# Quantum superintegrable spin systems on graph connections

Nicolai Reshetikhin  N.R.: YMSC, Tsinghua University, Beijing, China & BIMSA,
Beijing, China & Department of Mathematics, University of California,
Berkeley, CA 94720, USA<EMAIL_ADDRESS>and  Jasper Stokman
J.S.: KdV Institute for Mathematics, University of Amsterdam, Science Park
105-107, 1098 XG Amsterdam, The Netherlands<EMAIL_ADDRESS>
###### Abstract.

In this paper we construct certain quantum spin systems on moduli spaces of
$G$-connections on a connected oriented finite graph, with $G$ a simply
connected compact Lie group. We construct joint eigenfunctions of the
commuting quantum Hamiltonians in terms of local invariant tensors. We
determine sufficient conditions ensuring superintegrability of the quantum
spin system using irreducibility criteria for Harish-Chandra modules due to
Harish-Chandra and Lepowsky & McCollum.

The resulting class of quantum superintegrable spin systems includes the
quantum periodic and open spin Calogero-Moser spin chains as special cases. In
the periodic case the description of the joint eigenfunctions in terms of
local invariant tensors are multipoint generalised trace functions, in the
open case multipoint spherical functions on compact symmetric spaces.

###### Key words and phrases:

Dedicated to the memory of Gerrit van Dijk

##  1\. Introduction

### 1.1.

Let $\Gamma$ be a connected oriented finite graph with vertex set $V$, edge
set $E$, and source and target maps $s,t:E\rightarrow V$. Let $G$ be a
connected compact Lie group. The product group $G^{E}=\\{\bm{g}=(g_{e})_{e\in
E}\,\,|\,\,g_{e}\in G\\}$ of graph $G$-connections (or lattice gauge fields)
on $\Gamma$ consists of colorings $\bm{g}$ of the edges of $\Gamma$ by group
elements $g_{e}\in G$ ($e\in E$). We view $G^{E}$ both as compact Lie group
and as algebraic group (via Tannaka duality).

The group $G^{V}=\\{\bm{k}=(k_{v})_{v\in V}\,\,|\,\,k_{v}\in G\\}$ of lattice
gauge transformations acts on $G^{E}$ by111With this convention of the action,
$g_{e}\in G$ describes the holonomy along $e$ in the reverse direction.

(1) $(\bm{k}\cdot\bm{g})_{e}=k_{s(e)}g_{e}k_{t(e)}^{-1}.$

The resulting space $G^{E}/G^{V}$ of $G^{V}$-orbits in $G^{E}$ is the moduli
space of graph $G$-connections on $\Gamma$ introduced by Fock and Rosly [11]
to describe moduli spaces of flat connections on surfaces, see also [2, 3].
See [1, 4, 5, 22] and references therein for the associated quantization
problem. In this paper we construct quantum systems with Hamiltonians being
differential operators on the moduli space $G^{E}/\mathbf{K}$ of graph
$G$-connections modulo gauge groups $\mathbf{K}$ of the form
$\mathbf{K}=\prod_{v\in V}K_{v}\subseteq G^{V}$, with $K_{v}$ arbitrary
subgroups of $G$.

### 1.2.

Let $\mathcal{D}(G^{E})$ be the algebra of algebraic differential operators on
$G^{E}$. The contragredient $\mathbf{K}$-action on the space of algebraic
functions on $G^{E}$ induces a $\mathbf{K}$-action on $\mathcal{D}(G^{E})$ by
algebra automorphisms. We denote by
$\mathcal{D}(G^{E})^{\mathbf{K}}\subseteq\mathcal{D}(G^{E})$ the subalgebra of
$\mathbf{K}$-invariant differential operators. In this paper we consider its
subalgebras

(2)
$\mathcal{D}_{\textup{biinv}}(G^{E})\subseteq\mathcal{D}_{\textup{inv}}(G^{E})^{\mathbf{K}}\subseteq\mathcal{D}(G^{E})^{\mathbf{K}},$

with $\mathcal{D}_{\textup{inv}}(G^{E})\subseteq\mathcal{D}(G^{E})$ the
subalgebra generated by the left and right $G^{E}$-invariant differential
operators and $\mathcal{D}_{\textup{biinv}}(G^{E})\subseteq\mathcal{D}(G^{E})$
the subalgebra of $G^{E}$-biinvariant differential operators.

### 1.3.

Let $\sigma:\mathbf{K}\rightarrow\textup{GL}(S)$ be a finite dimensional
$\mathbf{K}$-representation. Functions in the space
$\mathcal{H}=\mathcal{H}_{\Gamma,G,S}$ of algebraic sections of the associated
vector bundle over $G^{E}/\mathbf{K}$ are called spin graph functions. They
are algebraic functions $f:G^{E}\rightarrow S$ satisfying

$f(\bm{k}\cdot\bm{g})=\sigma(\bm{k})f(\bm{g})$

for $\bm{k}\in\mathbf{K}$ and $\bm{g}\in G^{E}$. Here spin refers to the
interpretation of $S$ as spin space for the associated quantum spin system,
see §1.4 and §1.9.

### 1.4.

The algebra $\mathcal{D}(G^{E})^{\mathbf{K}}$ acts as scalar valued
$\mathbf{K}$-invariant differential operators on the space $\mathcal{H}$ of
spin graph functions. The resulting homomorphic image of the inclusions (2) of
algebras gives rise to an inclusion

(3) $I\subseteq J\subseteq A$

of subalgebras of $\textup{End}(\mathcal{H})$. We view (3) as a quantum spin
system with quantum state space $\mathcal{H}$, and with the homomorphic image
$A=A_{\Gamma,G,S}\subseteq\textup{End}(\mathcal{H})$ of
$\mathcal{D}(G^{E})^{\mathbf{K}}$ as algebra of quantum observables, the
homomorphic image $J=J_{\Gamma,G,S}$ of
$\mathcal{D}_{\textup{inv}}(G^{E})^{\mathbf{K}}$ as algebra of quantum
integrals, and the homomorphic image $I=I_{\Gamma,G,S}$ of
$\mathcal{D}_{\textup{biinv}}(G^{E})$ as the algebra of quantum Hamiltonians.

### 1.5.

The quantum state space $\mathcal{H}$ breaks up in an algebraic direct sum of
finite dimensional simultaneous eigenspaces for the action of the quantum
Hamiltonians,

$\mathcal{H}=\bigoplus_{\chi\in I^{\wedge}}\mathcal{H}_{\chi},$

with $I^{\wedge}$ the set of characters of $I$. We say that the quantum spin
system is superintegrable if for all $\chi\in I^{\wedge}$, the simultaneous
eigenspace $\mathcal{H}_{\chi}$ is either $\\{0\\}$ or an irreducible
$J$-module. Equivalently, the quantum spin system is superintegrability when
eigenstates $f,g\in\mathcal{H}_{\chi}$ with the same energy eigenvalues
$\chi\in I^{\wedge}$ are related by a quantum integral: $g=D(f)$ for some
$D\in J$.

We say that the superintegrable quantum spin system is integrable when $I=J$.
In this case $\textup{dim}(\mathcal{H}_{\chi})\leq 1$ for all $\chi\in
I^{\wedge}$, i.e., an eigenstate is determined by its energy eigenvalues up to
normalisation.

### 1.6.

The main result of this paper is as follows.

######  Theorem.

The quantum spin system on $\mathcal{H}=\mathcal{H}_{\Gamma,G,S}$ is
superintegrable when the following three conditions are satisfied:

  1. (a)

$G$ is simply connected,

  2. (b)

the local gauge group $K_{v}$ is a closed connected subgroup of $G$ for all
$v\in V$,

  3. (c)

$\sigma:\mathbf{K}\rightarrow\textup{GL}(S)$ is irreducible.

We will prove this theorem using a result of Harish-Chandra [13] and Lepowsky
& McCollum [14] relating irreducible $\mathfrak{g}$-modules to irreducible
$U(\mathfrak{g})^{K}$-modules for appropriate compact Lie groups $K$ (this
result plays an important role in the proof of the subquotient theorem for
Harish-Chandra modules).

### 1.7.

We will say that a spin graph function $f$ is elementary if it is a
simultaneous eigenfunction of the quantum Hamiltonians, i.e., when
$f\in\mathcal{H}_{\chi}$ for some $\chi\in I^{\wedge}$. For tensor product
$\mathbf{K}$-representations $S$ we construct spanning sets of
$\mathcal{H}_{\chi}$ using the data of the following colored version of
$\Gamma$.

The colors at the vertices $v\in V$ are the local representations
$\sigma_{v}:K_{v}\rightarrow\textup{GL}(S_{v})$ of the tensor product
representation $S$. To obtain the colors of the edges, we use the fact that
$\mathcal{H}_{\chi}\not=\\{0\\}$ if and only if $\chi$ is the central
character of an irreducible representation of $G^{E}$. The irreducible
$G^{E}$-representation provides the colors of the edges of $\Gamma$ by local
irreducible $G$-representations.

We construct the spanning set of the elementary spin graph functions in
$\mathcal{H}_{\chi}$ in terms of local invariant tensors (local in the sense
that they only depend on the star of some vertex $v$ of the colored graph
$\Gamma$).

### 1.8.

If $\Gamma$ is the directed cycle graph with $n$ edges with $K_{v}=G$ for all
$v\in V$, then we show that the resulting elementary spin graph functions are
essentially the generalised (or $n$-point) trace functions from Etingof &
Schiffmann [8].

If $\Gamma$ is the linearly ordered linear graph with $n$ edges and the local
gauge groups $K_{v}$ are $G$ (resp. $K$) for $2$-valent (resp. $1$-valent)
vertices $v\in V$, then we show that the resulting elementary spin graph
functions are the $n$-point spherical functions from [21]. For $n=1$ these are
the usual elementary $\sigma$-spherical functions on $G$, see, e.g., [12, 24].

In both cases the local invariant tensors may be identified with topological
degenerations of vertex operators, cf. [24, 23].

### 1.9.

The explicit desciption of the elementary spin graph functions as multipoint
trace functions and multipoint spherical functions for the two special cases
in §1.8, connects the associated quantum spin systems to the periodic and open
quantum spin Calogero-Moser chains from [8, 21, 20] and [24, 21],
respectively.

This can be made concrete on the level of quantum Hamiltonians. It requires a
parametrisation of the moduli space $G^{E}/\mathbf{K}$ of $G$-graph
connections in terms of an appropriate subtorus $T$ of $G$, as well as Harish-
Chandra’s radial component techniques to describe the action of the edge-
coordinate quadratic Casimirs on $\mathcal{H}_{\Gamma,G,S}$ in terms of
explicit second-order $\textup{End}(S)$-valued differential operators $H_{e}$
($e\in E$) on $T$ (which are the quadratic Hamiltonians of the quantum
Calogero-Moser spin chain up to a gauge). The differences
$H_{e}-H_{e^{\prime}}$ for neighboring edges $e$ and $e^{\prime}$ then form an
explicit commuting family of first order differential operators, called
asymptotic Knizhnik-Zamolodchikov-Bernard operators. See [8] and [24, 21] for
the details.

### 1.10.

Combining the results from §1.6 and §1.9 we obtain explicit conditions
ensuring the superintegrability of the periodic and open quantum spin
Calogero-Moser chains. For the special case of the directed cycle graph
$\Gamma$ with one vertex and $\mathbf{K}=G$, the superintegrability of the
associated periodic quantum spin Calogero-Moser system was considered before
in [18]. The classical superintegrability of the periodic and open Calogero-
Moser spin chains is discussed in [19].

### 1.11.

The contents of the paper is as follows.

In Section 2 we describe the type of quantum systems that we consider in this
paper, and discuss the concept of superintegrability in this context.

In Section 3 we formulate a result of Harish-Chandra [13] and Lepowsky &
McCollum [14] (Corollary 3.17) that will play the key role in establishing the
superintegrability of the quantum spin systems on moduli spaces of graph
connections. This involves the concept of reductive extensions of Lie
algebras, which we discuss in detail.

In Section 4 we introduce the space of spin graph functions on graph
connections, and provide a spanning set in terms of local invariant tensors
(Theorem 4.15). For the directed cycle graph and the linearly ordered linear
graph we relate the spin graph functions to multipoint trace functions and
multipoint spherical functions (see §4.17 and §4.18).

In Section 5 we provide a detailed introduction of the quantum spin systems on
moduli spaces of graph connections. We state the conditions ensuring
superintegrability of the quantum spin system and discuss the
superintegrability of the quantum periodic and open spin Calogero-Moser chains
(see §5.12 and §5.13).

In Section 6 we give the proof of the main result (Theorem 1.6/5.11). The
crucial intermediate step, which will allow us to use the result of Harish-
Chandra and Lepowsky & McCollum in this context, is the translation of the
condition of superintegrability in terms of irreducibility conditions of local
intertwining spaces at the stars of the vertices of the graph (see Corollary
6.9).

Conventions: The ground field will be $\mathbb{C}$ unless explicitly stated
otherwise. Lie algebras are finite dimensional unless stated explicitly
otherwise. We use $\textup{Hom}(V,W)$ for the Hom-space in the category of
complex vector spaces. For $G$ a group, $A$ an associative algebra and
$\mathfrak{g}$ a Lie algebra we write $\textup{Hom}_{G}(V,W)$,
$\textup{Hom}_{A}(V,W)$, $\textup{Hom}_{\mathfrak{g}}(V,W)$ for the Hom-space
in the category of $G$-representations, left $A$-modules and
$\mathfrak{g}$-modules, respectively. We write $U(\mathfrak{k})$ for the
universal enveloping algebra of a complex Lie algebra $\mathfrak{k}$, and
$Z(\mathfrak{k})$ for its center.

For sets $X,\mathcal{I}$ with $\mathcal{I}$ finite, we write $X^{\mathcal{I}}$
for the direct product of $\\#\mathcal{I}$-copies of $X$. In case $X$ is a Lie
group/algebra, we endow $X^{\mathcal{I}}$ with the direct product Lie
group/algebra structure.

For a finite family $\\{M_{i}\\}_{i\in\mathcal{I}}$ of vector spaces $M_{i}$
with index set $\mathcal{I}=\\{i_{1},\ldots,i_{s}\\}$, totally ordered by
$i_{1}<\cdots<i_{s}$, we write

$\bigotimes_{i\in\mathcal{I}}M_{i}:=M_{i_{1}}\otimes\cdots\otimes M_{i_{s}}.$

Acknowledgements: both authors were supported by the Dutch Research Council
(NWO), project number 613.009.1260. In addition, the work of N.R. was
supported by the NSF grant DMS-1902226, by the RSF grant 18-11-00-297 and by
the Changjiang fund.

##  2\. Centraliser algebras

In this section we derive some elementary properties of centraliser algebras,
with an eye towards the application to quantum superintegrable systems. The
starting point is a complex vector space $\mathcal{H}$ and an inclusion

$I\subseteq A\subseteq\textup{End}(\mathcal{H})$

of unital algebras, with $I$ being commutative. In applications to quantum
mechanics $\mathcal{H}$ is the quantum state space, $A$ the algebra of
observables, and $I$ its subalgebra of quantum Hamiltonians. We do not fix a
particular $H\in I$ as the quantum Hamiltonian of the system, since we are not
considering quantum dynamics at this point.

### 2.1.

Denote by $I^{\wedge{}}$ the set of characters of $I$. For an element $\chi\in
I^{\wedge{}}$, i.e., for an unital algebra homomorphism
$\chi:I\rightarrow\mathbb{C}$, we write

$\mathcal{H}_{\chi}:=\\{h\in\mathcal{H}\,\,|\,\,y\cdot
h=\chi(y)h\quad\forall\,y\in I\\}$

for the corresponding joint eigenspace (it may be zero).

### 2.2.

Denote by

$C_{A}(I):=\\{x\in A\,\,|\,\,xy=yx\quad\forall y\in I\\}$

the centraliser of $I$ in $A$. It is a subalgebra of $A$ containing $I$. It
stabilises $\mathcal{H}_{\chi}$ for all $\chi\in I^{\wedge}$.

### 2.3.

Suppose that $J\subseteq\textup{End}(\mathcal{H})$ is a sub-algebra
stabilising $\mathcal{H}_{\chi}\subseteq\mathcal{H}$. Then
$\mathcal{H}_{\chi}$ is a $J$-module, and

$J_{\chi}:=\\{x|_{\mathcal{H}_{\chi}}\,\,|\,\,x\in J\\}$

is a sub-algebra of $\textup{End}(\mathcal{H}_{\chi})$. If
$\mathcal{H}_{\chi}$ is a finite dimensional irreducible $J$-module, then
$J_{\chi}=\textup{End}(\mathcal{H}_{\chi})$ by the density theorem. If $J=I$
then we have $I_{\chi}=\mathbb{C}\textup{id}_{\mathcal{H}_{\chi}}$.

### 2.4.

Let $J$ be a sub-algebra of $A$ containing $I$. Then

$C_{A}(J)\subseteq C_{A}(I).$

If in addition $J$ stabilises $\mathcal{H}_{\chi}$ for some $\chi\in
I^{\wedge}$, then $C_{A}(J)$ stabilises $\mathcal{H}_{\chi}$ in view of §2.2.
The fact that both $J$ and $C_{A}(J)$ stabilise $\mathcal{H}_{\chi}$ implies
that $C_{A}(J)_{\chi}$ is contained in the commutant of $J_{\chi}$ in
$\textup{End}(\mathcal{H}_{\chi})$.

If in addition $\mathcal{H}_{\chi}$ is an irreducible finite dimensional
$J$-module (in particular, $\mathcal{H}_{\chi}\not=\\{0\\}$), then

$C_{A}(J)_{\chi}=\mathbb{C}\,\textup{id}_{\mathcal{H}_{\chi}}=I_{\chi}$

by Schur’s lemma.

### 2.5.

Suppose that $J\subseteq A$ is a subalgebra satisfying

$I\subseteq J\subseteq C_{A}(I).$

For such an algebra $J$ the joint eigenspace $\mathcal{H}_{\chi}$ is
$J$-stable for all $\chi\in I^{\wedge}$, in view of §2.2.

If in addition $\mathcal{H}$ is a semisimple $I$-module (i.e.,
$\mathcal{H}=\bigoplus_{\chi\in I^{\wedge}}\mathcal{H}_{\chi}$), then the map

$J\rightarrow\prod_{\chi\in I^{\wedge}}\textup{End}(\mathcal{H}_{\chi}),\qquad
x\mapsto\bigl{(}x|_{\mathcal{H}_{\chi}}\bigr{)}_{\chi\in I^{\wedge}}$

is an injective algebra homomorphism, with $\prod_{\chi\in
I^{\wedge}}\textup{End}(\mathcal{H}_{\chi})$ the direct product of the family
$\\{\textup{End}(\mathcal{H}_{\chi})\,\,|\,\,\chi\in I^{\wedge}\\}$ of
algebras. Its image is contained in $\prod_{\chi\in I^{\wedge}}J_{\chi}$.

### 2.6.

Suppose that $J\subseteq A$ is a subalgebra satisfying

(4) $I\subseteq J\subseteq C_{A}(I)\subseteq
A\subseteq\textup{End}(\mathcal{H}).$

Assume furthermore that the following two spectral properties hold true:

  1. (a)

$\mathcal{H}$ is a semisimple $I$-module.

  2. (b)

For $\chi\in I^{\wedge}$, either $\mathcal{H}_{\chi}=\\{0\\}$ or
$\mathcal{H}_{\chi}$ is an irreducible finite dimensional $J$-module.

By §2.3 and §2.4 we then have

(5)
$\begin{split}J_{\chi}&=\textup{End}(\mathcal{H}_{\chi})=C_{A}(I)_{\chi},\\\
I_{\chi}&=\mathbb{C}\textup{id}_{\mathcal{H}_{\chi}}=C_{A}(J)_{\chi}\end{split}$

for all $\chi\in I^{\wedge}$. Informally speaking, $J$ is “locally” of maximal
size and equal to $C_{A}(I)$, and $I$ is “locally” the center of $J$.

### 2.7.

The setup of §2.6 provides the mathematical framework for superintegrability
of quantum systems in this paper. From this perspective (4) is defining a
quantum system with quantum state space $\mathcal{H}$, algebra of quantum
observables $A$, algebra of quantum Hamiltonians $I$ and algebra of quantum
integrals $J$.

######  Definition.

The quantum system (4) is said to be superintegrable if the two spectral
conditions 2.6(a)&(b) hold true.

The resulting properties (5) for the quantum superintegrable system provide
the link with the notion of a core structure of a quantum superintegrable
system considered in [18, §2].

A quantum superintegrable system is said to be quantum integrable if $I=J$. In
this case $\textup{dim}(\mathcal{H}_{\chi})\leq 1$ for all $\chi\in
I^{\wedge}$, i.e., the eigenvalues of the quantum Hamiltonians determine the
corresponding joint eigenvector up to a multiplicative constant.

For quantum superintegrable systems eigenstates this is no longer true. But
when $f,g\in\mathcal{H}_{\chi}$ then there exists a quantum integral $D\in J$
such that $g=D(f)$. Here we use that by the density theorem, the
irreducibility of the $J$-module $\mathcal{H}_{\chi}$ is equivalent to

$J_{\chi}=\textup{End}(\mathcal{H}_{\chi}).$

For the examples of quantum superintegrable systems we thus have the weaker
condition that simultaneous eigenspaces are finite dimensional, but two joint
eigenvectors with the same eigenvalues can always be related through the
action of a quantum integral.

### 2.8.

From the perspective of quantisation, quantum superintegrability requires the
algebras $I,J$ and $A$ to be quantisations of the Poisson algebras of
Hamiltonians, integrals and observables for a classical superintegrable system
(which is also sometimes called a degenerate integrable system). This is known
in the case of periodic and open quantum spin Calogero-Moser chains [17, 19].
For a discussion of classical superintegrability, see [17] and references
therein.

##  3\. Preservation of irreducibility

In this section we focus on a purely representation theoretic result due to
Lepowsky and McCollum [14, Thm. 5.5] (in special cases it goes back to Harish-
Chandra [13, Thm. 2]). It will be the crucial ingredient in proving
superintegrability of the quantum spin systems on graph connections in Section
6.

### 3.1.

Let $\mathfrak{g}$ be a Lie algebra. Recall that a $\mathfrak{g}$-module $M$
is said to be semisimple if $M$ is the sum of its irreducible
$\mathfrak{g}$-submodules. If furthermore all the irreducible
$\mathfrak{g}$-submodules of $M$ are finite dimensional, then we say that $M$
is a finitely semisimple $\mathfrak{g}$-module.

### 3.2.

Let $G$ be a real Lie group and $K\subseteq G$ a connected compact Lie
subgroup. Denote by $\mathfrak{g}_{0}$ the Lie algebra of $G$, and
$\mathfrak{g}$ its complexification. If $\pi$ is a Hilbert space
representation, then its (dense) subspace $M$ of smooth $K$-finite vectors
becomes a $(\mathfrak{g},K)$-module with $x\in\mathfrak{g}_{0}$ acting by

$x\cdot m:=\frac{d}{dt}\bigg{|}_{t=0}\pi(\exp(tx))m$

(see, e.g., [25, §3.3.1] for the definition of a $(\mathfrak{g},K)$-module).
The $(\mathfrak{g},K)$-module $M$ is finitely semisimple as a
$\mathfrak{k}$-module. Furthermore, if $\pi$ is irreducible and unitary, then
the associated $(\mathfrak{g},K)$-module $M$ is irreducible as
$\mathfrak{g}$-module. This in fact holds true under the weaker assumption
that $\pi$ is irreducible and admissible (see, e.g., [25, §3.3-4] for further
details).

### 3.3.

Let $\mathfrak{k}\subseteq\mathfrak{g}$ be an inclusion of Lie algebras and
$M$ a $\mathfrak{g}$-module. Denote by $\mathfrak{k}^{\wedge}$ the set of
isomorphism classes of finite dimensional irreducible $\mathfrak{k}$-modules.
For $\alpha\in\mathfrak{k}^{\wedge}$ the $\alpha$-isotypical component
$M_{\alpha}$ of $M$ is the subspace of $M$ generated by the finite dimensional
irreducible $\mathfrak{k}$-submodules of $M$ from the isomorphism class
$\alpha$. The sum $\sum_{\alpha\in\mathfrak{k}^{\wedge}}M_{\alpha}\subseteq M$
is direct (see, e.g., [6, §1.2.8]). Furthermore,
$M=\bigoplus_{\alpha\in\mathfrak{k}^{\wedge}}M_{\alpha}$ if and only if $M$ is
finitely semisimple as a $\mathfrak{k}$-module.

### 3.4.

A Lie subalgebra $\mathfrak{k}\subseteq\mathfrak{g}$ is said to be reductive
in $\mathfrak{g}$ when $\mathfrak{g}$ is a semisimple
$\textup{ad}(\mathfrak{k})$-module.

Note that if $\mathfrak{k}$ is reductive in $\mathfrak{g}$, then
$\mathfrak{k}$ is a reductive Lie algebra. On the other hand, if
$\mathfrak{k}$ is a semisimple Lie subalgebra of $\mathfrak{g}$, then
$\mathfrak{k}$ is reductive in $\mathfrak{g}$ by Weyl’s complete reducibility
theorem.

### 3.5.

Let $G$ be a real Lie group with Lie algebra $\mathfrak{g}_{0}$, and
$K\subseteq G$ a connected compact Lie subgroup. Denote by $\mathfrak{k}$ and
$\mathfrak{g}$ the complexified Lie algebras of $K$ and $G$, respectively.
Then $\mathfrak{k}$ is reductive in $\mathfrak{g}$.

### 3.6.

Let $\mathfrak{g}$ be a Lie algebra and $\theta\in\textup{Aut}(\mathfrak{g})$
an automorphism of finite order $n$. The associated fix-point Lie algebra is

$\mathfrak{g}^{\theta}:=\\{x\in\mathfrak{g}\,\,|\,\,\theta(x)=x\\}.$

######  Proposition.

Suppose that $\theta$ is an automorphism of a semisimple Lie algebra
$\mathfrak{g}$ of finite order $m$. Then $\mathfrak{g}^{\theta}$ is reductive
in $\mathfrak{g}$.

###### Proof.

The proof is a rather straightforward adjustment of the proof of the statement
for involutions, see [6, Prop. 1.13.3]. We give the proof for convenience of
the reader.

Denote by $\mathfrak{g}_{\overline{r}}$
($\overline{r}\in\mathbb{Z}/m\mathbb{Z}$) the eigenspace of $\theta$ with
eigenvalue $e^{2\pi i/r}$. Then
$\mathfrak{g}^{\theta}=\mathfrak{g}_{\overline{0}}$ and
$\mathfrak{g}_{\overline{r}}$ are
$\textup{ad}(\mathfrak{g}^{\theta})$-invariant subspaces of $\mathfrak{g}$.
Since $\theta$ is of order $m$, the assignment $\overline{1}\mapsto\theta$
defines a representation
$\mathbb{Z}/m\mathbb{Z}\rightarrow\textup{GL}(\mathfrak{g})$ of the finite
abelian group $\mathbb{Z}/m\mathbb{Z}$. By Maschke’s theorem, we conclude that

$\mathfrak{g}=\bigoplus_{\overline{r}\in\mathbb{Z}/m\mathbb{Z}}\mathfrak{g}_{\overline{r}}.$

Write
$\mathfrak{p}:=\bigoplus_{\overline{r}\not=\overline{0}}\mathfrak{g}_{\overline{r}}$,
so that

$\mathfrak{g}=\mathfrak{g}^{\theta}\oplus\mathfrak{p}.$

Let $\kappa:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{C}$ be the
Killing form of $\mathfrak{g}$. Then $\kappa(\theta(x),\theta(y))=\kappa(x,y)$
for all $x,y\in\mathfrak{g}$, hence
$\kappa(\mathfrak{g}^{\theta},\mathfrak{p})=0$. Since $\mathfrak{g}$ is
semisimple, we conclude that
$\kappa|_{\mathfrak{g}^{\theta}\times\mathfrak{g}^{\theta}}$ is nondegenerate.
Furthermore, if $x\in\mathfrak{g}^{\theta}$ and $x=s+n$ is the abstract
Chevalley decomposition of $x$ in $\mathfrak{g}$, with $s\in\mathfrak{g}$
(resp. $n\in\mathfrak{g}$) the semisimple (resp. nilpotent) part of $x$, then
$s,n\in\mathfrak{g}^{\theta}$ (this holds true for any automorphism $\theta$
of $\mathfrak{g}$). Then [6, Prop. 1.7.6] implies that $\mathfrak{g}^{\theta}$
is reductive in $\mathfrak{g}$. ∎

### 3.7.

If $\mathfrak{k}$ is reductive in $\mathfrak{g}$ and $M$ is a finitely
semisimple $\mathfrak{g}$-module, then $M$ is finitely semisimple as a
$\mathfrak{k}$-module by [6, Prop. 1.7.9(ii)]. In particular, suppose that we
have inclusions of Lie algebras

$\mathfrak{l}\subseteq\mathfrak{m}\subseteq\mathfrak{g}$

where $\mathfrak{m}$ is reductive in $\mathfrak{g}$ and $\mathfrak{l}$ is
reductive in $\mathfrak{m}$, then $\mathfrak{l}$ is reductive in
$\mathfrak{g}$.

### 3.8.

For a homomorphic image of a Lie algebra $\mathfrak{k}\subseteq\mathfrak{g}$
which is reductive in $\mathfrak{g}$, we have the following result.

######  Lemma.

Suppose that $\mathfrak{k}$ is reductive in $\mathfrak{g}$. Let
$\phi:\mathfrak{g}\twoheadrightarrow\mathfrak{l}$ be an epimorphism of Lie
algebras. Then $\phi(\mathfrak{k})$ is reductive in $\mathfrak{l}$.

###### Proof.

Let $\mathfrak{g}=\bigoplus_{i=1}^{m}S_{i}$ be a decomposition as a direct sum
of finite dimensional irreducible $\textup{ad}(\mathfrak{k})$-modules. Then
$\mathfrak{l}=\sum_{i=1}^{m}\phi(S_{i})$. Either $\phi(S_{i})=\\{0\\}$ or
$\phi(S_{i})$ is an irreducible $\textup{ad}(\phi(\mathfrak{k}))$-module. By a
straightforward induction argument it follows that
$\mathfrak{l}=\bigoplus_{i\in\mathcal{I}}\phi(S_{i})$ for some subset
$\mathcal{I}\subseteq\\{1,\ldots,m\\}$. This completes the proof. ∎

### 3.9.

For $m\in\mathbb{Z}_{>0}$ denote by
$\delta_{\mathfrak{g}}^{(m)}:\mathfrak{g}\rightarrow\mathfrak{g}^{\times m}$
the Lie algebra homomorphism mapping $x\in\mathfrak{g}$ to the $m$-tuple
$(x,\ldots,x)$. If $\mathfrak{k}$ is a Lie subalgebra of $\mathfrak{g}$, then
we denote by $\mathfrak{k}^{(m)}\subseteq\mathfrak{g}^{\times m}$ its image
under $\delta_{\mathfrak{g}}^{(m)}$.

######  Proposition.

Suppose that $\mathfrak{g}$ is semisimple and that $\mathfrak{k}$ is reductive
in $\mathfrak{g}$. Then $\mathfrak{k}^{(m)}$ is reductive in
$\mathfrak{g}^{\times m}$.

###### Proof.

By Lemma 3.8, $\mathfrak{k}^{(m)}$ is reductive in $\mathfrak{g}^{(m)}$. Note
that

$\mathfrak{g}^{(m)}=(\mathfrak{g}^{\times m})^{\theta_{m}}$

with $\theta_{m}$ the automorphism of $\mathfrak{g}^{\times m}$ of order $m$
defined by

$\theta_{m}(x_{1},\ldots,x_{m}):=(x_{m},x_{1},\ldots,x_{m-1}).$

Proposition 3.6 then shows that $\mathfrak{g}^{(m)}$ is reductive in
$\mathfrak{g}^{\times m}$. Hence $\mathfrak{k}^{(m)}$ is reductive in
$\mathfrak{g}^{\times m}$ by §3.7. ∎

### 3.10.

Let $\mathfrak{k}\subseteq\mathfrak{g}$ be an inclusion of Lie algebras. We
say that $\mathfrak{g}$ is a reductive extension of $\mathfrak{k}$ when the
inclusion map $\mathfrak{k}\hookrightarrow\mathfrak{g}$ is a section of
$\textup{ad}(\mathfrak{k})$-modules and the quotient module
$\mathfrak{g}/\mathfrak{k}$ is a semisimple $\mathfrak{k}$-module. We then
typically write $\mathfrak{p}$ for a choice of an
$\textup{ad}(\mathfrak{k})$-invariant complement of $\mathfrak{k}$ in
$\mathfrak{g}$ (which is finitely semisimple as
$\textup{ad}(\mathfrak{k})$-module).

### 3.11.

Let $\mathfrak{k}\subseteq\mathfrak{g}$ be a Lie subalgebra. The following two
statements are equivalent:

  1. (a)

$\mathfrak{k}$ is reductive in $\mathfrak{g}$.

  2. (b)

$\mathfrak{k}$ is a reductive Lie algebra and $\mathfrak{g}$ is a reductive
extension of $\mathfrak{k}$.

In particular, §3.5, §3.6 and §3.7 provide examples of reductive extensions.

### 3.12.

The following result should be compared to the transitivity property in §3.7.

######  Lemma.

Let $\mathfrak{l}\subseteq\mathfrak{m}\subseteq\mathfrak{g}$ be inclusions of
finite dimensional Lie algebras. Suppose that $\mathfrak{g}$ is a reductive
extension of $\mathfrak{m}$ and that $\mathfrak{l}$ is reductive in
$\mathfrak{m}$. Then $\mathfrak{l}$ is reductive in $\mathfrak{g}$.

###### Proof.

Let $\mathfrak{p}\subseteq\mathfrak{g}$ be an
$\textup{ad}(\mathfrak{m})$-invariant subspace such that
$\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{p}$. By the assumptions,
$\mathfrak{m}$ is a finite dimensional semisimple
$\textup{ad}(\mathfrak{l})$-module and $\mathfrak{p}$ is a finite dimensional
semisimple $\textup{ad}(\mathfrak{m})$-module. It then follows from [6, Prop.
1.7.9(ii)] (see also §3.7) that $\mathfrak{p}$ is also semisimple as an
$\textup{ad}(\mathfrak{l})$-module. Since $\mathfrak{l}$ is reductive in
$\mathfrak{m}$, it follows that $\mathfrak{l}$ is also reductive in
$\mathfrak{g}$. ∎

### 3.13.

The following lemma is the analog of Lemma 3.8 for reductive extensions.

######  Lemma.

Let $\mathfrak{g}$ be a reductive extension of $\mathfrak{k}$. Let
$\phi:\mathfrak{g}\twoheadrightarrow\mathfrak{l}$ be an epimorphism of Lie
algebras. Then $\mathfrak{l}$ is a reductive extension of
$\phi(\mathfrak{k})$.

###### Proof.

Let $\mathfrak{p}\subseteq\mathfrak{g}$ be an
$\textup{ad}(\mathfrak{k})$-invariant subspace such that
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$. Let
$\mathfrak{p}=\bigoplus_{i=1}^{m}S_{i}$ be a decomposition as a direct sum of
finite dimensional irreducible $\textup{ad}(\mathfrak{k})$-modules. Then
$\mathfrak{l}=\phi(\mathfrak{k})+\sum_{i=1}^{m}\phi(S_{i})$, and either
$\phi(S_{i})=\\{0\\}$ or $\phi(S_{i})$ is an irreducible
$\textup{ad}(\phi(\mathfrak{k}))$-module. A straightforward induction argument
then shows that
$\mathfrak{l}=\phi(\mathfrak{k})\oplus\bigoplus_{i\in\mathcal{I}}\phi(S_{i})$
for some subset $\mathcal{I}\subseteq\\{1,\ldots,m\\}$. This completes the
proof. ∎

### 3.14.

Let $\mathfrak{k}\subseteq\mathfrak{g}$ be an inclusion of Lie algebras and
$\mathfrak{g}$ a reductive extension of $\mathfrak{k}$. Lepowsky and McCollum
[14, Prop. 4.2] obtained the following criterion to detect whether an
irreducible $\mathfrak{g}$-module is finitely semisimple as a
$\mathfrak{k}$-module.

######  Proposition.

Let $\mathfrak{g}$ be a reductive extension of $\mathfrak{k}$, and $M$ an
irreducible $\mathfrak{g}$-module. Then $M$ is finitely semisimple as a
$\mathfrak{k}$-module unless $M_{\alpha}=0$ for all
$\alpha\in\mathfrak{k}^{\wedge}$.

### 3.15.

Let $\mathfrak{k}\subseteq\mathfrak{g}$ be an inclusion of Lie algebras. Using
the associated canonical inclusion $U(\mathfrak{k})\subseteq U(\mathfrak{g})$
of universal enveloping algebras, we set

$U(\mathfrak{g})^{\mathfrak{k}}:=C_{U(\mathfrak{g})}\bigl{(}U(\mathfrak{k})\bigr{)}$

for the centraliser subalgebra of $U(\mathfrak{k})$ in $U(\mathfrak{g})$
(which equals the centraliser of $\mathfrak{k}$ in $U(\mathfrak{g})$).

Let $M$ be a $\mathfrak{g}$-module, and view it as an
$U(\mathfrak{g})$-module. The corresponding homomorphism
$U(\mathfrak{g})\rightarrow\textup{End}(M)$ restricts to an algebra map

$U(\mathfrak{g})^{\mathfrak{k}}\rightarrow\textup{End}_{\mathfrak{k}}(M).$

As a consequence, for a $\mathfrak{k}$-module $S$ the space
$\textup{Hom}_{\mathfrak{k}}(S,M)$ of $\mathfrak{k}$-linear maps $S\rightarrow
M$ becomes a left $U(\mathfrak{g})^{\mathfrak{k}}$-module, with
$U(\mathfrak{g})^{\mathfrak{k}}$ acting on the codomain $M$.

### 3.16.

Let $\mathfrak{k}\subseteq\mathfrak{g}$ be an inclusion of Lie algebras. Let
$S^{\alpha}$ be a finite dimensional irreducible $\mathfrak{k}$-module of
isomorphism class $\alpha\in\mathfrak{k}^{\wedge}$. For a
$\mathfrak{g}$-module $M$, the space

$\textup{Hom}_{\mathfrak{k}}\bigl{(}S^{\alpha},M)$

models the multiplicity space of $S^{\alpha}$ in $M$. In fact,
$\textup{Hom}_{\mathfrak{k}}\bigl{(}S^{\alpha},M)$ is isomorphic to
$\textup{Hom}_{\mathfrak{k}}\bigl{(}S^{\alpha},M_{\alpha})$ as a complex
vector space, and the $\mathfrak{k}$-module $M_{\alpha}$ is isomorphic to an
algebraic direct sum of
$\textup{dim}(\textup{Hom}_{\mathfrak{k}}(S^{\alpha},M))$ copies of
$S^{\alpha}$ (see [6, §1.2.8]). In particular,
$\textup{Hom}_{\mathfrak{k}}\bigl{(}S^{\alpha},M)=0$ if and only if
$M_{\alpha}=0$. Hence Proposition 3.14 can be restated as follows:

######  Proposition.

Let $\mathfrak{g}$ be a reductive extension of $\mathfrak{k}$, and $M$ an
irreducible $\mathfrak{g}$-module. Then $M$ is finitely semisimple as a
$\mathfrak{k}$-module unless $\textup{Hom}_{\mathfrak{k}}(S^{\alpha},M)=0$ for
all $\alpha\in\mathfrak{k}^{\wedge}$.

The multiplicity space $\textup{Hom}_{\mathfrak{k}}\bigl{(}S^{\alpha},M)$
“remembers” the $\mathfrak{g}$-action on $M$ through the left
$U(\mathfrak{g})^{\mathfrak{k}}$-action from §3.15. Up to isomorphism of
$U(\mathfrak{g})^{\mathfrak{k}}$-modules, the multiplicity space
$\textup{Hom}_{\mathfrak{k}}(S^{\alpha},M)$ does not depend on the choice of
$S^{\alpha}$.

### 3.17.

Let $\mathfrak{g}$ be a reductive extension of $\mathfrak{k}$ and
$\alpha\in\mathfrak{k}^{\wedge}$ an isomorphism class of a finite dimensional
irreducible $\mathfrak{k}$-module. Lepowsky and McCollum [14, Thm. 5.5] showed
that $M\mapsto\textup{Hom}_{\mathfrak{k}}(S^{\alpha},M)$ gives rise to a
bijective correspondence between the isomorphism classes of irreducible
$\mathfrak{g}$-modules $M$ with $M_{\alpha}\not=0$ and the isomorphism classes
of irreducible modules over the quotient algebra
$U(\mathfrak{g})^{\mathfrak{k}}/(U(\mathfrak{g)}^{\mathfrak{k}}\cap
U(\mathfrak{g})\mathcal{J}^{\alpha})$, where $\mathcal{J}^{\alpha}\subseteq
U(\mathfrak{k})$ is the annihilator of $S^{\alpha}$ in $U(\mathfrak{k})$. In
the context of §3.2, this correspondence goes back to Harish-Chandra [13].

In view of §3.16, we have the following immediate consequence of this
correspondence.

######  Corollary.

Let $\mathfrak{g}$ be a reductive extension of $\mathfrak{k}$. Let $M$ be an
irreducible $\mathfrak{g}$-module.

For each $\alpha\in\mathfrak{k}^{\wedge}$, the multiplicity space
$\textup{Hom}_{\mathfrak{k}}(S^{\alpha},M)$ is either $\\{0\\}$ or it is an
irreducible $U(\mathfrak{g})^{\mathfrak{k}}$-module.

##  4\. Spin graph functions

In this section we introduce the space $\mathcal{H}=\mathcal{H}_{G,\Gamma,S}$
of spin graph functions and construct spanning sets of $\mathcal{H}$ using
local tensor invariants. Here $G$ is a connected compact Lie group, and
$\Gamma=(V,E,s,t)$ is a finite oriented graph with vertices
$V=\\{v_{1},\ldots,v_{r}\\}$, edges $E=\\{e_{1},\ldots,e_{n}\\}$ and source
and target maps $s,t:E\rightarrow V$.

### 4.1.

Let $C(G)$ be the space of continuous complex-valued functions on $G$, viewed
as $G^{\times 2}$-representation by the left-and right-regular $G$-action,

$\bigl{(}(g_{1},g_{2})\cdot f\bigr{)}(g):=f(g_{1}^{-1}gg_{2}).$

Let $\mathcal{R}(G)\subset C(G)$ be the subalgebra of representative functions
on $G$. In other words, $\mathcal{R}(G)$ consists of the functions $f\in C(G)$
which generate a finite dimensional $G^{\times 2}$-subrepresentation of
$C(G)$.

### 4.2.

Let $\pi:G\rightarrow\textup{GL}(M)$ be a finite dimensional continuous
$G$-representation, and denote by $\pi^{*}:G\rightarrow\textup{GL}(M^{*})$ its
dual representation. For $m\in M$ and $\phi\in M^{*}$ we write
$c_{\phi,m}^{\pi}\in C(G)$ for the associated matrix coefficient

$c_{\phi,m}^{\pi}(g):=\phi\bigl{(}\pi(g)m\bigr{)}=(\pi^{*}(g^{-1})\phi)(m)$

of $\pi$ (and $\pi^{*}$). Then $c_{\phi,m}^{\pi}\in\mathcal{R}(G)$. Moreover,
the space $\mathcal{R}^{\pi}(G)$ spanned by the matrix coefficients
$c_{\phi,m}^{\pi}$ ($m\in M$, $\phi\in M^{*}$) is a $G^{\times 2}$-invariant
subspace of $\mathcal{R}(G)$. In fact, the map

$M^{*}\otimes
M\overset{\sim}{\longrightarrow}\mathcal{R}^{\pi}(G),\qquad\phi\otimes
m\mapsto c_{\phi,m}^{\pi}$

is an isomorphism of $G^{\times 2}$-representations, where $M^{*}\otimes M$ is
endowed with the natural tensor product action of $G^{\times 2}$.

### 4.3.

Let $G^{\wedge}$ be the set of isomorphism classes of irreducible finite
dimensional continuous $G$-representations. We denote the isomorphism class of
an irreducible finite dimensional $G$-representation simply by its
representation map $\pi$. The associated representation space is then denoted
by $M^{\pi}$.

The Peter-Weyl theorem yields the decomposition

$\mathcal{R}(G)=\bigoplus_{\pi\in G^{\wedge}}\mathcal{R}^{\pi}(G)$

of $\mathcal{R}(G)$ in irreducible $G^{\times 2}$-subrepresentations.

### 4.4.

For two $G$-representations $M$ and $N$ we identify $M^{*}\otimes
N^{*}\simeq(M\otimes N)^{*}$ as $G^{\times 2}$-representations, where
$M\otimes N$ and $M^{*}\otimes N^{*}$ are endowed with the tensor product
$G^{\times 2}$-action. Under this correspondence, $\phi\otimes\psi$ for
$\phi\in M^{*}$ and $\psi\in N^{*}$ corresponds to the linear functional on
$M\otimes N$ satisfying $m\otimes n\mapsto\phi(m)\psi(n)$ for $m\in M$ and
$n\in N$. In particular, if $\\{m_{i}\\}_{i}$ and $\\{n_{j}\\}_{j}$ are bases
of $M$ and $N$ and $\\{m_{i}^{*}\\}_{i}$ and $\\{n_{j}^{*}\\}_{j}$ are the
respective dual bases of $M^{*}$ and $N^{*}$, then the basis $\\{m_{i}\otimes
n_{j}\\}_{i,j}$ of $M\otimes N$ has dual basis $\\{m_{i}^{*}\otimes
n_{j}^{*}\\}_{i,j}$.

### 4.5.

We write $G^{E}$ for the compact product Lie group $G^{E}$. Its elements are
denoted either by $\bm{g}=(g_{e})_{e\in E}$ or by
$\bm{g}=(g_{1},\ldots,g_{n})$, with $g_{j}=g_{e_{j}}$ the group element
attached to the edge $e_{j}$. The group $G^{E}$ is called the group of graph
$G$-connections on $\Gamma$.

### 4.6.

For each vertex $v\in V$ we fix a Lie subgroup $K_{v}\subseteq G$. It will
play the role as local gauge group at the vertex $v$. The product group
$\mathbf{K}:=\prod_{v\in V}K_{v}$ is the associated gauge group. It is a
subgroup of the group $G^{V}$ of lattice gauge transformations.

A group element in $\mathbf{K}$ is denoted by $\bm{k}=(k_{v})_{v\in V}$ with
$k_{v}\in K_{v}$. We will sometimes write $\bm{k}=(k_{1},\ldots,k_{r})$ with
$k_{i}=k_{v_{i}}$. The gauge action of $\mathbf{K}$ on $G^{E}$ is defined by

$\bm{k}\cdot\bm{g}:=\bigl{(}k_{s(e)}g_{e}k_{t(e)}^{-1}\bigr{)}_{e\in
E}\qquad\qquad\bigl{(}\bm{k}\in\mathbf{K},\,\,\bm{g}\in G^{E}\bigr{)}.$

### 4.7.

Let $\sigma:\mathbf{K}\rightarrow\textup{GL}(S)$ be a finite dimensional
representation. The space of global algebraic sections of the associated
vector bundle over $G^{E}/\mathbf{K}$ is denoted by
$\mathcal{H}=\mathcal{H}_{\Gamma,G,S}$. Concretely, $\mathcal{H}$ is the space

$\bigl{(}\mathcal{R}(G^{E})\otimes S\bigr{)}^{\mathbf{K}}$

of $\mathbf{K}$-invariant $S$-valued representative functions $f$ on $G^{E}$,
relative to the $\mathbf{K}$-action

$\bigl{(}\bm{k}\cdot f)(\bm{g}):=\sigma(\bm{k})f(\bm{k}^{-1}\cdot\bm{g})$

on $\mathcal{R}(G^{E})\otimes S$. In other words, $\mathcal{H}$ consists of
the $\mathbf{S}$-valued representative functions $f$ on $G^{E}$ satisfying the
equivariance property

$f(\bm{k}\cdot\bm{g})=\sigma(\bm{k})f(\bm{g})$

for $\bm{k}\in\mathbf{K}$ and $\bm{g}\in G^{E}$. We call functions
$f\in\mathcal{H}$ spin graph functions (spin refers to the interpretation of
$S$ as spin space for the associated quantum spin system, see §5.9, §5.12 and
§5.13).

### 4.8.

Fix $\pi\in(G^{E})^{\wedge}$ an isomorphism class of a finite dimensional
irreducible representation of $G^{E}$ and fix a finite dimensional
representation $\sigma:\mathbf{K}\rightarrow\textup{GL}(S)$. Then the subspace

$\mathcal{R}^{\pi}(G^{E})\otimes S\subseteq\mathcal{R}(G^{E})\otimes S$

of $\pi$-elementary spin graph functions is $\mathbf{K}$-invariant.

We call a spin graph function $f\in\mathcal{H}$ elementary if $f$ is
$\pi$-elementary for some $\pi\in(G^{E})^{\wedge}$. We denote

$\mathcal{H}^{\pi}=\mathcal{H}^{\pi}_{\Gamma,G,S}$

for the space $(\mathcal{R}^{\pi}(G^{E})\otimes S)^{\mathbf{K}}$ of
$\pi$-elementary spin graph functions.

Note that the elementary spin graph functions span $\mathcal{H}$, since

$(\mathcal{R}(G^{E})\otimes
S)^{\mathbf{K}}=\bigoplus_{\pi\in(G^{E})^{\wedge}}(\mathcal{R}^{\pi}(G^{E})\otimes
S)^{\mathbf{K}}$

by the Peter-Weyl theorem (see §4.3).

### 4.9.

Let $\pi_{e_{j}}=\pi_{j}:G\rightarrow\textup{GL}(M_{j})$ be finite dimensional
$G$-representations, attached to the edges of $\Gamma$. The associated tensor
product representation $\bm{\pi}:G^{E}\rightarrow\textup{GL}(\mathbf{M})$ is
defined by

$\bm{\pi}(\bm{g}):=\pi_{1}(g_{1})\otimes\cdots\otimes\pi_{n}(g_{n}),$

where $\mathbf{M}:=M_{1}\otimes\cdots\otimes M_{n}$. It is convenient to think
of the local representations $\pi_{e}$ ($e\in E$) as a choice of coloring of
the edges of $\Gamma$.

We will identify $\mathbf{M}^{*}\simeq M_{1}^{*}\otimes\cdots\otimes
M_{n}^{*}$ as in §4.9. In particular, for pure tensors
$\bm{m}:=m_{1}\otimes\cdots\otimes m_{n}\in\mathbf{M}$ and
$\bm{\phi}:=\phi_{1}\otimes\cdots\otimes\phi_{n}\in
M_{1}^{*}\otimes\cdots\otimes M_{n}^{*}$, the matrix coefficient
$c_{\bm{\phi},\bm{m}}^{\bm{\pi}}$ of $\mathbf{M}$ is

$c_{\bm{\phi},\bm{m}}^{\bm{\pi}}(\bm{g})=c_{\phi_{1},m_{1}}^{\pi_{1}}(g_{1})\cdots
c_{\phi_{n},m_{n}}^{\pi_{n}}(g_{n}),\qquad\quad\bm{g}\in\mathbf{G}.$

When the $\pi_{j}:G\rightarrow\textup{GL}(M^{\pi_{j}})$ are all irreducible,
then $\bm{\pi}$ is irreducible and its representation space will be denoted by
$\mathbf{M}^{\bm{\pi}}$. The assigment $(\pi_{e})_{e\in E}\mapsto\bm{\pi}$
induces a bijection

$(G^{\wedge})^{\times n}\overset{\sim}{\longrightarrow}(G^{E})^{\wedge}.$

If $\pi\in(G^{E})^{\wedge}$ then we call the $\pi_{e}\in G^{\wedge}$ such that
$\pi\simeq\bm{\pi}$ the local components of $\pi$. The local components of
$\pi^{*}$ are $\pi_{e}^{*}$.

### 4.10.

Similarly we denote tensor product representations of the gauge group
$\mathbf{K}=\prod_{v\in V}K_{v}$ by
$\bm{\sigma}:\mathbf{K}\rightarrow\textup{GL}(\mathbf{S})$ with

$\bm{\sigma}(\bm{k}):=\sigma_{1}(k_{1})\otimes\cdots\otimes\sigma_{r}(k_{r})$

and $\mathbf{S}:=S_{1}\otimes\cdots\otimes S_{r}$, where
$\sigma_{v_{i}}=\sigma_{i}:K_{v_{i}}\rightarrow\textup{GL}(S_{i})$ are finite
dimensional representations of the local gauge groups $K_{v_{i}}$. We now
think of the local representations $\sigma_{v}$ ($v\in V$) as a choice of
coloring of the vertices of $\Gamma$.

If $\sigma:\mathbf{K}\rightarrow\textup{GL}(S)$ is a finite dimensional
irreducible representation then $\sigma$ is isomorphic to a tensor product
representation with finite dimensional irreducible local
$K_{v}$-representations $\sigma_{v}:K_{v}\rightarrow\textup{GL}(S_{v})$.

### 4.11.

The product group $G^{E}\times G^{E}$ acts on $\mathcal{R}(G)^{\otimes n}$ by

$(\bm{g}^{\prime},\bm{g}^{\prime\prime})\cdot(f_{1}\otimes\cdots\otimes
f_{n}):=(g_{1}^{\prime},g_{1}^{\prime\prime})\cdot
f_{1}\otimes\cdots\otimes(g_{n}^{\prime},g_{n}^{\prime\prime})\cdot f_{n},$

where $\bm{g}^{\prime}=(g_{1}^{\prime},\ldots,g_{n}^{\prime})\in G^{E}$ and
$\bm{g}^{\prime\prime}=(g_{1}^{\prime\prime},\ldots,g_{n}^{\prime\prime})\in
G^{E}$. The linear isomorphism

(6) $\begin{split}&\mathcal{R}(G)^{\otimes
n}\overset{\sim}{\longrightarrow}\mathcal{R}(G^{E})\\\
&f_{1}\otimes\cdots\otimes f_{n}\mapsto\bm{f},\end{split}$

with $\bm{f}\in\mathcal{R}(G^{E})$ defined by

$\bm{f}(\bm{g}):=f_{1}(g_{1})\cdots f_{n}(g_{n}),$

intertwines the $G^{E}\times G^{E}$-actions.

### 4.12.

For $\pi_{j}\in G^{\wedge}$ ($1\leq j\leq n$) the isomorphism (6) restricts to
an isomorphism

$\mathcal{R}^{\pi_{1}}(G)\otimes\cdots\otimes\mathcal{R}^{\pi_{n}}(G)\overset{\sim}{\longrightarrow}\mathcal{R}^{\bm{\pi}}(G^{E}).$

It maps $c_{\phi_{1},m_{1}}^{\pi_{1}}\otimes\cdots\otimes
c_{\phi_{n},m_{n}}^{\pi_{n}}$ to $c_{\bm{\phi},\bm{m}}^{\bm{\pi}}$, where
$\bm{\phi}:=\phi_{1}\otimes\cdots\otimes\phi_{n}$ and
$\bm{m}:=m_{1}\otimes\cdots\otimes m_{n}$, cf. §4.9.

### 4.13.

The star $\mathcal{S}(v)$ of $v\in V$ is the set of edges $e\in E$ with source
and/or target equal to $v$. Then

$\mathcal{S}(v)=\mathcal{S}(v|s)\cup\mathcal{S}(v|t)$

where $\mathcal{S}(v|s)$ is the set of edges oriented outward of $v$ and
$\mathcal{S}(v|t)$ the set of edges oriented toward $v$. Note that the union
may not be disjoint since we allow loops in the graph $\Gamma$.

We consider the $K_{v}$-representation

$\pi_{\mathcal{S}(v)}:K_{v}\rightarrow\textup{GL}(\mathbf{M}^{\pi_{\mathcal{S}(v)}})$

with

$\mathbf{M}^{\pi_{\mathcal{S}(v)}}:=\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|s)}M^{\pi_{e}^{*}}\Bigr{)}\otimes\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|t)}M^{\pi_{e}}\Bigr{)}$

and $K_{v}$ acting diagonally,

$\pi_{\mathcal{S}(v)}(k_{v}):=\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|s)}\pi_{e}^{*}(k_{v})\Bigr{)}\otimes\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|t)}\pi_{e}(k_{v})\Bigr{)}.$

Here the tensor factors are ordered using the total order on
$\mathcal{S}(v|s)$ and $\mathcal{S}(v|t)$ induced from the total order on $E$.

We will identify the dual $K_{v}$-representation
$\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v)}}\bigr{)}^{*}$ with

$\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|s)}M^{\pi_{e}}\Bigr{)}\otimes\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|t)}M^{\pi_{e}^{*}}\Bigr{)},$

with $K_{v}$ acting diagonally, cf. §4.4.

### 4.14.

Fix finite dimensional $K_{v}$-representations
$\sigma_{v}:K_{v}\rightarrow\textup{GL}(S_{v})$ for $v\in V$ and finite
dimensional $G$-representations $\pi_{e}:G\rightarrow\textup{GL}(M_{e})$ for
$e\in E$, thus providing the vertices and the edges of $\Gamma$ with
representation colors.

At vertex $v\in V$ we assign to the colored graph $\Gamma$ the
$K_{v}$-representation

$\mathbf{M}^{\pi_{\mathcal{S}(v)}}\otimes S_{v},$

with $K_{v}$ acting diagonally, and we endow

$\bigotimes_{v\in V}\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v)}}\otimes
S_{v}\bigr{)}$

with the tensor product action of the gauge group $\mathbf{K}=\prod_{v\in
V}K_{v}$. Concretely,

$\bm{k}\cdot\bigotimes_{v\in V}\bigl{(}C_{v}\otimes
u_{v}\bigr{)}:=\bigotimes_{v\in
V}\bigl{(}\pi_{\mathcal{S}(v)}(k_{v})C_{v}\otimes\sigma_{v}(k_{v})u_{v}\bigr{)}$

for $\bm{k}=(k_{v})_{v\in V}\in\mathbf{K}$,
$C_{v}\in\mathbf{M}^{\pi_{\mathcal{S}(v)}}$ and $u_{v}\in S_{v}$.

### 4.15.

For $\pi_{e}\in G^{\wedge}$ and
$\sigma_{v}:K_{v}\rightarrow\textup{GL}(S_{v})$ finite dimensional
$K_{v}$-representations, consider the linear map

$\Psi^{\bm{\pi}}:\,\bigotimes_{v\in
V}\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v)}}\otimes
S_{v}\bigr{)}\rightarrow\mathcal{R}^{\bm{\pi}}(G^{E})\otimes\mathbf{S}$

defined on pure tensors by

(7) $\Psi^{\bm{\pi}}\Bigl{(}\bigotimes_{v\in
V}\Bigl{(}\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|s)}\phi_{e}\Bigr{)}\otimes\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|t)}m_{e}\Bigr{)}\otimes
u_{v}\Bigr{)}\Bigr{)}:=c_{\bm{\phi},\bm{m}}^{\bm{\pi}}\otimes\bm{u}$

with $\bm{\phi}:=\bigotimes_{e\in E}\phi_{e}$, $\bm{m}:=\bigotimes_{e\in
E}m_{e}$ and $\bm{u}:=\bigotimes_{v\in V}u_{v}$. This is well defined due to
the disjoint union decompositions

$\bigsqcup_{v\in V}\mathcal{S}(v|s)=E=\bigsqcup_{v\in V}\mathcal{S}(v|t)$

of the edge set $E$ of $\Gamma$.

######  Theorem.

The linear map $\Psi^{\bm{\pi}}$ defines a $\mathbf{K}$-linear isomorphism

$\Psi^{\bm{\pi}}:\,\bigotimes_{v\in
V}\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v)}}\otimes
S_{v}\bigr{)}\overset{\sim}{\longrightarrow}\mathcal{R}^{\bm{\pi}}(G^{E})\otimes\mathbf{S}.$

It restricts to a linear isomorphism

$\Psi^{\bm{\pi}}:\,\bigotimes_{v\in
V}\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v)}}\otimes
S_{v}\bigr{)}^{K_{v}}\overset{\sim}{\longrightarrow}\bigl{(}\mathcal{R}^{\bm{\pi}}(G^{E})\otimes\mathbf{S}\bigr{)}^{\mathbf{K}}.$

###### Proof.

By §4.12 it is clear that $\Psi^{\bm{\pi}}$ is a linear isomorphism. It is
$\mathbf{K}$-linear, since for $\bm{k}=(k_{v})_{v\in V}\in\mathbf{K}$ and
$\bm{g}\in G^{E}$,

$\begin{split}\Psi^{\bm{\pi}}\Bigl{(}\bigotimes_{v\in
V}&\Bigl{(}\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|s)}\phi_{e}\pi_{e}(k_{v}^{-1})\Bigr{)}\otimes\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|t)}\pi_{e}(k_{v})m_{e}\Bigr{)}\otimes\sigma_{v}(k_{v})u_{v}\Bigr{)}\Bigr{)}(\bm{g})=\\\
&\qquad\qquad\quad=\Bigl{(}\prod_{e\in
E}\phi_{e}\bigl{(}\pi_{e}(k_{s(e)}^{-1}g_{e}k_{t(e)})m_{e}\bigr{)}\Bigr{)}\bm{\sigma}(\bm{k})\bm{u}\\\
&\qquad\qquad\quad=c_{\bm{\phi},\bm{m}}^{\bm{\pi}}(\bm{k}^{-1}\cdot\bm{g})\bm{\sigma}(\bm{k})\bm{u}.\end{split}$

The second statement of the theorem follows now immediately from the fact that
$\bigotimes_{v\in V}\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v)}}\otimes
S_{v}\bigr{)}$ is endowed with the tensor product action of
$\mathbf{K}=\prod_{v\in V}K_{v}$, see §4.14. ∎

### 4.16.

We keep the setup as in §4.15. For $e\in E$ let
$\\{m_{e,i_{e}}\\}_{i_{e}\in\mathcal{I}_{e}}$ be a basis of $M^{\pi_{e}}$. For
$v\in V$ set

$\mathcal{I}(v|s):=\\{\bm{i}=(i_{e})_{e\in\mathcal{S}(v|s)}\,\,|\,\,i_{e}\in\mathcal{I}_{e}\\}$

and write for $\bm{i}\in\mathcal{I}(v|s)$,

$\bm{m}_{\bm{i}}(v|s):=\bigotimes_{e\in\mathcal{S}(v|s)}m_{e,i_{e}}.$

In a similar way we define $\bm{m}_{\bm{i}}(v|t)$ for indices $\bm{i}$ from

$\mathcal{I}(v|t):=\\{\bm{i}=(i_{e})_{e\in\mathcal{S}(v|t)}\,\,|\,\,i_{e}\in\mathcal{I}_{e}\\}.$

Then

(8)
$\\{\,\bm{m}_{\bm{i}}(v|s)^{*}\otimes\bm{m}_{\bm{j}}(v|t)\,\,\,\,\,|\,\,\,\,\,(\bm{i},\bm{j})\in\mathcal{I}(v|s)\times\mathcal{I}(v|t)\,\\}$

is a basis of $\mathbf{M}^{\pi_{\mathcal{S}(v)}}$. We then have a linear
isomorphism

(9)
$\textup{Hom}_{K_{v}}\bigl{(}(\mathbf{M}^{\pi_{\mathcal{S}(v)}})^{*},S_{v}\bigr{)}\overset{\sim}{\longrightarrow}\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v)}}\otimes
S_{v}\bigr{)}^{K_{v}}$

mapping
$\Phi\in\textup{Hom}_{K_{v}}\bigl{(}(\mathbf{M}^{\pi_{\mathcal{S}(v)}})^{*},S_{v}\bigr{)}$
to the invariant tensor

$\sum_{\bm{i}\in\mathcal{I}(v|s)}\sum_{\bm{j}\in\mathcal{I}(v|t)}\bm{m}_{\bm{i}}(v|s)^{*}\otimes\bm{m}_{\bm{j}}(v|t)\otimes\Phi\bigl{(}\bm{m}_{\bm{i}}(v|s)\otimes\bm{m}_{\bm{j}}(v|t)^{*}\bigr{)}.$

Combined with Theorem 4.15 we thus obtain a parametrisation of the space
$\bigl{(}\mathcal{R}^{\bm{\pi}}(G^{E})\otimes\mathbf{S}\bigr{)}^{\mathbf{K}}$
of ${\bm{\pi}}$-elementary spin graph functions in terms of spaces of local
intertwiners (local in the sense that they only depend on the colors of
$\Gamma$ at the star of a vertex $v$).

### 4.17.

Let $n\geq 1$ and consider the directed cycle graph $\Gamma$ with $n$ edges.
We enumerate the vertices $v_{i}$ and edges $e_{j}$
($i,j\in\mathbb{Z}/n\mathbb{Z}$) in such a way that $s(e_{i})=v_{i}$ and
$t(e_{i})=v_{i+1}$ for $i\in\mathbb{Z}/n\mathbb{Z}$. The order $1<2<\cdots<n$
on $\mathbb{Z}/n\mathbb{Z}$ provides a total order on $V$ and $E$. We take
$\mathbf{K}=G^{V}$ as gauge group.

With these conventions $G^{E}\simeq G^{\times n}$ by $(g_{e})_{e\in
E}\mapsto(g_{e_{1}},\ldots,g_{e_{n}})$, and $\mathbf{K}\simeq G^{\times n}$ by
$(k_{v})_{v\in V}\mapsto(k_{v_{1}},\ldots,k_{v_{n}})$. The left gauge action
of $\mathbf{K}$ on $G^{E}$ then reads as

$\bm{k}\cdot\bm{g}:=(k_{1}g_{1}k_{2}^{-1},k_{2}g_{2}k_{3}^{-1},\ldots,k_{n}g_{n}k_{1}^{-1})$

for $\bm{k}=(k_{1},\ldots,k_{n})\in G^{\times n}\simeq\mathbf{K}$ and
$\bm{g}=(g_{1},\ldots,g_{n})\in G^{\times n}\simeq G^{E}$.

Let $\sigma_{i}:G\rightarrow\textup{GL}(S_{i})$ be finite dimensional
$K_{v_{i}}=G$-representations attached to the vertices $v_{i}$, and
$\pi_{i}:G\rightarrow\textup{GL}(M^{\pi_{i}})$ finite dimensional irreducible
$G$-representation attached to the edge $e_{i}$.

The partial trace $\textup{Tr}_{M^{\pi_{n}}}^{\mathbf{S}}(B)$ of
$B\in\textup{Hom}(M^{\pi_{n}},M^{\pi_{n}}\otimes\mathbf{S})$ is the unique
vector in $\mathbf{S}$ satisfying

$\phi\bigl{(}\textup{Tr}_{M^{\pi_{n}}}^{\mathbf{S}}(B)\bigr{)}=\textup{Tr}_{M^{\pi_{n}}}\bigl{(}(\textup{id}_{M^{\pi_{n}}}\otimes\phi)B\bigr{)}\qquad\quad\forall\,\phi\in\mathbf{S}^{*},$

where $\textup{Tr}_{M^{\pi_{n}}}$ is the usual trace on
$\textup{End}(M^{\pi_{n}})$. The elementary spin graph functions now have the
following description in terms of partial traces of compositions of
intertwiners.

######  Proposition.

Endow $M^{\pi_{i-1}}\otimes S_{i}$ with the diagonal $G$-action. We have a
linear isomorphism

$\begin{split}\bigotimes_{i\in\mathbb{Z}/n\mathbb{Z}}\textup{Hom}_{G}\bigl{(}M^{\pi_{i}},M^{\pi_{i-1}}\otimes\mathbf{S}_{i}\bigr{)}&\overset{\sim}{\longrightarrow}\bigl{(}\mathcal{R}^{\bm{\pi}}(G^{\times
n})\otimes\mathbf{S}\bigr{)}^{\mathbf{K}},\\\
\bigotimes_{i\in\mathbb{Z}/n\mathbb{Z}}\Phi_{i}&\mapsto
f_{\bm{\Phi}}^{\bm{\pi}}\end{split}$

with $\bm{\Phi}:=(\Phi_{1},\ldots,\Phi_{n})$ and
$f_{\bm{\Phi}}^{\bm{\pi}}\in\bigl{(}\mathcal{R}^{\pi}(G^{\times
n})\otimes\mathbf{S}\bigr{)}^{\mathbf{K}}$ the elementary $n$-point trace
function

$f_{\bm{\Phi}}^{\bm{\pi}}(\bm{g}):=\textup{Tr}_{M^{\pi_{n}}}^{\mathbf{S}}\Bigl{(}(\Phi_{1}\pi_{1}(g_{1})\otimes\textup{id}_{\mathbf{S}_{2}\otimes\cdots\otimes\mathbf{S}_{n}})\cdots(\Phi_{n-1}\pi_{n-1}(g_{n-1})\otimes\textup{id}_{\mathbf{S}_{n}})\Phi_{n}\pi_{n}(g_{n})\Bigr{)}.$

###### Proof.

In the current situation we have
$\mathbf{M}^{\pi_{\mathcal{S}(v_{i})}}=M^{\pi_{i}^{*}}\otimes M^{\pi_{i-1}}$.
Using (9) and the fact that $K_{v_{i}}=G$ we obtain

$\begin{split}\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v_{i})}}\otimes
S_{i}\bigr{)}^{K_{v_{i}}}&\simeq\textup{Hom}_{G}\bigl{(}M^{\pi_{i}}\otimes
M^{\pi_{i-1}^{*}},S_{i})\\\
&\simeq\textup{Hom}_{G}(M^{\pi_{i}},M^{\pi_{i-1}}\otimes S_{i}),\end{split}$

where $M^{\pi_{i}}\otimes M^{\pi_{i-1}^{*}}$ is considered as
$G$-representations via the diagonal $G$-action. The isomorphism
$\textup{Hom}_{G}(M^{\pi_{i}},M^{\pi_{i-1}}\otimes
S_{i})\overset{\sim}{\longrightarrow}\textup{Hom}_{G}(M^{\pi_{i}}\otimes
M^{\pi_{i-1}^{*}},S_{i})$ maps $\Phi_{i}$ to the $G$-intertwiner
$m_{i}\otimes\phi_{i-1}\mapsto(\phi_{i-1}\otimes\textup{id}_{S_{i}})\Phi_{i}(m_{i})$.

Under these identifications the intertwiner
$\Phi_{i}\in\textup{Hom}_{G}\bigl{(}M^{\pi_{i}},M^{\pi_{i-1}}\otimes
S_{i}\bigr{)}$ corresponds to the local invariant tensor

$\widetilde{\Phi}_{i}:=\sum_{k\in\mathcal{I}_{e_{i}}}m_{e_{i},k}^{*}\otimes\Phi_{i}(m_{e_{i},k})\in\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v_{i})}}\otimes
S_{i}\bigr{)}^{K_{v_{i}}}.$

Combined with Theorem 4.15 we thus obtain a linear isomorphism

$\bigotimes_{i\in\mathbb{Z}/n\mathbb{Z}}\textup{Hom}_{G}\bigl{(}M^{\pi_{i}},M^{\pi_{i-1}}\otimes
S_{i}\bigr{)}\overset{\sim}{\longrightarrow}\bigl{(}\mathcal{R}^{\bm{\pi}}(G^{\times
n})\otimes\mathbf{S}\bigr{)}^{\mathbf{K}},\qquad\bigotimes_{i\in\mathbb{Z}/n\mathbb{Z}}\Phi_{i}\mapsto\widetilde{f}_{\bm{\Phi}}^{\bm{\pi}}$

with
$\widetilde{f}_{\bm{\Phi}}^{\bm{\pi}}:=\Psi^{\bm{\pi}}\bigl{(}\bigotimes_{i\in\mathbb{Z}/n\mathbb{Z}}\widetilde{\Phi}_{i}\bigr{)}$.

Rewriting $\widetilde{\Phi}_{i}$ as

(10)
$\widetilde{\Phi}_{i}=\sum_{k_{i-1}\in\mathcal{I}_{e_{i-1}}}\sum_{\ell_{i}\in\mathcal{I}_{e_{i}}}m_{e_{i},\ell_{i}}^{*}\otimes
m_{e_{i-1},k_{i-1}}\otimes\bigl{(}(m_{e_{i-1},k_{i-1}}^{*}\otimes\textup{id}_{S_{i}})\Phi_{i}(m_{e_{i},\ell_{i}})\bigr{)}$

and applying (7), we obtain the explicit expression

$\widetilde{f}_{\bm{\Phi}}^{\bm{\pi}}(\bm{g})=\sum_{k_{1}\in\mathcal{I}_{e_{1}}}\cdots\sum_{k_{n}\in\mathcal{I}_{e_{n}}}\bigotimes_{i\in\mathbb{Z}/n\mathbb{Z}}\bigl{(}m_{e_{i-1},k_{i-1}}^{*}\pi_{i-1}(g_{i-1})\otimes\textup{id}_{\mathbf{S}_{i}}\bigr{)}\Phi_{i}(m_{e_{i},k_{i}}).$

For $i\not=n$ the sum over $k_{i}\in\mathcal{I}_{e_{i}}$ can be simplified
using the identity

$\begin{split}\sum_{k_{i}\in\mathcal{I}_{e_{i}}}\Phi_{i}(m_{e_{i},k_{i}})\otimes\bigl{(}(m_{e_{i},k_{i}}^{*}\pi_{i}(g_{i})\otimes\textup{id}_{S_{i+1}})&\Phi_{i+1}(m_{e_{i+1},k_{i+1}})\bigr{)}=\\\
&=(\Phi_{i}\pi_{i}(g_{i})\otimes\textup{id}_{S_{i+1}})\Phi_{i+1}(m_{e_{i+1},k_{i+1}})\end{split}$

in $M^{\pi_{i-1}}\otimes S_{i}\otimes S_{i+1}$. The expression for
$\widetilde{f}_{\bm{\Phi}}^{\bm{\pi}}(\bm{g})$ then reduces to

$\begin{split}\widetilde{f}_{\bm{\Phi}}^{\bm{\pi}}&(\bm{g})=\sum_{k_{n}\in\mathcal{I}_{e_{n}}}\left\\{\bigl{(}m_{e_{n},k_{n}}^{*}\pi_{n}(g_{n})\otimes\textup{id}_{\mathbf{S}}\bigr{)}\right.(\Phi_{1}\pi_{1}(g_{1})\otimes\textup{id}_{S_{2}\otimes\cdots\otimes
S_{n}})\cdots\\\
&\left.\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\cdots(\Phi_{n-1}\pi_{n-1}(g_{n-1})\otimes\textup{id}_{S_{n}})\Phi_{n}(m_{e_{n},k_{n}})\right\\}\\\
=&\textup{Tr}_{M^{\pi_{n}}}^{\bm{S}}\Bigl{(}(\pi_{n}(g_{n})\otimes\textup{id}_{\mathbf{S}})(\Phi_{1}\pi_{1}(g_{1})\otimes\textup{id}_{S_{2}\otimes\cdots\otimes
S_{n}})\cdots(\Phi_{n-1}\pi_{n-1}(g_{n-1})\otimes\textup{id}_{S_{n}})\Phi_{n}\Bigr{)}.\end{split}$

Hence $\widetilde{f}_{\bm{\Phi}}^{\bm{\pi}}=f_{\bm{\Phi}}^{\bm{\pi}}$ by the
cyclicity of the partial trace:

$\textup{Tr}_{M^{\pi_{n}}}^{\mathbf{S}}((A\otimes\textup{id}_{\mathbf{S}})B)=\textup{Tr}_{M^{\pi_{n}}}^{\mathbf{S}}(BA)$

for $A\in\textup{End}(M^{\pi_{n}})$ and
$B\in\textup{End}(M^{\pi_{n}}\otimes\mathbf{S})$. ∎

The study of $n$-point trace functions originates from the paper [8].
Intertwiners $\textup{Hom}_{G}\bigl{(}M^{\pi_{i}},M^{\pi_{i-1}}\otimes
S_{i}\bigr{)}$ may be viewed as a topological degenerations of vertex
operators. The class of (elementary) $n$-point trace functions and its
generalisation to the affine and quantum group level are particularly well
studied, see, e.g., [8, 10, 9].

### 4.18.

Let $n\in\mathbb{Z}_{\geq 1}$. As a next example we consider the linearly
ordered linear graph $\Gamma$ with $n$ edges. We denote the ordered vertex and
edge sets by $V=\\{v_{1},\ldots,v_{n+1}\\}$ and $E=\\{e_{1},\ldots,e_{n}\\}$.
The source and target maps are $s(e_{i})=v_{i}$ and $t(e_{i})=v_{i+1}$ for
$i=1,\ldots,n$. As local gauge groups we take

(11) $K_{v_{i}}=\begin{cases}H\qquad&\hbox{ for }\,i=1,\\\ G\qquad&\hbox{ for
}\,i=2,\ldots,n,\\\ K\qquad&\hbox{ for }\,i=n+1,\end{cases}$

where $H,K\subseteq G$ are subgroups.

The action of the associated gauge group

$\mathbf{K}=\prod_{i=1}^{n+1}K_{v_{i}}=H\times G^{\times(n-1)}\times K$

on $G^{E}\simeq G^{\times n}$ then becomes

$\bm{k}\cdot\bm{g}^{\prime}:=(hg_{1}^{\prime}g_{2}^{-1},g_{2}g_{2}^{\prime}g_{3}^{-1},\ldots,g_{n}g_{n}^{\prime}k^{-1})$

for $\bm{k}=(h,g_{2},\ldots,g_{n},k)\in\mathbf{K}$ and
$\bm{g}^{\prime}=(g_{1}^{\prime},\ldots,g_{n}^{\prime})\in G^{E}$.

Let $S_{i}$ ($1<i\leq n$) be finite dimensional $G$-representations, $L$ a
finite dimensional $H$-representation and $N$ a finite dimensional
$K$-representation. Denote by

$\mathbf{S}:=L\otimes S_{2}\otimes\cdots\otimes S_{n}\otimes N$

the resulting tensor product spin representation of $\mathbf{K}$. We also
write

$\underline{S}:=S_{2}\otimes\cdots\otimes S_{n},$

for the “bulk” $G^{\times(n-1)}$-representation associated to $\mathbf{S}$, so
that $\mathbf{S}=L\otimes\underline{S}\otimes N$. Denote by

$Q_{L\otimes\underline{S},N}:\textup{Hom}(N^{*},L\otimes\underline{S})\overset{\sim}{\longrightarrow}\mathbf{S}$

the linear isomorphism defined by

$Q_{L\otimes\underline{S},N}(T):=\sum_{j}T(n_{j}^{*})\otimes n_{j},$

with $\\{n_{j}\\}_{j}$ a basis of $N$ and $\\{n_{j}^{*}\\}_{j}$ the
corresponding dual basis of $N^{*}$.

Let $\bm{\pi}:G^{\times n}\rightarrow\textup{GL}(\mathbf{M}^{\bm{\pi}})$ be an
irreducible tensor product representation, with local components
$\pi_{i}:G\rightarrow\textup{GL}(M^{\pi_{i}})$ ($1\leq i\leq n$). In the
following proposition we will also view $M^{\pi_{1}}$ (resp. $M^{\pi_{n}}$) as
$H$-representation (resp. $K$-representation) by restriction.

######  Proposition.

We have a linear isomorphism

$\begin{split}\textup{Hom}_{H}(M^{\pi_{1}},L)\otimes\,\bigotimes_{i=2}^{n}\textup{Hom}_{G}\bigl{(}M^{\pi_{i}},M^{\pi_{i-1}}\otimes
S_{i}\bigr{)}&\otimes\textup{Hom}_{K}(N^{*},M^{\pi_{n}})\\\
&\qquad\overset{\sim}{\longrightarrow}\bigl{(}\mathcal{R}^{\bm{\pi}}(G^{\times
n})\otimes\mathbf{S}\bigr{)}^{\mathbf{K}}\end{split}$

mapping $\Theta\otimes\bigl{(}\bigotimes_{i=2}^{n}\Phi_{i}\bigr{)}\otimes\Xi$
to the spin graph function
$f_{\Theta,\bm{\Phi},\Xi}^{\bm{\pi}}\in\bigl{(}\mathcal{R}^{\bm{\pi}}(G^{\times
n})\otimes\mathbf{S}\bigr{)}^{\mathbf{K}}$, defined by

$\begin{split}f_{\Theta,\bm{\Phi},\Xi}^{\bm{\pi}}(\bm{g}):=&Q_{L\otimes\underline{S},N}\Bigl{(}(\Theta\pi_{1}(g_{1})\otimes\textup{id}_{\underline{S}})(\Phi_{2}\pi_{2}(g_{2})\otimes\textup{id}_{S_{3}\otimes\cdots\otimes
S_{n}})\cdots\\\
&\,\,\,\quad\qquad\qquad\qquad\qquad\qquad\quad\cdots(\Phi_{n-1}\pi_{n-1}(g_{n-1})\otimes\textup{id}_{S_{n}})\Phi_{n}\pi_{n}(g_{n})\Xi\Bigr{)}.\end{split}$

###### Proof.

At vertices $v_{i}$ with $1<i\leq n$ the analysis of the local space of
invariants $\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v_{i})}}\otimes
S_{i}\bigr{)}^{K_{v_{i}}}$ is as in the proof of Proposition 4.17. For $i=1$
we have

$\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v_{1})}}\otimes
L\bigr{)}^{K_{v_{1}}}=\bigl{(}M^{\pi_{1}^{*}}\otimes
L\bigr{)}^{H}\simeq\textup{Hom}_{H}\bigl{(}M^{\pi_{1}},L),$

with the isomorphism as in §4.16. For $i=n+1$ we analogously have

$\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v_{n+1})}}\otimes
N\bigr{)}^{K_{v_{n+1}}}=\bigl{(}M^{\pi_{n}}\otimes
N\bigr{)}^{K}\simeq\textup{Hom}_{K}(N^{*},M^{\pi_{n}}).$

Under these isomorphisms the intertwiner
$\Theta\in\textup{Hom}_{H}\bigl{(}M^{\pi_{1}},L)$ corresponds

$\widetilde{\Theta}:=\sum_{\ell_{1}\in\mathcal{I}_{e_{1}}}m_{e_{1},\ell_{1}}^{*}\otimes\Theta(m_{e_{1},\ell_{1}})\in\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v_{1})}}\otimes
L\bigr{)}^{K_{v_{1}}}$

and $\Xi\in\textup{Hom}_{K}(N^{*},M^{\pi_{n}})$ to

$\widetilde{\Xi}:=\sum_{j}\Xi(n_{j}^{*})\otimes
n_{j}\in\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v_{n+1})}}\otimes
N\bigr{)}^{K_{v_{n+1}}}.$

Combined with Theorem 4.15 we thus obtain a linear isomorphism

$\begin{split}\textup{Hom}_{H}(M^{\pi_{1}},L)\otimes\,\bigotimes_{i=2}^{n}\textup{Hom}_{G}\bigl{(}M^{\pi_{i}},M^{\pi_{i-1}}\otimes
S_{i}\bigr{)}&\otimes\textup{Hom}_{K}(N^{*},M^{\pi_{n}})\\\
&\qquad\overset{\sim}{\longrightarrow}\bigl{(}\mathcal{R}^{\bm{\pi}}(G^{\times
n})\otimes\mathbf{S}\bigr{)}^{\mathbf{K}}\end{split}$

mapping $\Theta\otimes\bigl{(}\bigotimes_{i=2}^{n}\Phi_{i}\bigr{)}\otimes\Xi$
to

$\widetilde{f}_{\Theta,\bm{\Phi},\Xi}^{\bm{\pi}}:=\Psi^{\bm{\pi}}\Bigl{(}\widetilde{\Theta}\otimes\bigl{(}\bigotimes_{i=2}^{n}\widetilde{\Phi}_{i}\bigr{)}\otimes\widetilde{\Xi}\Bigr{)},$

where $\widetilde{\Phi}_{i}$ is given by (10).

A direct computation now shows that the spin graph function
$\widetilde{f}_{\Theta,\bm{\Phi},\Xi}^{\bm{\pi}}(\bm{g})$ is explicitly given
by

$\begin{split}\sum_{i_{1}\in\mathcal{I}_{e_{1}}}\cdots\sum_{i_{n-1}\in\mathcal{I}_{e_{n-1}}}\sum_{j}&\Theta(m_{e_{1},i_{1}})\otimes\bigl{(}(m_{e_{1},i_{1}}^{*}\pi_{1}(g_{1})\otimes\textup{id}_{S_{2}})\Phi_{2}(m_{e_{2},i_{2}})\bigr{)}\otimes\cdots\\\
&\cdots\otimes\bigl{(}(m_{e_{n-1},i_{n-1}}^{*}\pi_{n-1}(g_{n-1})\otimes\textup{id}_{S_{n}})\Phi_{n}(\pi_{n}(g_{n})\Xi(n_{j}^{*}))\bigr{)}\otimes
n_{j}.\end{split}$

Contracting the bulk intertwiners $\Phi_{i}$ ($i=2,\ldots,n$) as in the proof
of Proposition 4.17, we obtain the expression

$\begin{split}\sum_{i_{1}\in\mathcal{I}_{e_{1}}}\sum_{j}&\Theta(m_{e_{1},i_{1}})\otimes\left\\{(m_{e_{1},i_{1}}^{*}\pi_{1}(g_{1})\otimes\textup{id}_{\underline{S}})(\Phi_{2}\pi_{2}(g_{2})\otimes\textup{id}_{S_{3}\otimes\cdots\otimes
S_{n}})\cdots\right.\\\
&\left.\qquad\qquad\qquad\qquad\quad\cdots\bigl{(}\Phi_{n-1}\pi_{n-1}(g_{n-1})\otimes\textup{id}_{S_{n}}\bigr{)}\Phi_{n}(\pi_{n}(g_{n})\Xi(n_{j}^{*}))\right\\}\otimes
n_{j}\end{split}$

for $\widetilde{f}_{\Theta,\bm{\Phi},\Xi}^{\bm{\pi}}(\bm{g})$, which is easily
seen to be equal to $f_{\Theta,\bm{\Phi},\Xi}^{\bm{\pi}}(\bm{g})$. This
completes the proof. ∎

For $H=K$ the modified spin graph functions
$Q_{L\otimes\underline{S},N}^{-1}f_{\Theta,\bm{\Phi},\Xi}^{\bm{\pi}}$ are the
elementary $n$-point spherical functions from [24, 21]. For $n=1$, they reduce
to the elementary spherical functions on compact symmetric spaces.

##  5\. Quantum spin systems on graph connections

Let $\Gamma$ be a finite oriented graph and $G$ a connected compact Lie group.
We write $V=\\{v_{1},\ldots,v_{r}\\}$ and $E=\\{e_{1},\ldots,e_{n}\\}$ for the
totally ordered vertex and edge set of $\Gamma$. We denote by $\mathfrak{g}$
the complexification of the Lie algebra $\mathfrak{g}_{0}$ of $G$. In this
section we introduce a quantum spin system on the spaces
$\mathcal{H}=\mathcal{H}_{\Gamma,G,S}$ of spin graph functions.

### 5.1.

We call a linear differential operator $D$ on $G$ algebraic if it preserves
the space $\mathcal{R}(G)$ of representative functions on $G$ (see §4.1). We
have an inclusion of algebras

$\mathcal{D}_{\textup{biinv}}(G)\subseteq\mathcal{D}_{\textup{inv}}(G)\subseteq\mathcal{D}(G)$

with $\mathcal{D}(G)$ the algebra of algebraic differential operators on $G$,
with $\mathcal{D}_{\textup{inv}}(G)\subseteq\mathcal{D}(G)$ the subalgebra
generated by the left and right $G$-invariant differential operators on $G$,
and with
$\mathcal{D}_{\textup{biinv}}(G)\subseteq\mathcal{D}_{\textup{inv}}(G)$ the
subalgebra of $G$-biinvariant differential operators on $G$.

### 5.2.

We have a surjective algebra map

(12) $U(\mathfrak{g}^{\times
2})\twoheadrightarrow\mathcal{D}_{\textup{inv}}(G),\qquad X\mapsto D_{X}$

defined by

$\begin{split}\bigl{(}D_{(x,y)}f\bigr{)}(g):=&\frac{d}{ds}\bigg{|}_{s=0}f\bigl{(}\exp(-sx)g\exp(sy)\bigr{)}\\\
=&\frac{d}{ds}\bigg{|}_{s=0}f\bigl{(}\exp(-sx)g\bigr{)}+\frac{d}{dt}\bigg{|}_{t=0}f\bigl{(}g\exp(ty)\bigr{)}\end{split}$

for $(x,y)\in\mathfrak{g}_{0}^{\times 2}$. Identify $U(\mathfrak{g}^{\times
2})\simeq U(\mathfrak{g})\otimes U(\mathfrak{g})$ as algebras, with the
isomorphism $U(\mathfrak{g}^{\times
2})\overset{\sim}{\longrightarrow}U(\mathfrak{g})\otimes U(\mathfrak{g})$
induced by $(x,y)\mapsto x\otimes 1+1\otimes y$ for $x,y\in\mathfrak{g}$. We
then have the balancing condition

$D_{X\otimes ZY}=D_{X\iota(Z)\otimes Y}$

for $X,Y\in U(\mathfrak{g})$ and $Z\in Z(\mathfrak{g})$, where $\iota$ is the
antipode of $U(\mathfrak{g})$ (i.e., $\iota$ is the unique anti-algebra
automorphism of $U(\mathfrak{g})$ such that $x\mapsto-x$ for
$x\in\mathfrak{g}$). Hence the algebra map (12) descends to an isomorphism of
algebras

(13)
$U(\mathfrak{g})\otimes_{Z(\mathfrak{g})}U(\mathfrak{g})\overset{\sim}{\longrightarrow}\mathcal{D}_{\textup{inv}}(G)$

with the balanced tensor product over $Z(\mathfrak{g})$ relative to the
$\iota$-twisted right regular $Z(\mathfrak{g})$-action on $U(\mathfrak{g})$

$X\cdot Z:=X\iota(Z)\qquad\quad(X\in U(\mathfrak{g}),\,\,Z\in
Z(\mathfrak{g}))$

and the left regular $Z(\mathfrak{g})$-action on $U(\mathfrak{g})$
(injectivity of the map (13) was shown in [15]).

### 5.3.

The algebra $\mathcal{D}_{\textup{biinv}}(G)$ of $G$-biinvariant differential
operators on $G$ is isomorphic to $Z(\mathfrak{g})$ via the map

$Z(\mathfrak{g})\overset{\sim}{\longrightarrow}\mathcal{D}_{\textup{biinv}}(G),\qquad\quad
Z\mapsto D_{1\otimes Z}=D_{\iota(Z)\otimes 1}.$

In particular, $\mathcal{D}_{\textup{biinv}}(G)$ is contained in the center of
$\mathcal{D}_{\textup{inv}}(G)$.

### 5.4.

Consider the algebra $\mathcal{D}(G^{E})$ of algebraic differential operators
on the connected compact Lie group $G^{E}$, and recall the gauge action of
$\mathbf{K}$ on $G^{E}$ (see §4.6). The corresponding contragredient
$\mathbf{K}$-action on $\mathcal{R}(G^{E})$ is

$(\mathbf{k}\cdot f)(\mathbf{g}):=f(\mathbf{k}^{-1}\cdot\mathbf{g})$

for $\mathbf{k}\in\mathbf{K}$, $f\in\mathcal{R}(G^{E})$ and
$\mathbf{g}\in\mathbf{G}$ (this is the special case of the $\mathbf{K}$-action
on $\mathcal{R}(G^{E})\otimes S$ from §4.7 when $S$ is the trivial
$\mathbf{K}$-representation). This action induces an $\mathbf{K}$-action

$\mathbf{K}\times\mathcal{D}(G^{E})\rightarrow\mathcal{D}(G^{E}),\qquad\quad(\mathbf{k},D)\mapsto\mathbf{k}\bullet
D$

on $\mathcal{D}(G^{E})$ by algebra automorphisms such that

(14) $\mathbf{k}\cdot(Df)=(\mathbf{k}\bullet D)(\mathbf{k}\cdot f)$

for $\mathbf{k}\in\mathbf{K}$, $D\in\mathcal{D}(G^{E})$ and
$f\in\mathcal{R}(G^{E})$. We denote by
$\mathcal{D}(G^{E})^{\mathbf{K}}\subseteq\mathcal{D}(G^{E})$ the subalgebra of
$\mathbf{K}$-invariant differential operators on $G^{E}$.

### 5.5.

As in §5.2, we identify $U((\mathfrak{g}^{E})^{\times 2})\simeq
U(\mathfrak{g}^{E})\otimes U(\mathfrak{g}^{E})$. Furthermore, we identify
$U(\mathfrak{g}^{E})\simeq U(\mathfrak{g})^{\otimes\\#E}$ as algebras, with
the isomorphism induced by

$(x_{e})_{e\in E}\mapsto\sum_{i=1}^{n}1^{\otimes(i-1)}\otimes x_{e_{i}}\otimes
1^{\otimes(n-i)}$

for $(x_{e})_{e\in E}\in\mathfrak{g}^{E}$. It restricts to an isomorphism
$Z(\mathfrak{g}^{E})\simeq Z(\mathfrak{g})^{\otimes\\#E}$.

We will use the notation

$X^{(i)}:=1^{\otimes(i-1)}\otimes X\otimes 1^{\otimes(n-i)}\in
U(\mathfrak{g})^{\otimes\\#E}$

for $X\in U(\mathfrak{g})$ and $i\in\\{1,\ldots,n\\}$, and a pure tensor in
$U((\mathfrak{g}^{E})^{\times 2})$ will be denoted by
$\mathbf{X}\otimes\mathbf{Y}$ with

$\mathbf{X}=\bigotimes_{e\in
E}X_{e},\qquad\quad\mathbf{Y}=\bigotimes_{e^{\prime}\in E}Y_{e^{\prime}}.$

and $X_{e},Y_{e^{\prime}}\in U(\mathfrak{g})$.

######  Lemma.

The formula

(15)
$\bm{k}\bullet\bigl{(}\mathbf{X}\otimes\mathbf{Y}\bigr{)}:=\Bigl{(}\bigotimes_{e\in
E}\textup{Ad}(k_{s(e)})X_{e}\Bigr{)}\otimes\Bigl{(}\bigotimes_{e^{\prime}\in
E}\textup{Ad}(k_{t(e^{\prime})})Y_{e^{\prime}}\Bigr{)}$

defines an action of $\mathbf{K}$ on $U((\mathfrak{g}^{E})^{\times 2})$ by
algebra automorphisms. Furthermore,

(16) $\bm{k}\bullet
D_{\mathbf{X}\otimes\mathbf{Y}}=D_{\bm{k}\bullet(\mathbf{X}\otimes\mathbf{Y})}.$

###### Proof.

The first statement is immediate. For the second statement, it suffices to
check (16) when $\mathbf{X}=x^{(i)}$ and $\mathbf{Y}=1_{U(\mathfrak{g}^{E})}$
and when $\mathbf{X}=1_{U(\mathfrak{g}^{E})}$ and $\mathbf{Y}=y^{(i)}$, where
$x,y\in\mathfrak{g}_{0}$. When $\mathbf{X}=x^{(i)}$ and
$\mathbf{Y}=1_{U(\mathfrak{g}^{E})}$ we have

$\begin{split}\bigl{(}\bm{k}\cdot(D_{\mathbf{X}\otimes\mathbf{Y}}f)\bigr{)}(\bm{g})&=\frac{d}{dt}\bigg{|}_{t=0}f(\cdots,\exp(-tx)k_{s(e_{i})}^{-1}g_{i}k_{t(e_{i})},\cdots)\\\
&=\frac{d}{dt}\bigg{|}_{t=0}f(\cdots,k_{s(e_{i})}^{-1}\exp(-t\textup{Ad}(k_{s(e_{i})})x)g_{i}k_{t(e_{i})},\cdots)\\\
&=\bigl{(}D_{\mathbf{k}\bullet(\mathbf{X}\otimes\mathbf{Y})}(\bm{k}\cdot
f)\bigr{)}(\bm{g}),\end{split}$

as desired. A similar computation proves (16) when
$\mathbf{X}=1_{U(\mathfrak{g}^{E})}$ and $\mathbf{Y}=y^{(i)}$. ∎

### 5.6.

Lemma 5.5 shows that $\mathcal{D}_{\textup{inv}}(G^{E})$ is a
$\mathbf{K}$-invariant subalgebra of $\mathcal{D}(G^{E})$. Denote by

$\mathcal{D}_{\textup{inv}}(G^{E})^{\mathbf{K}}\subseteq\mathcal{D}_{\textup{inv}}(G^{E})$

the subalgebra of $\mathbf{K}$-invariant differential operators in
$\mathcal{D}_{\textup{inv}}(G^{E})$.

By §5.3 and Lemma 5.5 we then have the inclusion

(17)
$\mathcal{D}_{\textup{biinv}}(G^{E})\subseteq\mathcal{D}_{\textup{inv}}(G^{E})^{\mathbf{K}}\subseteq\mathcal{D}(G^{E})^{\mathbf{K}}$

of algebras.

### 5.7.

Let $S$ be a finite dimensional $\mathbf{K}$-representation. The space
$\mathcal{R}(G^{E})\otimes S$ of $S$-valued representative functions on
$G^{E}$ becomes a $\mathcal{D}(G^{E})$-module by

$D(h\otimes u):=D(h)\otimes u$

for $D\in\mathcal{D}(G^{E})$, $h\in\mathcal{R}(G^{E})$ and $u\in S$. In
addition we have the restricted gauge group $\mathbf{K}$ acts on
$\mathcal{R}(G^{E})\otimes S$ by the twisted $\mathbf{K}$-action
$(\bm{k},f)\mapsto\bm{k}\cdot f$ from §4.7. Then formula (14) remains true in
this more general context,

$\mathbf{k}\cdot(Df)=(\mathbf{k}\bullet D)(\mathbf{k}\cdot f)$

for $\mathbf{k}\in\mathbf{K}$, $D\in\mathcal{D}(G^{E})$ and
$f\in\mathcal{R}(G^{E})\otimes S$.

### 5.8.

As a consequence of §5.7, the algebra $\mathcal{D}(G^{E})^{\mathbf{K}}$ of
$\mathbf{K}$-invariant algebraic differential operators on $G^{E}$ acts on the
space $\mathcal{H}_{\Gamma,G,S}=(\mathcal{R}(G^{E})\otimes S)^{\mathbf{K}}$ of
spin graph functions. The resulting homomorphic image of the inclusions (17)
of algebras in $\textup{End}(\mathcal{H}_{\Gamma,G,S})$ gives rise to the
inclusion

$I_{\Gamma,G,S}\subseteq J_{\Gamma,G,S}\subseteq A_{\Gamma,G,S}$

of subalgebras of $\textup{End}(\mathcal{H}_{\Gamma,G,S})$. We omit the labels
$\Gamma,G,S$ if they are clear from context. Note that $I$ is contained in the
center of $J$, in view of §5.3.

### 5.9.

Following §2.7 we view the inclusion of algebras

$I\subseteq J\subseteq C_{A}(I)\subseteq A\subseteq\textup{End}(\mathcal{H})$

as a quantum spin system with quantum state space $\mathcal{H}$, algebra of
quantum observables $A$, algebra of quantum integrals $J$, and commutative
algebra of quantum Hamiltonians $I$. For $i\in\\{1,\ldots,n\\}$ and $\Omega\in
Z(\mathfrak{g})$ the quadratic Casimir element, the action of

$\Omega^{(i)}\in Z(\mathfrak{g}^{E})\simeq\mathcal{D}_{\textup{biinv}}(G^{E})$

on $\mathcal{H}$ is a quantum Hamiltonian $H_{i}\in I$ of the quantum spin
system.

We call $H_{i}$ ($i\in\\{1,\ldots,n\\}$) the edge-component quadratic
Hamiltonians of the quantum spin system.

### 5.10.

Write $\chi_{\pi}:Z(\mathfrak{g})\rightarrow\mathbb{C}$ for the central
character of $\pi\in G^{\wedge}$. Then

$(G^{\wedge})^{E}\hookrightarrow
I^{\wedge},\qquad\quad\bm{\pi}\mapsto\bm{\chi}_{\bm{\pi}}$

with $\bm{\chi}_{\bm{\pi}}\in I^{\wedge}$ determined by the formula

$\bm{\chi}_{\bm{\pi}}(Z_{i})=\chi_{\pi_{i}}(Z)$

for $i=1,\ldots,n$ and $Z\in Z(\mathfrak{g})$ (here the $\pi_{j}$ are the
local components of the tensor product representation $\bm{\pi}$, see §4.9).
It follows from §4.12 that the space $\mathcal{H}^{\bm{\pi}}$ of
$\bm{\pi}$-elementary spin graph functions can alternatively be described as
the simultaneous $I$-eigenspace for the one-dimensional $I$-module
$\bm{\chi}_{\bm{\pi}}\in I^{\wedge}$,

$\mathcal{H}^{\bm{\pi}}=\mathcal{H}_{\bm{\chi}_{\bm{\pi}}}.$

Hence condition (a) from §2.6 always holds true for the quantum spin system,

$\mathcal{H}=\bigoplus_{\bm{\pi}\in(G^{\wedge})^{E}}\mathcal{H}_{\bm{\chi}_{\bm{\pi}}},$

with $\mathcal{H}_{\bm{\chi}_{\bm{\pi}}}=\mathcal{H}^{\bm{\pi}}$ the finite
dimensional space of $\bm{\pi}$-elementary spin graph functions.

### 5.11.

The following is the main result of the paper.

######  Theorem.

Let $\Gamma$ be a finite connected oriented graph, $G$ a connected compact Lie
group, $\mathbf{K}=\prod_{v\in V}K_{v}$ with $K_{v}\subseteq G$ subgroups, and
$\sigma:\mathbf{K}\rightarrow\textup{GL}(S)$ a finite dimensional
representation.

The quantum spin system on $\mathcal{H}=\mathcal{H}_{\Gamma,G,S}$ as defined
in §5.9 is superintegrable if the following three conditions hold true:

  1. (a)

$G$ is simply connected.

  2. (b)

For each $v\in V$, the local gauge group $K_{v}\subseteq G$ is closed and
connected.

  3. (c)

The representation $\sigma:\mathbf{K}\rightarrow\textup{GL}(S)$ is
irreducible.

We give the proof of the theorem in §6.10.

### 5.12.

Consider the quantum spin system with $\Gamma$ the oriented cycle graph with
$n$ edges, $\mathbf{K}=G^{V}$ and $\sigma:\mathbf{K}\rightarrow\textup{GL}(V)$
a finite dimensional representation, see §4.17. By Theorem 5.11 it is
superintegrable when $G$ is simply connected and $\sigma$ is irreducible. The
condition on $\sigma$ implies that $\sigma$ is equivalent to a tensor product
representation $\bm{\sigma}$ with its local representations
$\sigma_{v}:G\rightarrow\textup{GL}(S_{v})$ irreducible for all $v\in V$.

This quantum spin system can be made more explicit using the parametrisation
of its moduli space $\mathcal{M}$ of graph $G$-connections in terms of a
maximal torus $T\subset G$. The edge-component quadratic Hamiltonians $H_{i}$
then become explicit second-order $\textup{End}(S)$-valued differential
operator on $T$ of Calogero-Moser type. The differences $H_{i}-H_{i-1}$ are
first-order commuting differential operators called asymptotic Knizhnik-
Zamolodchikov operators, which can be entirely described in terms of Felder’s
classical trigonometric dynamical $r$-matrix (see [8, 23, 20]). This provides
the interpretation of this quantum spin system as a quantum periodic spin
Calogero-Moser chain [21].

For the special case $n=1$, the superintegrability of the quantum periodic
Calogero-Moser spin system was discussed in [18].

### 5.13.

Consider now the quantum spin system with $\Gamma$ the linearly ordered linear
graph with $n$ edges, local gauge groups of the form (11) with $H,K\subseteq
G$ closed connected subgroups, and
$\sigma:\mathbf{K}\rightarrow\textup{GL}(S)$ a finite dimensional
representation of the associated gauge group $\mathbf{K}$ (see §4.18). By
Theorem 5.11 this quantum spin system is superintegrable when $G$ is simply
connected and $\sigma$ is irreducible. The condition on $\sigma$ implies that
$S\simeq L\otimes S_{2}\otimes\cdots\otimes S_{n}\otimes N$ with $L$ an
irreducible $H$-representation, $N$ an irreducible $K$ representation and
$S_{j}$ irreducible $G$-representations.

This quantum spin system can be made more concrete when $H=K$ is the connected
component of the identity of a fix-point subgroup $G^{\Theta}$ of an
involution $\Theta$ of $G$, using an appropriate parametrisation of its moduli
space $\mathcal{M}$ of graph $G$-connections in terms of an appropriate
subtorus $A\subset G$. The edge-component quadratic Hamiltonians $H_{i}$ then
become second-order $\textup{End}(S)$-valued differential operator on $A$ of
Calogero-Moser type and $H_{i}-H_{i-1}$ are asymptotic boundary Knizhnik-
Zamolodchikov operators, which are first order differential operators
involving folded classical dynamical $r$-matrices and associated dynamical
$k$-matrices (see [24, 21, 23, 20]). This provides the interpretation of this
quantum spin system as a quantum open spin Calogero-Moser chain [21].

##  6\. Conditions for superintegrability

In this section we provide a proof of the sufficient conditions ensuring
superintegrability of the quantum spin systems defined in §5 (see Theorem
5.11). We retain the notations and conventions of §5. In particular, $\Gamma$
is an oriented finite graph, $G$ is a connected compact Lie group, and
$\mathbf{K}=\prod_{v\in V}K_{v}$ with subgroups $K_{v}\subseteq G$.

We take as finite dimensional $\mathbf{K}$-representation of the quantum
system a tensor product representation
$\bm{\sigma}:\mathbf{K}\rightarrow\textup{GL}(\mathbf{S})$ (see §4.10). We
furthermore fix an irreducible finite dimensional tensor product
representation $\bm{\pi}:G^{E}\rightarrow\textup{GL}(\mathbf{M}^{\bm{\pi}})$,
with local irreducible $G$-representations
$\pi_{e}:G\rightarrow\textup{GL}(M^{\pi_{e}})$.

Finally, we write $\mathfrak{g}_{0}$ for the Lie algebra of $G$, and
$\mathfrak{g}$ for its complexification.

### 6.1.

For $v\in V$ consider the linear isomorphism

(18)
$\tau_{v}:\textup{Hom}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})\overset{\sim}{\longrightarrow}\mathbf{M}^{\pi_{\mathcal{S}(v)}}\otimes
S_{v}$

defined by

$\tau_{v}(\phi_{v}):=\sum_{t_{v}}\phi_{v}(u_{t_{v}}^{(v),*})\otimes
u_{t_{v}}^{(v)}$

where $\\{u_{t}^{(v)}\\}_{t}$ is a basis of $S_{v}$ and
$\\{u_{t}^{(v),*}\\}_{t}$ is the corresponding dual basis of $S_{v}^{*}$.

It is often convenient to expand $\phi_{v}(u_{t_{v}}^{(v),*})$ in terms of the
tensor product basis of $\mathbf{M}^{\pi_{\mathcal{S}(v)}}$ (see §4.16). Its
expansion coefficients will be denoted by
$\phi_{v}[t_{v};\bm{i},\bm{j}]\in\mathbb{C}$,

$\phi_{v}(u_{t_{v}}^{(v),*})=\sum_{\bm{i}\in\mathcal{I}(v|s)}\sum_{\bm{j}\in\mathcal{I}(v|t)}\phi_{v}[t_{v};\bm{i},\bm{j}]\,\bigl{(}\bm{m}_{\bm{i}}(v|s)^{*}\otimes\bm{m}_{\bm{j}}(v|t)\bigr{)},$

so that

(19)
$\tau_{v}(\phi_{v})=\sum_{\bm{i}\in\mathcal{I}(v|s)}\sum_{\bm{j}\in\mathcal{I}(v|t)}\sum_{t_{v}}\phi_{v}[t_{v};\bm{i},\bm{j}]\,\bigl{(}\bm{m}_{\bm{i}}(v|s)^{*}\otimes\bm{m}_{\bm{j}}(v|t)\bigr{)}\otimes
u_{t_{v}}^{(v)}.$

### 6.2.

Turn $\textup{Hom}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$ into a
$K_{v}$-representation, with action

$k_{v}\cdot T:=\pi_{\mathcal{S}(v)}(k_{v})\circ
T\circ\sigma_{v}^{*}(k_{v}^{-1}),\qquad\quad k_{v}\in K_{v}.$

The linear map $\tau_{v}$ (see (19)) is $K_{v}$-linear, with the
$K_{v}$-action on the codomain of $\tau_{v}$ as defined in §4.14. Hence
$\tau_{v}$ restricts to a linear isomorphism

$\tau_{v}:\textup{Hom}_{K_{v}}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})\overset{\sim}{\longrightarrow}\bigl{(}\mathbf{M}^{\pi_{\mathcal{S}(v)}}\otimes
S_{v}\bigr{)}^{K_{v}}.$

### 6.3.

Recall the isomorphism $\Psi^{\bm{\pi}}$ defined in §4.15. It follows from
§6.2 that

$\Upsilon^{\bm{\pi}}:=\Psi^{\bm{\pi}}\circ\Bigl{(}\bigotimes_{v\in
V}\tau_{v}\Bigr{)}:\,\,\bigotimes_{v\in
V}\textup{Hom}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})\overset{\sim}{\longrightarrow}\mathcal{R}^{\bm{\pi}}(G^{E})\otimes\mathbf{S}$

is a $\mathbf{K}$-linear isomorphism, where the domain of
$\Upsilon^{\bm{\pi}}$ is viewed as $\mathbf{K}$-representation relative to the
tensor product action of $\mathbf{K}=\prod_{v\in V}K_{v}$. The map
$\Upsilon^{\bm{\pi}}$ restricts to a linear isomorphism

$\Upsilon^{\bm{\pi}}:\bigotimes_{v\in
V}\textup{Hom}_{K_{v}}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})\overset{\sim}{\longrightarrow}\mathcal{H}^{\bm{\pi}}.$

### 6.4.

Let $\mathcal{I}$ be the set of sequences $(i_{e})_{e\in E}$ with
$i_{e}\in\mathcal{I}_{e}$. Consider the tensor product basis
$\\{\mathbf{m}_{\bm{i}}\\}_{\bm{i}\in\mathcal{I}}$ of $\mathbf{M}^{\bm{\pi}}$,
where

$\mathbf{m}_{\bm{i}}:=\bigotimes_{e\in E}m_{i_{e},e},$

and write $\mathbf{m}_{\bm{i}}^{*}:=\bigotimes_{e\in E}m_{i_{e},e}^{*}$ for
the corresponding dual basis elements of
$(\mathbf{M}^{\bm{\pi}})^{*}\simeq\otimes_{e\in E}M^{\pi_{e}^{*}}$, cf. §4.9.
For $\bm{i}\in\mathcal{I}$ write

$\bm{i}_{v|s}:=(i_{e})_{e\in\mathcal{S}(v|s)}\in\mathcal{I}(v|s),\qquad\quad\bm{i}_{v|t}:=(i_{e})_{e\in\mathcal{S}(v|t)}\in\mathcal{I}(v|t).$

A direct computation using (19) then leads to the formula

(20) $\Upsilon^{\bm{\pi}}\Bigl{(}\bigotimes_{v\in
V}\phi_{v}\Bigr{)}=\sum_{\bm{i},\bm{j}\in\mathcal{I}}c_{{}\bm{m}_{\bm{i}}^{*},\bm{m}_{\bm{j}}}^{\bm{\pi}}\otimes\Bigl{(}\bigotimes_{v\in
V}\bigl{(}\sum_{t_{v}}\phi_{v}[t_{v};\bm{i}_{v|s},\bm{j}_{v|t}]\,u_{t_{v}}^{(v)}\bigr{)}\Bigr{)}$

for $\phi_{v}\in\textup{Hom}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$.

### 6.5.

Consider the tensor product algebra

$U(\mathfrak{g})^{(v)}:=U(\mathfrak{g})^{\otimes\\#\mathcal{S}(v|s)}\otimes
U(\mathfrak{g})^{\otimes\\#\mathcal{S}(v|t)}.$

A pure tensor in $U(\mathfrak{g})^{(v)}$ is denoted by
$\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t}$ with

$\mathbf{X}_{v|s}=\bigotimes_{e\in\mathcal{S}(v|s)}X_{e},\qquad\mathbf{Y}_{v|t}=\bigotimes_{e^{\prime}\in\mathcal{S}(v|t)}Y_{e^{\prime}},$

where we order the tensor products along the total orders on
$\mathcal{S}(v|s)$ and $\mathcal{S}(v|t)$ induced by the total order on $E$.
In §4.13 we considered the space

$\mathbf{M}^{\pi_{\mathcal{S}(v)}}=\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|s)}M^{\pi_{e}^{*}}\Bigr{)}\otimes\Bigl{(}\bigotimes_{e^{\prime}\in\mathcal{S}(v|t)}M^{\pi_{e^{\prime}}}\Bigr{)},$

as $K_{v}$-representation space relative to the diagonal $K_{v}$-action
$\pi_{\mathcal{S}(v)}$. Differentiating the $G$-action turns $M^{\pi_{e}^{*}}$
and $M^{\pi_{e^{\prime}}}$ into irreducible $U(\mathfrak{g})$-modules, and
hence $\mathbf{M}^{\pi_{\mathcal{S}(v)}}$ into an irreducible
$U(\mathfrak{g})^{(v)}$-module via the diagonal
$U(\mathfrak{g})^{(v)}$-action.

We view the linear space

$\textup{Hom}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$

as $U(\mathfrak{g})^{(v)}$-module, with $U(\mathfrak{g})^{(v)}$ acting on its
co-domain,

(21) $\bigl{(}(\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t})\cdot
T\bigr{)}(\xi):=(\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t})\cdot(T(\xi))$

for $\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t}\in U(\mathfrak{g})^{(v)}$,
$T\in\textup{Hom}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$ and $\xi\in
S_{v}^{*}$.

### 6.6.

The local gauge group $K_{v}$ acts by algebra automorphisms on
$U(\mathfrak{g})^{(v)}$ via the diagonal adjoint action,

$k_{v}\bullet_{v}\bigl{(}\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t}\bigr{)}:=\Bigl{(}\bigotimes_{e\in\mathcal{S}(v|s)}\textup{Ad}(k_{v})X_{e}\Bigr{)}\otimes\Bigl{(}\bigotimes_{e^{\prime}\in\mathcal{S}(v|t)}\textup{Ad}(k_{v})Y_{e^{\prime}}\Bigr{)}.$

We then have

$k_{v}\cdot\bigl{(}(\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t})\cdot
B\bigr{)}=\bigl{(}k_{v}\bullet_{v}(\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t})\bigr{)}\cdot\bigl{(}k_{v}\cdot
B\bigr{)}$

for $k_{v}\in K_{v}$, $\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t}\in
U(\mathfrak{g})^{(v)}$ and $B\in\mathbf{M}^{\pi_{\mathcal{S}(v)}}$.

Let $(U(\mathfrak{g})^{(v)})^{K_{v}}$ be the algebra of $K_{v}$-invariant
elements in $U(\mathfrak{g})^{(v)}$ relative to the $K_{v}$-action
$\bullet_{v}$. It follows from §6.2 and §6.5 that the space

$\textup{Hom}_{K_{v}}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$

of $K_{v}$-intertwiners is a $(U(\mathfrak{g})^{(v)})^{K_{v}}$-module, with
the action on
$\textup{Hom}_{K_{v}}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$ given by
(21).

### 6.7.

Consider the algebra isomorphism

(22) $U((\mathfrak{g}^{E})^{\times
2})\overset{\sim}{\longrightarrow}\bigotimes_{v\in V}U(\mathfrak{g})^{(v)}$

defined by

$\mathbf{X}\otimes\mathbf{Y}\mapsto\bigotimes_{v\in
V}\bigl{(}\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t}\bigr{)}$

for $\mathbf{X}=\bigotimes_{e\in E}X_{e}$ and
$\mathbf{Y}=\bigotimes_{e^{\prime}\in E}Y_{e^{\prime}}$, where

$\mathbf{X}_{v|s}=\bigotimes_{e\in\mathcal{S}(v|s)}X_{e},\qquad\qquad\mathbf{Y}_{v|t}=\bigotimes_{e^{\prime}\in\mathcal{S}(v|t)}Y_{e^{\prime}}.$

Consider the tensor product action of the gauge group $\mathbf{K}=\prod_{v\in
V}K_{v}$ on the co-domain $\bigotimes_{v\in V}U(\mathfrak{g})^{(v)}$ of the
algebra isomorphism (22), with the $K_{v}$-action on $U(\mathfrak{g})^{(v)}$
as defined in §6.6. A direct check shows that the algebra isomorphism (22) is
$\mathbf{K}$-linear, with $\mathbf{K}$ acting on $U((\mathfrak{g}^{E})^{\times
2})$ according to Lemma 5.5. The algebra isomorphism (22) thus restricts to an
algebra isomorphism

(23) $U((\mathfrak{g}^{E})^{\times
2})^{\mathbf{K}}\overset{\sim}{\longrightarrow}\bigotimes_{v\in
V}(U(\mathfrak{g})^{(v)})^{K_{v}}.$

### 6.8.

Endow

(24) $\bigotimes_{v\in
V}\textup{Hom}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$

with the tensor product action of $\bigotimes_{v\in V}U(\mathfrak{g})^{(v)}$.
We reinterpret this as an action of $U((\mathfrak{g}^{E})^{\times 2})$ via the
algebra isomorphism (22).

By §5.2 and §5.7 the universal enveloping algebra
$U((\mathfrak{g}^{E})^{\times 2})$ also acts on the space
$\mathcal{R}^{\bm{\pi}}(G^{E})\otimes\mathbf{S}$ of $\mathbf{S}$-valued
representative functions on $G^{E}$ by

$(\mathbf{X}\otimes\mathbf{Y})\cdot(f\otimes
u):=D_{\mathbf{X}\otimes\mathbf{Y}}(f)\otimes u$

for $\mathbf{X},\mathbf{Y}\in U(\mathfrak{g}^{E})$,
$f\in\mathcal{R}^{\bm{\pi}}(G^{E})$ and $u\in\mathbf{S}$.

######  Lemma.

The $\mathbf{K}$-linear isomorphism

$\Upsilon^{\bm{\pi}}:\bigotimes_{v\in
V}\textup{Hom}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})\overset{\sim}{\longrightarrow}\mathcal{R}^{\bm{\pi}}(G^{E})\otimes\mathbf{S}$

as defined in §6.3, is $U((\mathfrak{g}^{E})^{\times 2})$-linear.

###### Proof.

Using the notations from §6.7 we have

$\begin{split}\Upsilon^{\bm{\pi}}\Bigl{(}(\mathbf{X}\otimes\mathbf{Y})\cdot\Bigl{(}\bigotimes_{v\in
V}\phi_{v}\Bigr{)}\Bigr{)}&=\Upsilon^{\bm{\pi}}\Bigl{(}\bigotimes_{v\in
V}(\mathbf{X}_{v|s}\otimes\mathbf{X}_{v|t})\cdot\phi_{v}\Bigr{)}\\\
&=\sum_{\bm{i},\bm{j}\in\mathcal{I}}c_{\mathbf{X}\cdot\mathbf{m}_{\bm{i}}^{*},\mathbf{Y}\cdot\mathbf{m}_{\bm{j}}}^{\bm{\pi}}\otimes\Bigl{(}\bigotimes_{v\in
V}\bigl{(}\sum_{t_{v}}\phi_{v}[t_{v};\bm{i}_{v|s},\bm{j}_{v|t}]\,u_{t_{v}}^{(v)}\bigr{)}\Bigr{)}\end{split}$

(here $\mathbf{X}\cdot\mathbf{m}_{\bm{i}}^{*}$ and
$\mathbf{Y}\cdot\mathbf{m}_{\bm{j}}$ refer to the $U(\mathfrak{g}^{E})$-action
on $(\mathbf{M}^{\bm{\pi}})^{*}$ and $\mathbf{M}^{\bm{\pi}}$, obtained by
differentiating the $G^{E}$-action). The second equality follows from the
expansion formula

$\begin{split}&\bigl{(}(\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t})\cdot\phi_{v}\bigr{)}[t_{v};\bm{i}_{v|s}^{\prime},\bm{j}_{v|t}^{\prime}]=\\\
&\,\,=\sum_{\bm{i}_{v|s},\bm{j}_{v|t}}\bigl{(}\mathbf{X}_{v|s}\cdot\mathbf{m}_{\bm{i}_{v|s}}(v|s)^{*}\bigr{)}(\mathbf{m}_{\bm{i}_{v|s}^{\prime}}(v|s))\mathbf{m}_{\bm{j}_{v|t}^{\prime}}(v|t)^{*}(\mathbf{Y}_{v|t}\cdot\mathbf{m}_{\bm{j}_{v|t}}(v|t))\phi_{v}[t_{v};\bm{i}_{v|s},\bm{j}_{v|t}]\end{split}$

with the sums in the left hand side taken over
$\bm{i}_{v|s}\in\mathcal{I}(v|s)$ and $\bm{j}_{v|t}\in\mathcal{I}(v|t)$, and
the fact that

$\begin{split}\sum_{\bm{i}^{\prime}\in\mathcal{I}}\Bigl{(}\prod_{v\in
V}\bigl{(}\mathbf{X}_{v|s}\cdot\mathbf{m}_{\bm{i}_{v|s}}(v|s)^{*}\bigr{)}(\mathbf{m}_{\bm{i}_{v|s}^{\prime}}(v|s))\Bigr{)}\mathbf{m}_{\bm{i}^{\prime}}^{*}&=\sum_{\bm{i}^{\prime}\in\mathcal{I}}(\mathbf{X}\cdot\mathbf{m}_{\bm{i}}^{*})(\mathbf{m}_{\bm{i}^{\prime}})\mathbf{m}_{\bm{i}^{\prime}}^{*}=\mathbf{X}\cdot\mathbf{m}_{\bm{i}}^{*},\\\
\sum_{\bm{j}^{\prime}\in\mathcal{I}}\Bigl{(}\prod_{v\in
V}\mathbf{m}_{\bm{j}_{v|t}^{\prime}}(v|t)^{*}\bigl{(}\mathbf{Y}_{v|t}\cdot\mathbf{m}_{\bm{j}_{v|t}}(v|t)\bigr{)}\Bigr{)}\mathbf{m}_{\bm{j}^{\prime}}&=\sum_{\bm{j}^{\prime}\in\mathcal{I}}\mathbf{m}_{\bm{j}^{\prime}}^{*}(\mathbf{Y}\cdot\mathbf{m}_{\bm{j}})\mathbf{m}_{\bm{j}^{\prime}}=\mathbf{Y}\cdot\mathbf{m}_{\bm{j}}.\end{split}$

The result now follows from the fact that

$c_{\mathbf{X}\cdot\mathbf{m}_{\bm{i}}^{*},\mathbf{Y}\cdot\mathbf{m}_{\bm{j}}}^{\bm{\pi}}=D_{\mathbf{X}\otimes\mathbf{Y}}(c_{\mathbf{m}_{\bm{i}}^{*},\mathbf{m}_{\bm{j}}}^{\bm{\pi}}).$

∎

### 6.9.

The results of §6.6-§6.8 immediately lead to the following conclusion.

######  Corollary.

  1. (a)

The isomorphism $\Upsilon^{\bm{\pi}}$ restricts to a
$U((\mathfrak{g}^{E})^{\times 2})^{\mathbf{K}}$-linear isomorphism

$\Upsilon^{\bm{\pi}}:\,\,\bigotimes_{v\in
V}\textup{Hom}_{K_{v}}\bigl{(}S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}}\bigr{)}\overset{\sim}{\longrightarrow}\mathcal{H}^{\bm{\pi}}.$

  2. (b)

$\mathcal{H}^{\bm{\pi}}$ is an irreducible $U((\mathfrak{g}^{E})^{\times
2})^{\mathbf{K}}$-module iff
$\textup{Hom}_{K_{v}}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$ is an
irreducible $(U(\mathfrak{g})^{(v)})^{K_{v}}$-module for all $v\in V$.

### 6.10.

We have now all in the required ingredients for the proof of Theorem 5.11.

###### Proof of Theorem 5.11.

Suppose that the three conditions (a)-(c) in Theorem 5.11 hold true. In view
of §4.10, we may assume without loss of generality that the
$\mathbf{K}$-representation $\sigma$ is a tensor product representation
$\bm{\sigma}$ with irreducible local representations
$\sigma_{v}:K_{v}\rightarrow\textup{GL}(S_{v})$. Let
$\bm{\pi}\in(G^{\wedge})^{E}$ such that $\mathcal{H}^{\bm{\pi}}\not=0$. We
need to show that $\mathcal{H}^{\bm{\pi}}$ is an irreducible
$U((\mathfrak{g}^{E})^{\times 2})^{\mathbf{K}}$-module.

Denote by $\mathfrak{k}_{v}$ the complexified Lie algebra of $K_{v}$. Set

$e(v):=\\#\mathcal{S}(v|s)+\\#\mathcal{S}(v|t)$

(which might be strictly larger than $\\#\mathcal{S}(v)$ since $\Gamma$ may
have loops). Note that $e(v)>0$ since $\Gamma$ is connected.

By Corollary 6.9 and the fact that $K_{v}$ is connected, it suffices to show
that
$\textup{Hom}_{\mathfrak{k}_{v}}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$
is an irreducible $U(\mathfrak{g}^{\times
e(v)})^{\mathfrak{k}_{v}^{(e(v))}}$-module, where
$\mathfrak{k}_{v}^{(e(v))}\subseteq\mathfrak{g}^{\times e(v)}$ is the image
$\mathfrak{k}_{v}$ under the diagonal embedding
$\delta_{\mathfrak{g}}^{(e(v))}:\mathfrak{g}\hookrightarrow\mathfrak{g}^{\times
e(v)}$, see §3.9.

Note that $\mathfrak{g}$ is semisimple since $G$ is simply connected, and
$\mathfrak{k}_{v}$ is reductive in $\mathfrak{g}$ by §3.5. Hence
$\mathfrak{g}^{\times e(v)}$ is a reduction extension of
$\mathfrak{k}_{v}^{(e(v))}$, see Proposition 3.9. Corollary 3.17 then implies
that
$\textup{Hom}_{\mathfrak{k}_{v}}(S_{v}^{*},\mathbf{M}^{\pi_{\mathcal{S}(v)}})$
is an irreducible $U(\mathfrak{g}^{\times
e(v)})^{\mathfrak{k}_{v}^{(e(v))}}$-module. ∎

## References

  * [1] A. Yu. Alekseev, H. Grosse, V. Schomerus, Combinatorial quantization of the Hamiltonian Chern-Simons Theory, Comm. Math. Phys. 172 (1995), 317–358.
  * [2] J.E. Andersen, J. Mattes, N. Reshetikhin, The Poisson structure on the moduli space of flat connections and chord diagrams, Topology 15 (1996), 1069–1083.
  * [3] S. Arthamonov, N. Reshetikhin, Superintegrable systems on moduli spaces of flat connections, Comm. Math. Phys. 386 (2021), 1337–1381.
  * [4] E. Buffenoir, Ph. Roche, Two dimensional lattice gauge theory based on a quantum group, Comm. Math. Phys. 170 (1995), 669–698.
  * [5] E. Buffenoir, Ph. Roche, Link invariants and combinatorial quantization of Hamiltonian Chern Simons theory, Comm. Math. Phys. 181 (1996), 331–365.
  * [6] J. Dixmier, Enveloping algebras, Graduate studes in Math., 11. Amer. Math. Soc., Providence, RI, 1996.
  * [7] P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina, Introduction to Representation Theory, Student Math. Library 59, Amer. Math. Soc. (2011).
  * [8] P. Etingof, O. Schiffmann, Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical $r$-matrices corresponding to Belavin-Drinfeld triples, Math. Res. Lett. 6 (1999), 593–612.
  * [9] P. Etingof, O. Schiffmann, A. Varchenko, Traces of intertwiners for quantum groups and difference equations, Lett. Math. Phys. 62 (2002), 143–158.
  * [10] P. Etingof, A. Varchenko, Traces of intertwiners for quantum groups and difference equations. I, Duke Math. J 104 (2000), 391–432.
  * [11] V.V. Fock, A.A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and the $r$-matrix, Moscow Seminar in Math. Phys., 67–86, Amer. Math. Soc. Transl. Ser. 2, 191, Adv. Math. Sci., 43, Amer. Math. Soc., Providence, RI, 1999.
  * [12] Harish-Chandra, Spherical functions on a semisimple Lie group I, Amer. J. Math. 80 (1958), 241–310.
  * [13] Harish-Chandra, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26–65.
  * [14] J. Leposwky, G.W. McCollum, On the determination of irreducible modules by restriction to a subalgebra, Trans. Amer. Math. Soc. 176 (1973), 45–57.
  * [15] Ju. A. Neretin, Differential operators on Lie groups (in russian), Uspekhi Mat. Nauk 36, no. 2(218) (1981), 195–196.
  * [16] E.M. Opdam, Root systems and hypergeometric functions IV, Compositio Math. 67 (1988), 191–209.
  * [17] N. Reshetikhin, Degenerate integrability of the Spin Calogero-Moser Systems and the Duality with the Spin Ruijsenaars Systems, Lett. Math. Phys. 63 (2003), 55–71.
  * [18] N. Reshetikhin, Degenerate integrability of quantum spin Calogero-Moser systems, Lett. Math. Phys. 107 (2017), 187–200.
  * [19] N. Reshetikhin, Periodic and open classical spin Calogero-Moser chains, arXiv:2302.14281.
  * [20] N. Reshetikhin, Spin Calogero-Moser periodic chains and two dimensional Yang-Mills theory with corners, arXiv:2303.10579.
  * [21] N. Reshetikhin, J.V. Stokman, Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains, in “Hypergeometry, integrability and Lie theory”, 205–241, Contemp. Math. 780, Amer. Math. Soc., Providence, RI, 2022.
  * [22] P. Roche, A. Szenes, Functionals on noncommutative deformations of moduli spaces of flat connections, Adv. Math. 168 (2002), 133–192.
  * [23] J.V. Stokman, Folded and contracted solutions of coupled classical dynamical Yang-Baxter and reflection equations, Indag. Math. (N.S.) 32 (2021), 1372–1411.
  * [24] J.V. Stokman, N. Reshetikhin, $N$-point spherical functions and asymptotic boundary KZB equations, Invent. Math. 229 (2022), 1–86.
  * [25] N. Wallach, Real reductive groups I. Pure and Applied Mathematics, 132. Academic Press, Inc., Boston, MA, 1988.