. $(v,v)=0$,
then the complex $n$-dimensional vector space

$\mathcal{E}_{v}=\\{\phi_{a}:\text{\bf SO}(n,{\mathbb{C}})\to{\mathbb{C}}\ |\
\phi_{a}(z)=\operatorname{trace}(v^{t}az^{t}),\ a\in{\mathbb{C}}^{n}\\}$

is an eigenfamily on $\text{\bf SO}(n,{\mathbb{C}})$ such that for all
$\phi,\psi\in\mathcal{E}_{v}$ we have

$\tau(\phi)=-\,(n-1)\cdot\phi,\ \ \kappa(\phi,\psi)=-\,\phi\cdot\psi.$

###### Proof.

The Lie algebra $\mathfrak{so}(n,{\mathbb{C}})$ of the complex special linear
group $\text{\bf SO}(n,{\mathbb{C}})$ is the complexification of
$\mathfrak{so}(n)$ and we have the orthogonal decomposition

$\mathfrak{so}(n,{\mathbb{C}})=\mathfrak{so}(n)\oplus i\cdot\mathfrak{so}(n).$

Hence the statement is an immediate consequence of Theorem 4.3 of [10]. ∎

##  12\. The Semi-Riemannian Lie Group $\text{\bf Sp}(n,{\mathbb{C}})$

The semisimple complex symplectic group $\text{\bf Sp}(n,{\mathbb{C}})$ is the
subgroup of $\text{\bf SL}_{2n}({\mathbb{C}})$ with

$\text{\bf Sp}(n,{\mathbb{C}})=\\{z\in\text{\bf
SL}_{2n}({\mathbb{C}})\,|\,z\,J_{n}\,z^{t}=J_{n}\\}$

and Lie algebra

$\mathfrak{sp}(n,{\mathbb{C}})=\\{Z\in\mathfrak{sl}_{2n}({\mathbb{C}})\,|\,Z\,J_{n}+J_{n}\,Z^{t}=0\\}.$

The maximal compact subgroup of $\text{\bf Sp}(n,{\mathbb{C}})$ is the
quaternionic unitary group

$\text{\bf Sp}(n)=\\{z\in\text{\bf Sp}(n,{\mathbb{C}})\,|\
z\,\bar{z}^{t}=I_{2n}\\}$

with Lie algebra

$\mathfrak{sp}(n)=\\{Z\in\mathfrak{sp}(n,{\mathbb{C}})\
|\,Z+\bar{Z}^{t}=0\\}.$

For $\mathfrak{sp}(n,{\mathbb{C}})$ we have the orthogonal decomposition

$\mathfrak{sp}(n,{\mathbb{C}})=\mathfrak{sp}(n)\oplus i\cdot\mathfrak{sp}(n).$

The restriction of the semi-Riemannian metric $g$ on $\text{\bf
Sp}(n,{\mathbb{C}})$ to $\text{\bf Sp}(n)$ is its standard Riemannian metric.
For this we have the following result, see Lemma 6.1 of [10] and Lemma 6.1 of
[9].

######  Lemma 12.1.

Let $z_{j\alpha},w_{j\alpha}:\text{\bf Sp}(n)\to\mathbb{C}$ be the complex-
valued matrix elements of the standard representation of the quaternionic
unitary group $\text{\bf Sp}(n)$. Then the tension field $\tau$ and the
conformality operator $\kappa$ on $\text{\bf Sp}(n)$ satisfy the following
relations

$\tau(z_{j\alpha})=-\,\frac{2n+1}{2}\cdot z_{j\alpha},\ \
\tau(w_{k\beta})=-\,\frac{2n+1}{2}\cdot w_{k\beta},$
$\kappa(z_{j\alpha},z_{k\beta})=-\,\frac{1}{2}\cdot z_{k\alpha}\cdot
z_{j\beta},\ \ \kappa(w_{j\alpha},w_{k\beta})=-\,\frac{1}{2}\cdot
w_{k\alpha}\cdot w_{j\beta},$
$\kappa(z_{j\alpha},w_{k\beta})=-\,\frac{1}{2}\cdot z_{k\alpha}\cdot
w_{j\beta}.$

With this at hand we then yield the following result.

######  Proposition 12.2.

Let $z_{j\alpha},w_{j\alpha}:\text{\bf Sp}(n,{\mathbb{C}})\to\mathbb{C}$ be
the complex-valued matrix elements of the standard representation of the
quaternionic unitary group $\text{\bf Sp}(n)$. Then the tension field $\tau$
and the conformality operator $\kappa$ on $\text{\bf Sp}(n)$ satisfy the
following relations

$\tau(z_{j\alpha})=-\,(2n+1)\cdot z_{j\alpha},\ \
\tau(w_{k\beta})=-\,(2n+1)\cdot w_{k\beta},$
$\kappa(z_{j\alpha},z_{k\beta})=-\,z_{k\alpha}\cdot z_{j\beta},\ \
\kappa(w_{j\alpha},w_{k\beta})=-\,w_{k\alpha}\cdot w_{j\beta},$
$\kappa(z_{j\alpha},w_{k\beta})=-\,z_{k\alpha}\cdot w_{j\beta}.$

###### Proof.

This is an immediate consequence of Lemma 12.1 and how the semi-Riemannian
metric is defined on the complex Lie algebra

$\mathfrak{sp}(n,{\mathbb{C}})=\mathfrak{sp}(n)\oplus i\cdot\mathfrak{sp}(n).$

∎

######  Theorem 12.3.

Let $u,v\in{\mathbb{C}}^{n}$ be a non-zero elements of ${\mathbb{C}}^{n}$,
then the complex $2n$-dimensional vector space

$\mathcal{E}_{uv}=\\{\phi_{ab}:\text{\bf Sp}(n,{\mathbb{C}})\to{\mathbb{C}}\
|\ \phi_{ab}(g)=\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t}),\
a,b\in{\mathbb{C}}^{n}\\}$

is an eigenfamily on $\text{\bf Sp}(n,{\mathbb{C}})$ such that for all
$\phi,\psi\in\mathcal{E}_{uv}$ we have

$\tau(\phi)=-\,(2n+1)\cdot\phi,\ \ \kappa(\phi,\psi)=-\,\phi\cdot\psi.$

###### Proof.

The statement is an immediate consequence of Proposition 12.2. ∎

##  13\. The Semi-Riemannian Lie Group $\text{\bf Sp}(n,{\mathbb{R}})$

The semisimple real symplectic group $\text{\bf Sp}(n,{\mathbb{R}})$ is the
subgroup of the complex symplectic group $\text{\bf Sp}(n,{\mathbb{C}})$ given
by

$\text{\bf Sp}(n,{\mathbb{R}})=\\{x\in\text{\bf
SL}_{2n}({\mathbb{R}})\,|\,x\,J_{n}\,x^{t}=J_{n}\\}$

with Lie algebra

$\mathfrak{sp}(n,{\mathbb{R}})=\\{X\in\mathfrak{gl}_{2n}({\mathbb{R}})\,|\,X\,J_{n}+J_{n}\,X^{t}=0\\}.$

For the Lie algebra $\mathfrak{sp}(n,{\mathbb{C}})$ we have the orthogonal
decomposition

$\mathfrak{sp}(n,{\mathbb{C}})=\mathfrak{sp}(n,{\mathbb{R}})\oplus
i\,\mathfrak{sp}(n,{\mathbb{R}}).$

######  Theorem 13.1.

Let $v$ be a non-zero element of ${\mathbb{C}}^{n}$, then the complex
$n$-dimensional vector space

$\mathcal{E}_{v}=\\{\phi_{a}:\text{\bf Sp}(n,{\mathbb{R}})\to{\mathbb{C}}\ |\
\phi_{a}(x)=\operatorname{trace}(v^{t}ax^{t}),\ a\in{\mathbb{C}}^{n}\\}$

is an eigenfamily on $\text{\bf Sp}(n,{\mathbb{R}})$ such that for all
$\phi,\psi\in\mathcal{E}_{v}$ we have

$\tau(\phi)=-\,\frac{2n+1}{2}\cdot\phi,\ \
\kappa(\phi,\psi)=-\,\frac{1}{2}\cdot\phi\cdot\psi.$

###### Proof.

The result follows directly from Theorem 12.3 and the fact that
$\mathfrak{sp}(n,{\mathbb{C}})=\mathfrak{sp}(n,{\mathbb{R}})\oplus
i\,\mathfrak{sp}(n,{\mathbb{R}})$. ∎

##  14\. The Semi-Riemannian Lie Group $\text{\bf SO}^{*}(2n)$

In this section we construct eigenfamilies of complex-valued functions on the
semisimple non-compact Lie group

$\text{\bf SO}^{*}(2n)=\\{g\in\text{\bf SU}(n,n)\ |\ g\cdot I_{nn}\cdot
J_{n}\cdot g^{t}=I_{nn}\cdot J_{n}\\},$

where

$\text{\bf SU}(n,n)=\\{z\in\text{\bf SL}_{2n}({\mathbb{C}})\ |\ z\cdot
I_{n,n}\cdot z^{*}=I_{n,n}\\}.$

For the Lie algebra

$\mathfrak{so}^{*}(2n)=\Big{\\{}\begin{bmatrix}Z&W\\\
-\bar{W}&\bar{Z}\end{bmatrix}\in{\mathbb{C}}^{2n\times 2n}\Big{|}\ Z+Z^{*}=0\
\text{and}\ W+W^{t}=0\Big{\\}}$

of $\text{\bf SO}^{*}(2n)$ we have the orthogonal splitting
${\mathfrak{so}_{+}^{*}}(2n)\oplus{\mathfrak{so}_{-}^{*}}(2n)$, where the
subspaces ${\mathfrak{so}_{+}^{*}}(2n)$ and ${\mathfrak{so}_{-}^{*}}(2n)$ have
the orthonormal basis $\mathcal{B}^{+}$ and $\mathcal{B}^{-}$, respectively,
with

$\displaystyle\mathcal{B}^{+}$ $\displaystyle=$
$\displaystyle\Big{\\{}\frac{1}{\sqrt{2}}\begin{bmatrix}Y_{rs}&0\\\
0&Y_{rs}\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}iX_{rs}&0\\\
0&-iX_{rs}\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}iD_{t}&0\\\
0&-iD_{t}\end{bmatrix}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\Big{|}\ 1\leq r<s\leq n\ \
\text{and}\ \ 1\leq t\leq n\Big{\\}}$

and

$\mathcal{B}^{-}=\left\\{\frac{1}{\sqrt{2}}\begin{bmatrix}0&Y_{rs}\\\
-Y_{rs}&0\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}0&iY_{rs}\\\
iY_{rs}&0\end{bmatrix}\Big{|}\ 1\leq r<s\leq n\right\\}.$

Here $g(Z,Z)=1$ for all $Z\in\mathcal{B}^{+}$ and $g(Z,Z)=-1$ for all
$Z\in\mathcal{B}^{-}$.

######  Proposition 14.1.

Let $z_{j\alpha},w_{k\beta}:\text{\bf SO}^{*}(2n)\to\mathbb{C}$ be the
complex-valued matrix coefficients of the standard representation of
$\text{\bf SO}^{*}(2n)$. Then the tension field $\tau$ and the conformality
operator $\kappa$ on $\text{\bf SO}^{*}(2n)$ satisfy the following relations

$\tau(z_{j\alpha})=-\,\frac{2n-1}{2}\cdot z_{j\alpha},\ \
\tau(w_{j\alpha})=-\,\frac{2n-1}{2}\cdot w_{j\alpha},$
$\kappa(z_{j\alpha},z_{k\beta})=-\,\frac{1}{2}\cdot z_{k\alpha}\cdot
z_{j\beta},\ \ \kappa(w_{j\alpha},w_{k\beta})=-\,\frac{1}{2}\cdot
w_{k\alpha}\cdot w_{j\beta},$
${\kappa}(z_{j\alpha},w_{k\beta})=-\frac{1}{2}\cdot z_{k\alpha}\cdot
w_{j\beta}.$

###### Proof.

Here we can apply exactly the same strategy as for the proof of Proposition
7.1. ∎

######  Theorem 14.2.

Let $u,v\in{\mathbb{C}}^{n}$ be a non-zero elements of ${\mathbb{C}}^{n}$,
then the complex $2n$-dimensional vector space

$\mathcal{E}_{uv}=\\{\phi_{ab}:\text{\bf SO}^{*}(2n)\to{\mathbb{C}}\ |\
\phi_{ab}(g)=\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t}),\
a,b\in{\mathbb{C}}^{n}\\},$

is an eigenfamily on $\text{\bf SO}^{*}(2n)$ such that for all
$\phi,\psi\in\mathcal{E}_{uv}$ we have

$\tau(\phi)=-\,\frac{2n-1}{2}\cdot\phi,\ \
\kappa(\phi,\psi)=-\,\frac{1}{2}\cdot\phi\cdot\psi.$

###### Proof.

The statement is an immediate consequence of Proposition 14.1. ∎

##  15\. The Semi-Riemannian Lie Group $\text{\bf SU}(p,q)$

In this section we construct eigenfamilies on the non-compact semisimple Lie
group

$\text{\bf SU}(p,q)=\\{z\in\text{\bf SL}_{p+q}({\mathbb{C}})\ |\ z\cdot
I_{p,q}\cdot z^{*}=I_{p,q}\\}.$

For its Lie algebra

$\mathfrak{su}(p,q)=\\{Z\in\mathfrak{sl}_{p+q}({\mathbb{C}})\ |\ Z\cdot
I_{p,q}+I_{p,q}\cdot Z^{*}=0\\}$

we have a natural orthogonal splitting

$\mathfrak{su}(p,q)=\mathfrak{s}(\mathfrak{u}(p)+\mathfrak{u}(q))\oplus
i\cdot\mathfrak{m}$

such that

$\mathfrak{su}(p+q)=\mathfrak{s}(\mathfrak{u}(p)+\mathfrak{u}(q))\oplus\mathfrak{m}$

is the Lie algebra of the special orthogonal group $\text{\bf SU}(p+q)$.

######  Proposition 15.1.

Let $z_{j\alpha}:\text{\bf SU}(p,q)\to\mathbb{C}$ be the complex-valued matrix
elements of the standard representation of the special unitary group
$\text{\bf SU}(p,q)$. Then the tension field $\tau$ and the conformality
operator $\kappa$ on $\text{\bf SU}(p,q)$ satisfy the following relations

$\displaystyle\tau(z_{j\alpha})$ $\displaystyle=$
$\displaystyle-\,\frac{(p+q)^{2}-1}{(p+q)}\cdot z_{j\alpha},$
$\displaystyle\kappa(z_{j\alpha},z_{k\beta})$ $\displaystyle=$
$\displaystyle-\,(z_{k\alpha}\cdot z_{j\beta}-\frac{1}{(p+q)}\cdot
z_{j\alpha}\cdot z_{k\beta}).$

###### Proof.

This is an immediate consequence of Lemma 8.1, the above relationship between
the Lie algebras $\mathfrak{su}(p+q)$ and $\mathfrak{su}(p,q)$ and how the
semi-Riemannian metric $g$ is defined on the complex Lie algebra
$\mathfrak{gl}_{p+q}({\mathbb{C}})$. ∎

######  Theorem 15.2.

Let $v$ be a non-zero element of $\mathbb{C}^{p+q}$. Then the complex
$(p+q)$-dimensional vector space

$\mathcal{E}_{v}=\\{\phi_{a}:\text{\bf
SU}(p,q)\to\mathbb{C}\,|\,\phi_{a}(z)=\operatorname{trace}(v^{t}az^{t}),\
a\in\mathbb{C}^{p+q}\\}$

is an eigenfamily on $\text{\bf SU}(p,q)$ such that for all
$\phi,\psi\in\mathcal{E}_{v}$ we have

$\tau(\phi)=-\,\frac{(p+q)^{2}-1}{(p+q)}\cdot\phi,\ \
\kappa(\phi,\psi)=-\frac{(p+q-1)}{(p+q)}\cdot\phi\cdot\psi.$

###### Proof.

The statement follows directly from Proposition 15.1. ∎

##  16\. The Semi-Riemannian Lie Group $\text{\bf SO}(p,q)$

In this section we construct eigenfamilies on the non-compact semisimple Lie
group

$\text{\bf SO}(p,q)=\\{x\in\text{\bf SL}_{p+q}({\mathbb{R}})\ |\ x\cdot
I_{p,q}\cdot x^{t}=I_{p,q}\\}.$

For its Lie algebra

$\mathfrak{so}(p,q)=\\{X\in\mathfrak{sl}_{p+q}({\mathbb{R}})\ |\ X\cdot
I_{p,q}+I_{p,q}\cdot X^{t}=0\\}$

we have a natural orthogonal spilling

$\mathfrak{so}(p,q)\cong(\mathfrak{so}(p)\oplus\mathfrak{so}(q))\oplus
i\cdot\mathfrak{m}$

such that

$\mathfrak{so}(p+q)=(\mathfrak{so}(p)\oplus\mathfrak{so}(q))\oplus\mathfrak{m}$

is the Lie algebra of the special orthogonal group $\text{\bf SO}(p+q)$.

######  Proposition 16.1.

Let $x_{j\alpha}:\text{\bf SO}(p,q)\to\mathbb{R}$ be the real-valued matrix
elements of the standard representation of the special orthogonal group
$\text{\bf SO}(p,q)$. Then the tension field $\tau$ and the conformality
operator $\kappa$ on $\text{\bf SO}(p,q)$ satisfy the following relations

$\displaystyle\tau(x_{j\alpha})$ $\displaystyle=$
$\displaystyle-\,\frac{(p+q-1)}{2}\cdot x_{j\alpha},$
$\displaystyle\kappa(x_{j\alpha},x_{k\beta})$ $\displaystyle=$
$\displaystyle-\,\frac{1}{2}\cdot(x_{j\beta}\cdot
x_{k\alpha}-\delta_{jk}\cdot\delta_{\alpha\beta}).$

###### Proof.

This is an immediate consequence of Lemma 11.1, the above relationship between
the Lie algebras $\mathfrak{so}(p+q)$ and $\mathfrak{so}(p,q)$ and how the
semi-Riemannian metric $g$ is defined on the complex Lie algebra
$\mathfrak{gl}_{p+q}({\mathbb{R}})$. ∎

######  Theorem 16.2.

Let $v\in{\mathbb{C}}^{p+q}$ be a non-zero isotropic element i.e. $(v,v)=0$,
then the complex $(p+q)$-dimensional vector space

$\mathcal{E}_{v}=\\{\phi_{a}:\text{\bf SO}(p,q)\to{\mathbb{C}}\ |\
\phi_{a}(z)=\operatorname{trace}(v^{t}az^{t}),\ a\in{\mathbb{C}}^{p+q}\\}$

is an eigenfamily on $\text{\bf SO}(p,q)$ such that for all
$\phi,\psi\in\mathcal{E}_{v}$ we have

$\tau(\phi)=-\,\frac{(p+q-1)}{2}\cdot\phi,\ \
\kappa(\phi,\psi)=-\,\frac{1}{2}\cdot\phi\cdot\psi.$

###### Proof.

The statement follows directly from Proposition 16.1. ∎

Lie group Eigenfunctions $\phi$ $\lambda$ $\mu$ Conditions $\text{\bf
SO}(n,{\mathbb{C}})$ $\operatorname{trace}(v^{t}az^{t})$ $-\,(n-1)$ $-1$
$a\in{\mathbb{C}}^{n},\ (v,v)=0$ $\text{\bf Sp}(n,{\mathbb{C}})$
$\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t})$ $-\,(2n+1)$ $-1$
$a,b\in{\mathbb{C}}^{n}$ $\text{\bf Sp}(n,{\mathbb{R}})$
$\operatorname{trace}(v^{t}ax^{t})$ $-\,\frac{2n+1}{2}$ $-\frac{1}{2}$
$a\in{\mathbb{C}}^{n}$ $\text{\bf SO}^{*}(2n)$
$\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t})$ $-\frac{2n-1}{2}$
$-\,\frac{1}{2}$ $a,b\in{\mathbb{C}}^{n}$ $\text{\bf SU}(p,q)$
$\operatorname{trace}(v^{t}az^{t})$ $-\frac{(p+q)^{2}-1}{(p+q)}$
$-\frac{(p+q-1)}{(p+q)}$ $a\in{\mathbb{C}}^{n}$ $\text{\bf SO}(p,q)$
$\operatorname{trace}(v^{t}ax^{t})$ $-\frac{(p+q-1)}{2}$ $-\frac{1}{2}$
$a\in{\mathbb{C}}^{n}$ $\text{\bf Sp}(p,q)$
$\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t})$ $-\frac{2(p+q)+1}{2}$
$-\frac{1}{2}$ $a,b\in{\mathbb{C}}^{n}$

Table 2. Eigenfunctions on classical non-compact Lie groups.

##  17\. The Semi-Riemannian Lie Group $\text{\bf Sp}(p,q)$

In this section we construct eigenfamilies on the non-compact semisimple Lie
group

$\text{\bf Sp}(p,q)=\\{g\in\text{\bf SL}_{p+q}({\mathbb{H}})\ |\ g\cdot
I_{p,q}\cdot g^{*}=I_{p,q}\\}.$

For its Lie algebra

$\mathfrak{sp}(p,q)=\\{Z\in\mathfrak{sl}_{p+q}({\mathbb{H}})\ |\ Z\cdot
I_{p,q}+I_{p,q}\cdot Z^{*}=0\\}$

we have a natural orthogonal decomposition

$\mathfrak{sp}(p,q)=(\mathfrak{sp}(p)\oplus\mathfrak{sp}(q))\oplus
i\cdot\mathfrak{m}$

such that

$\mathfrak{sp}(p+q)=(\mathfrak{sp}(p)\oplus\mathfrak{sp}(q))\oplus\mathfrak{m}$

is the Lie algebra of the quaternionic unitary group $\text{\bf Sp}(p+q)$.

######  Proposition 17.1.

Let $z_{j\alpha},w_{j\alpha}:\text{\bf Sp}(p,q)\to\mathbb{C}$ be the complex-
valued matrix elements of the standard representation of the quaternionic
unitary group $\text{\bf Sp}(p,q)$. Then the tension field $\tau$ and the
conformality operator $\kappa$ on $\text{\bf Sp}(p,q)$ satisfy the following
relations

$\tau(z_{j\alpha})=-\,\frac{2(p+q)+1}{2}\cdot z_{j\alpha},\ \
\tau(w_{k\beta})=-\,\frac{2(p+q)+1}{2}\cdot w_{k\beta},$
$\kappa(z_{j\alpha},z_{k\beta})=-\,\frac{1}{2}\cdot z_{k\alpha}\cdot
z_{j\beta},\ \ \kappa(w_{j\alpha},w_{k\beta})=-\,\frac{1}{2}\cdot
w_{k\alpha}\cdot w_{j\beta},$
$\kappa(z_{j\alpha},w_{k\beta})=-\,\frac{1}{2}\cdot z_{k\alpha}\cdot
w_{j\beta}.$

###### Proof.

This is an immediate consequence of Lemma 12.1, the above relationship between
the Lie algebras $\mathfrak{sp}(p+q)$ and $\mathfrak{sp}(p,q)$ and how the
semi-Riemannian metric $g$ is defined on the complex Lie algebra
$\mathfrak{gl}_{p+q}({\mathbb{H}})$. ∎

######  Theorem 17.2.

Let $u,v\in{\mathbb{C}}^{p+q}$ be a non-zero element of ${\mathbb{C}}^{p+q}$,
then the complex $(p+q)$-dimensional vector space

$\mathcal{E}_{uv}=\\{\phi_{ab}:\text{\bf Sp}(p,q)\to{\mathbb{C}}\ |\
\phi_{ab}(g)=\operatorname{trace}(u^{t}az^{t}+v^{t}bw^{t}),\
a,b\in{\mathbb{C}}^{p+q}\\}$

is an eigenfamily on $\text{\bf Sp}(p,q)$ such that for all
$\phi,\psi\in\mathcal{E}_{uv}$ we have

$\tau(\phi)=-\,\frac{2(p+q)+1}{2}\cdot\phi,\ \
\kappa(\phi,\psi)=-\,\frac{1}{2}\cdot\phi\cdot\psi.$

###### Proof.

The statement follows directly from Proposition 17.1. ∎

##  18\. Acknowledgements

The first author would like to thank the Department of Mathematics at Lund
University for its great hospitality during her time there as a postdoc.

The authors would like to thank Albin Ingelström for useful discussions on
case of $\text{\bf SU}(p,q)$.

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